# Geometric approach to stable homotopy groups of spheres II; Arf-Kervaire
Invariants

P.M. Akhmetiev

###### Abstract

The Kervaire Invariant 1 Problem until recently was an open problem in
algebraic topology. Hill, Hopkins, and Ravenel solved this problem for all
dimensions except $n=126$, [H-H-R]. In dimension $n=126$, the problem has not
been solved and has the status of a hypothesis by V.P. Snaith (2009). We
consider an alternative (with respect to the approach in [H-H-R]) geometric to
the Kervaire Invariant 1 Problem.

A notion of Abelian (for skew-framed immersions), bi-cyclic (for
${\mathbb{Z}}/2^{[3]}$-framed immersions) and quaternion-cyclic structure (for
${\mathbb{Z}}/2^{[4]}$-framed immersions) are introduced. Proofs are
particularly based on common results with Th.Yu.Popelenskii and are presented
in Section8.

##  1 Self-intersections of generic immersions and the Kervaire invariant; the
problem statement

The Kervaire Invariant 1 Problem until recently was an open problem in
algebraic topology. For setting and applications, see [A1]. In dimension
$n=126$, the problem has not been solved and has the status of a hypothesis by
V.P. Snaith (2009). A geometrical approach toward the solution is proposed.
With this the Arf-Kervaire invariant one Problem in all dimensions, including
the dimension $126$, is solved. Our approach develops an approach by
L.S.Pontrjagin [P], and by V.A.Rokhlin [Ro].

Let us consider a smooth generic immersion
$f:M^{n-1}\looparrowright{\mathbb{R}}^{n}$, $n=2^{\ell}-2$, $\ell>1$ of the
codimension $1$. Denote by $g:N^{n-2}\looparrowright{\mathbb{R}}^{n}$ the
immersion of self-intersection manifold of $f$.

Let us recall a definition of the cobordism group $Imm^{sf}(n-k,k)$, a
particular case $k=1$ is better known: $Imm^{sf}(n-1,1)$. The cobordism group
is defined as equivalent classes of triples up to the standard cobordism
relation, equipped with a disjoint union operation;

$\bullet$ $f:M^{n-k}\looparrowright{\mathbb{R}}^{n}$ is a codimension $k$
immersion;

$\bullet$ $\Xi:\nu(f)\cong k\varkappa_{M}^{\ast}(\gamma)$ is a bundle map,
which is invertible, an invertible bundle map is a fibrewise isomorphism;

$\bullet$ $\varkappa_{M}\in H^{1}(M^{n-k};{\mathbb{Z}}/2)$ is a prescribed
cohomology class, which is a mapping $M^{n-k}\to{\mathbb{R}}\\!{\rm
P}^{\infty}=K({\mathbb{Z}}/2,1)$.

By $\nu(f)$ is denoted the normal bundle of the immersion $f$, by
$\varkappa_{M}^{\ast}(\gamma)$ is denoted the pull-back of the universal line
bundle $\gamma$ over ${\mathbb{R}}\\!{\rm P}^{\infty}$ by the mapping
$\varkappa_{M}$, by $k\varkappa_{M}^{\ast}(\gamma)$ is denoted the Whitney sum
of the $k$ copies of the line bundles, below for short we will write
$k\varkappa_{M}$. The isomorphism $\Xi$ is called a skew-framing of the
immersion $f$.

The Kervaire invariant is an invariant of cobordism classes, which is
homomorphism

$\displaystyle\Theta^{sf}:Imm^{sf}(n-k,k)\to{\mathbb{Z}}/2.$ (1)

Let us recall the homomorphism $(\ref{1a})$ in the case $k=1$.

The normal bundle $\nu_{g}$ of the immersion
$g:N^{n-2}\looparrowright{\mathbb{R}}^{n}$ is a $2$-dimensional bundle over
$N^{n-2}$, which is equipped by a ${\bf D}$–framing $\Xi$, where ${\bf D}$ is
the dihedral group of the order $8$. The classifying map of this bundle (and
also the corresponding characteristic class) is denoted by
$\eta_{N}:N^{n-2}\to K({\bf D},1)$. Triples $(g,\eta_{N},\Xi)$ up to a
cobordism relation represents an element in the group $Imm^{{\bf D}}(n-2,2)$.
The correspondence $(f,\varkappa,\Psi)\mapsto(g,\eta_{N},\Xi)$ defines a
homomorphism

$\displaystyle\delta^{{\bf D}}:Imm^{sf}(n-1,1)\to Imm^{{\bf D}}(n-2,2).$ (2)

######  Definition 1.

The Kervaire invariant of an immersion $f$ is defined by the formula:

$\displaystyle\Theta_{sf}(f)=\langle\eta_{N}^{\frac{n-2}{2}};[N^{n-2}]\rangle.$
(3)

It is not difficult to prove that the formula $(\ref{arf})$ determines a
homomorphism

$\displaystyle\Theta^{{\bf D}}:Imm^{{\bf D}}(n-2,2)\to{\mathbb{Z}}/2.$ (4)

The homomorphism (4) is called the Kervaire invariant of a ${\bf D}$–framed
immersion. The composition of the homomorphisms $(\ref{2a})$, $(\ref{3a})$
determines the homomorphism $(\ref{1a})$.

### Main Result

Assume an arbitrary element $x\in Imm^{sf}(n-1,1)$ admits a compression of the
order $7$ (see Definition (216)). In the case $n=2^{\ell}-2$, $\ell\geq 8$ the
homomorphism $\Theta_{sf}(f)$ (3) is trivial. In the case $n\geq 8$ this
result is proved in [H-H-R]. Because of the following result by the author:

#### Compression Theorem

For an arbitrary positive integer $d$ there exists $\ell=\ell(d)$, such that
an arbitrary element in $Imm^{sf}(n-1,1)$ admits a compression of the order
$d$.

we have an alternative proof of Hill-Hopkins-Ravenel theorem for all $n\geq
256$, probably, except a finite number of exceptional cases. Lemma (23) case
the $n=126$ is not proved.

The group $Imm^{{\bf D}}(n-2,2)$ admits a standard generalization to the
cobordism group $Imm^{{\bf D}}(n-2k,2k)$ with a parameter $k\geq 1$. Each
element in $Imm^{{\bf D}}(n-2k,2k)$ is represented by a triple
$(g,\Psi,\eta_{N})$, where $g:N^{n-2k}\looparrowright{\mathbb{R}}^{n}$ is an
immersion, $\Psi$– is a dihedral framing in the codimension $2k$, i.e. an
isomorphism $\Xi:\nu_{g}\cong k\eta_{N}$, where $\eta_{N}:N\to K({\bf D},1)$
is the classifying mapping (this mapping is also will called a characteristic
mapping). Let us denote the universal $2$-dimensional ${\bf D}$-bundle over
$K({\bf D},1)$ by $\tau$.

The mapping $\eta_{N}$ classifies the normal bundle $\nu_{g}$, because
$\nu_{g}\cong k\eta_{N}$ by $\Xi$.

The Kervaire homomorphism is defined in a more general case using the
following generalizations of the homomorphisms (1) and (4):

$\displaystyle\Theta^{sf}_{k}:Imm^{sf}(n-k,k)\to{\mathbb{Z}}/2,\qquad\Theta^{sf}_{k}:=\Theta^{{\bf
D}}_{k}\circ\delta^{{\bf D}}_{k}.$ (5) $\displaystyle\Theta^{{\bf
D}}_{k}:Imm^{{\bf D}}(n-2k,2k)\to{\mathbb{Z}}/2,\qquad\Theta^{{\bf
D}}_{k}[(g,\Psi,\eta_{N})]=\langle\eta_{N}^{\frac{n-2k}{2}};[N^{n-2k}]\rangle.$
(6)

In the case $k=1$ the homomorphism (6) coincides with the homomorphism (4)
above, the following commutative diagram, in which the homomorphisms $J^{sf}$,
$J^{{\bf D}}$ are known, is well-defined.

$\displaystyle\begin{array}[]{ccccc}Imm^{sf}(n-1,1)&\stackrel{{\scriptstyle\delta^{{\bf
D}}}}{{\longrightarrow}}&Imm^{{\bf
D}}(n-2,2)&\stackrel{{\scriptstyle\Theta^{{\bf
D}}}}{{\longrightarrow}}&{\mathbb{Z}}/2\\\ \downarrow J^{sf}_{k}&&\downarrow
J^{{\bf D}}_{k}&&||\\\ Imm^{sf}(n-k,k)&\stackrel{{\scriptstyle\delta^{{\bf
D}}_{k}}}{{\longrightarrow}}&Imm^{{\bf
D}}(n-2k,2k)&\stackrel{{\scriptstyle\Theta^{{\bf
D}}_{k}}}{{\longrightarrow}}&{\mathbb{Z}}/2.\\\ \end{array}$ (10)

We will prove the following statement, which is interesting concerning the
paper [B-J-M].

######  Proposition 2.

Assume we get an element $x\in Imm^{sf}(n-1,1)$, $n=62$, with
$\Theta^{sf}_{31}(x)=1$, (5). Then the element $x$ is of an order, at least
$4$, $2x\neq 0$.

Let us remind, that an idea to investigate intersection points in Theorem 20
was communicated to me by A.V.Chernavskii (1996). S.A.Melikhov (2005) has
noted that a quaternion analogue of the Chernavskii mapping as in Lemma 25 is
required. An idea to make a control as in the formula (55) and in other
analogous formulas was realized after a talk by E.V.Scepin (2004) before his
business trip. As a conclusion, results was collected-out at the
A.S.Mishchenko Seminar in (2008)-(2022).

##  2 First step in proof

In the present and the next sections the cobordism groups $Imm^{sf}(n-k,k)$,
$Imm^{{\mathbb{Z}}/2^{[2]}}(n-2k,2k)$ will be used. In the case the first
argument in the bracket is strongly positive, the cobordism group is finite.

The dihedral group ${\mathbb{Z}}/2^{[2]}={\bf D}$ is defined by its
corepresentation

$\\{a,b\,|\,b^{4}=a^{2}=e,[a,b]=b^{2}\\}.$

This group is represented by rotations a subgroup in the group $O(2)$ of
orthogonal transformation of the standard plane. Elements transforms the base
$\\{{\rm e}_{1},{\rm e}_{2}\\}$ on the plane $Lin({\rm e}_{1},{\rm e}_{2})$ to
itself, a non-ordered pair of coordinate lines on the plane are preserved by
transformations. The element $b$ is represented by the rotation of the plane
by the angle $\frac{\pi}{2}$. The element $a$ is represented by the reflection
of the plane relative to the straight line $l_{1}=Lin({\rm e}_{1}+{\rm
e}_{2})$ parallel to the vector ${\rm e}_{1}+{\rm e}_{2}$.

Let us consider a subgroup ${\bf I}_{a\times\dot{{a}}}={\bf
I}_{a}\times{\dot{\bf I}}_{a}\subset{\mathbb{Z}}/2^{[2]}$ in the dihedral
group, which is generated by the elements $\\{a,b^{2}a\\}$. This is an
elementary $2$-group of the rank $2$. Transformations of this group keep each
line $l_{1}$, $l_{2}$ with the base vectors ${\bf f}_{1}={\rm e}_{1}+{\rm
e}_{2}$, ${\bf f}_{2}={\rm e}_{1}-{\rm e}_{2}$ correspondingly. The cohomology
group $H^{1}(K({\bf I}_{a}\times{\dot{\bf I}}_{a},1);{\mathbb{Z}}/2)$ contains
two generators $\varkappa_{a},\varkappa_{\dot{{a}}}$.

Let us define the cohomology classes

$\displaystyle\varkappa_{a}\in H^{1}(K({\bf I}_{a}\times{\dot{\bf
I}}_{a},1);{\mathbb{Z}}/2)),\quad\varkappa_{\dot{{a}}}\in H^{1}(K({\bf
I}_{a}\times{\dot{\bf I}}_{a},1);{\mathbb{Z}}/2)).$ (11)

Denote by $p_{a}:{\bf I}_{a}\times{\dot{\bf I}}_{a}\to{\bf I}_{a}$ a
projection, the kernel of $p_{a}$ consists the symmetry transformation with
respect to the bisector of the second coordinate angle and the identity.

Denote $\varkappa_{a}=p_{a}^{\ast}(t_{a})$, where $e\neq t_{a}\in H^{1}(K({\bf
I}_{a},1);{\mathbb{Z}}/2)\simeq{\mathbb{Z}}/2$. Let us denote by
$p_{\dot{{a}}}:{\bf I}_{a}\times{\dot{\bf I}}_{a}\to{\dot{\bf I}}_{a}$ the
projection, the kernel of $p_{\dot{{a}}}$ consists of the symmetry with
respect to the bisector of the first coordinate angle and the identity.

Let us denote $\varkappa_{\dot{{a}}}=p_{\dot{{a}}}^{\ast}(t_{\dot{{a}}})$,
where $e\neq t_{\dot{{a}}}\in H^{1}(K({\dot{\bf
I}}_{a},1);{\mathbb{Z}}/2)\cong{\mathbb{Z}}/2$.

### A standardized immersion with a dihedral framing

Let us consider a ${\bf D}$-framed immersion $(g,\Psi,\eta_{N})$ of
codimension $2k$. Let us assume that the image of $g$ contains in a regular
neighbourhood $U({\mathbb{R}}\\!{\rm P}^{2})$ of the embedding
${\mathbb{R}}\\!{\rm P}^{2}\subset{\mathbb{R}}^{n}$. The following mapping
$\pi\circ\eta_{N}:N^{n-2k}\to K({\bf D},1)\to K({\mathbb{Z}}/2,1)$ is well-
defined, where $K({\bf D},1)\to K({\mathbb{Z}}/2,1)$ is the epimorphism with
the kernel ${\bf I}_{a}\times{\dot{\bf I}}_{a}\subset{\bf D}$. It is required
that this mapping coincides to the composition $i\circ PROJ\circ\eta_{N}$,
where $PROJ:U({\mathbb{R}}\\!{\rm P}^{2})\to{\mathbb{R}}\\!{\rm P}^{2}$ is the
projection of the neighbourhood onto its central line, $i:{\mathbb{R}}\\!{\rm
P}^{2}\subset K({\mathbb{Z}}/2,1)$ is the standard inclusion, which transforms
the fundamental class into the generator.

Below we will clarify a small constant $d$ and denote an integer
$\delta=[\frac{n-k}{2}]+d\approx\frac{n-k}{2}$. The integer $\delta\in{\bf N}$
is called the denominator of a defect of a standardization.

C1 Let us consider the submanifold $N_{sing}^{n-2k-2}\subset N^{n-2k}$,
$N_{sing}=PROJ^{-1}({\mathbb{R}}\\!{\rm P}^{0}),{\mathbb{R}}\\!{\rm
P}^{0}\subset{\mathbb{R}}\\!{\rm P}^{2}$. Require that the mapping $\eta_{N}$,
restricted to $N_{sing}$, be skipped trough the skeleton of $K({\bf
I}_{a}\times{\dot{\bf I}}_{a},1)$ the dimension $\delta$ .

Geometrically, this means that the submanifold $N_{sing}$ is a defect of a
reduction of the structured mapping of the normal bundle with a control of the
subgroup ${\bf I}_{a}\times{\dot{\bf I}}_{a}\subset{\bf D}$ over
${\mathbb{R}}\\!{\rm P}^{2}$ to a mapping with a control over
${\mathbb{R}}\\!{\rm P}^{1}\subset{\mathbb{R}}\\!{\rm P}^{2}$.

In the case $k\to 0+$ structured mapping on the defect manifold is skipped
through a polyhedron of the dimension $\frac{n}{2}$, in the case
$2k\to\frac{n}{2}$ through the empty polyhedron: the defect manifold is
absent.

The complement $N\setminus N_{sing}$ to the defect we get an open manifold,
for which the classifying normal bundle mapping is given by
$\eta_{N}:N\setminus N_{sing}\to K({\bf I}_{a}\times{\dot{\bf
I}}_{a})\rtimes{\mathbb{R}}\\!{\rm P}^{1}$.

######  Definition 3.

Let us say that a ${\bf D}$-framed immersion $(g,\Psi,\eta_{N})$ of the
codimension $2k$ is standardized with a defect $\delta\approx\frac{n}{2}$, if
the condition $C1$ is satisfied.

Standardized ${\bf D}$-framed immersions with a given defect generates an
abelian group $Imm^{D;stand}(n-1,1)$, which is naturally mapped into
$Imm^{{\bf D}}(n-2k,2)$ after a control structure is omitted.

######  Definition 4.

Assume $[(f,\Xi,\varkappa_{M})]\in Imm^{sf}(n-k,k)$,
$f:M^{n-k}\looparrowright{\mathbb{R}}^{n}$, $\varkappa_{M}\in
H^{1}(M^{n-k};{\mathbb{Z}}/2)$, $\Xi$ is a skew-framing. Let us say that the
pair $(M^{n-k},\varkappa_{M})$ admits a compression of an order $q$, if the
mapping $\varkappa_{M}:M^{n-k}\to{\mathbb{R}}\\!{\rm P}^{\infty}$ is
represented (up to homotopy) by the following composition:
$\varkappa=I\circ\varkappa_{M}^{\prime}:M^{n-k}\to{\mathbb{R}}\\!{\rm
P}^{n-k-q-1}\subset{\mathbb{R}}\\!{\rm P}^{\infty}$. Let us say that the
element $[(f,\Xi,\varkappa_{M})]$ admits a compression of an order $q$, if in
its cobordism class exists a triple
$(f^{\prime},\Xi^{\prime},\varkappa_{M^{\prime}})$, which admits a compression
of the order $q$.

Below we assume $n_{\sigma}=14$, $k=\frac{n?1n_{\sigma}}{16}$. In particular,
in the case $n=126$ we get $k=7$ and in the case $n=254$ we get $k=15$.

######  Theorem 5.

1\. Assume $n=2^{\ell}-2$, $\ell\geq 7$. An arbitrary class of ${\bf
D}$-framed cobordism from the image of the homomorphism (2) (for $k>1$ from
the image of the left homomorphism on the bottom line of the diagram (10) ) is
represented by a standardized dihedral immersion $(g,\eta_{N},\Psi)$ with
$d=2$. This construction is well-defined up to regular cobordism in the caee
$k\geq 2$. 111The constant $d$ cannot be large

2\. Assume that an element $x\in
Imm^{sf}(n-\frac{n-m_{\sigma}}{16},\frac{n-m_{\sigma}}{16})$ admits a
compression of the order $q=\frac{m_{\sigma}}{2}$ (in particular, $q=7$,
$k=\frac{n-m_{\sigma}}{16}=7$, $m_{\sigma}=14$). 222Based on Theorem [R-L] by
Kee Yuen Lam and Duane Randall (2006), for $n=126$ the case $q=10$ is
sufficient for a proof of Main Result. By this additional assumption one may
get a standardization of $x$ with the following additional condition:

– a submanifold $N^{m_{\sigma}}_{a\times\dot{{a}}}\subset N^{n-2k}$,

$N^{m_{\sigma}}_{a\times\dot{{a}}}=\eta_{N}^{-1}(K(n-2k-m_{\sigma})\subset
K({\bf D},1))$

admits a structured subgroup ${\bf I}_{a}\times{\dot{\bf I}}_{a}\subset{\bf
D}$ of the normal bundle (equivalently, the mapping $\pi\circ
g:N^{n-2k}\looparrowright U({\mathbb{R}}\\!{\rm P}^{2})\to{\mathbb{R}}\\!{\rm
P}^{2}$, restricted on the submanifold $N^{m_{\sigma}}_{a\times\dot{{a}}}$, is
homotopy to the constant mapping).

3\. For the self-intersection manifold $N^{n-2k}$ of an immersion $f$ the
following formula (19) is satisfied, additionally, the structured mapping for
the second term at the right-hand side of the formula admits a reduction to
the subgroup ${\bf I}_{a}\times{\dot{\bf I}}_{a}$, the structured mapping for
the first term at the right-hand side of the formula admits a reduction to a
polyhedron of the dimension $\frac{n-3(k+q)}{2}+d=\frac{n-3k^{\prime}}{2}+d$.

To prove the theorem a preliminary construction is required.

Let

$\displaystyle d^{(2)}:{\mathbb{R}}\\!{\rm
P}^{n-k^{\prime}}\times{\mathbb{R}}\\!{\rm
P}^{n-k^{\prime}}\setminus{\mathbb{R}}\\!{\rm
P}^{n-k^{\prime}}_{diag}\to{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$ (12)

be an arbitrary $(T_{{\mathbb{R}}\\!{\rm
P}^{n-k^{\prime}}\times{\mathbb{R}}\\!{\rm
P}^{n-k^{\prime}}},T_{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}})$–equivariant
mapping, which is transversal along the diagonal
${\mathbb{R}}^{n}_{diag}\subset{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$. The
diagonal in the preimage is mapped into the diagonal of the image, by this
reason the equivariant mapping $d^{(2)}$ is defined on the open manifold
outside of the diagonal (in the pre-image).

(To Condition 1 in Theorem 5 the case $k^{\prime}=k$ is required, for
Condition 2 the codimension $k^{\prime}$ has to be defined by the formula:
$k^{\prime}=k+q+1$ (parameters $k$,$q$ correspond to denotations of Theorem
5).

Let us re-denote
$(d^{(2)})^{-1}({\mathbb{R}}^{n}_{diag})/T_{{\mathbb{R}}\\!{\rm
P}^{n-k^{\prime}}\times{\mathbb{R}}\\!{\rm P}^{n-k^{\prime}}}$ by ${\bf
N}_{circ}={\bf N}(d^{(2)})_{\circ}$ for short, this polyhedron is called a
polyhedron of (formal) self-intersection of the equivariant mapping $d^{(2)}$.

The polyhedron ${\bf N}_{\circ}$ is an open polyhedron, this polyhedron admits
a compactification, which is denoted by ${\bf N}$ with a boundary
$\partial{\bf N}$. The boundary consists of all critical (formal critical)
points of the mapping $d^{(2)}$. One has: ${\bf N}\setminus\partial{\bf N}$.
Let us denote by ${\bf N}_{\circ}$ an open polyhedron by $U(\partial{\bf
N})_{\circ}$ a thin regular neighbourhood of the diagonal $\partial{\bf N}$.

The polyhedron ${\bf N}_{\circ}$ is equipped by the mapping $d^{(2)}$, which
admits the following lift:

$\displaystyle\eta_{circ;Ab}:{\bf N}_{\circ}\to K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1).$ (13)

The restriction of the mapping $d^{(2)}$, called the structured mapping, is
the following:

$\displaystyle\eta_{\circ}:{\bf N}_{\circ}\to K({\mathbb{Z}}/2^{[2]},1).$ (14)

On the polyhedron $U(\partial{\bf N})_{\circ}$ the mapping $\eta_{circ;Ab}$
gets the values into the following subcomplex: $K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}2{\mathbb{Z}},1)\subset
K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$ (the
projection on $K({\mathbb{Z}},1)$ is the analogue of the projection of the
Moebius band onto its central line).

######  Definition 6.

Let us call that a formal (equivariant) mapping $d^{(2)}$, given by
$(\ref{d2})$, is holonomic, if this mapping is the formal extension of a
mapping

$\displaystyle d:{\mathbb{R}}\\!{\rm P}^{n-k^{\prime}}\to{\mathbb{R}}^{n}.$
(15)

######  Definition 7.

Assume a formal (equivariant) mapping $(\ref{d2})$ is holonomic. Let us say
$d^{(2)}$ admits an Abelian structure, if the following two conditions are
satisfied.

– 1. On the open polyhedron ${\bf N}_{\circ}$ the mapping (13) is well-
defined, which is a lift of the structured mapping (14), i.e. the composition
of the lift $\eta_{circ;Ab}$ with a natural mapping $K({\bf
I}_{a\times\dot{{a}}})\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)\to
K({\mathbb{Z}}/2^{[2]},1)$ coincides with the structured mapping (14).

–2. Let us consider the Moebius band $M^{2}$ and represent the Eilenberg-
MacLane space $K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$ as a
skew-product $K({\bf I}_{a\times\dot{{a}}},1)\tilde{\times}M^{2}\to M^{2}$;
the restriction of the fibration over the boundary circle
$S^{1}=\partial(M^{2})$ is identified with the subspace $K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}2{\mathbb{Z}},1)\subset
K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$. Let
us include the space $K({\bf I}_{a\times\dot{{a}}},1)\tilde{\times}M^{2}$ into
$K({\bf I}_{a\times\dot{{a}}},1)\tilde{\times}{\mathbb{R}}\\!{\rm P}^{2}$ by a
gluing of the trivial bundle over the boundary $S^{1}$ by the trivial bundle
over a small disk with a central point ${\bf x}_{\infty}\in{\mathbb{R}}\\!{\rm
P}^{2}$.

The resulting space is denoted by $K({\bf
I}_{a\times\dot{{a}}},1)\rtimes{\mathbb{R}}\\!{\rm P}^{2}$. The following
mapping:

$\displaystyle\eta_{\circ;Ab}:({\bf N}_{\circ},U(\partial{\bf
N}_{\circ}))\to(K({\bf I}_{a\times\dot{{a}}},1)\rtimes{\mathbb{R}}\\!{\rm
P}^{2},K({\bf I}_{a\times\dot{{a}}},1)\times{\bf x}_{\infty}),$ (16)

is well-defined, where the inverse image by $\eta_{\circ;Ab}$ of a fibre
$K({\bf I}_{a\times\dot{{a}}},1)\times{\bf y}_{\infty}\subset K({\bf
I}_{a\times\dot{{a}}},1)\rtimes{\mathbb{R}}\\!{\rm P}^{2}$, (${\bf
y}_{\infty}\approx{\bf x}_{\infty}$) is dominated by a subpolyhedron of the
dimension $\frac{n-3k^{\prime}}{2}=\frac{{\operatorname{dim}}({\bf
N}_{\circ})}{2}-k^{\prime}$ (up to a small constant $d=2$), which is equipped
by the structured mapping.

######  Remark 8.

The condition –2 in Definition 7 can be formulated, alternatively, as
following: in the polyhedron ${\bf N}_{\circ}$ there exists a subpolyhedron
${\bf N}_{\circ;reg}\subset U(\partial{\bf N}_{\circ})\subset{\bf N}_{\circ}$,
which is inside its proper thin neighbourhood, and there exists a polyhedron
$P$ of the dimension (up to a small constant $d=2$)
$dim(P)=\frac{{\operatorname{dim}}({\bf
N}_{\circ})}{2}-k^{\prime}=\frac{n-3k^{\prime}}{2}$; there exists a control
mapping ${\bf N}_{\circ;reg}\to P$ and the structured mapping $\eta_{P}:P\to
K({\bf I}_{a\times\dot{{a}}},1)\times S^{1}$, such that the composition ${\bf
N}_{\circ;reg}\to P\to K({\bf I}_{a\times\dot{{a}}},1)\times S^{1}\to K({\bf
I}_{a\times\dot{{a}}},1)\rtimes{\mathbb{R}}\\!{\rm P}^{1}\subset K({\bf
I}_{a\times\dot{{a}}},1)\rtimes{\mathbb{R}}\\!{\rm P}^{2}$ is the structured
mapping $\eta_{\circ;Ab}:{\bf N}_{\circ;reg}\to K({\bf
I}_{a\times\dot{{a}}},1)\rtimes_{\chi^{[2]}}{\mathbb{R}}\\!{\rm P}^{2}$ up to
homotopy. Consider the pair $(U(\partial{\bf N}_{\circ}),U(\partial{\bf
N}_{\circ})\setminus{\bf N}_{\circ;reg})$, and the restriction of
$\eta_{\circ;Ab}$ on this pair. The following mapping of pairs exists:

$\displaystyle\eta_{\circ;Ab}:(U(\partial{\bf N}_{\circ}),U(\partial{\bf
N}_{\circ})\setminus{\bf N}_{\circ;reg})\to(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}2{\mathbb{Z}},1),K({\bf
I}_{a\times\dot{{a}}},1)).$ (17)

In the formula above $K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}2{\mathbb{Z}},1)=K({\bf
I}_{a\times\dot{{a}}}\times{\mathbb{Z}},1)$; $K({\bf
I}_{a\times\dot{{a}}},1)\subset K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}2{\mathbb{Z}},1)$ is
the standard the inclusion.

The following lemma is proved in Section 8.

######  Lemma 9.

Small Lemma

For

$\displaystyle n-k^{\prime}\equiv 1\pmod{2},\quad n\equiv 0\pmod{2}$ (18)

there exist a holonomic formal mapping $d^{(2)}$, which admits an Abelian
structure, Definition 7.

### Mapping $d^{(2)}$ in Lemma 9

Let us consider the standard $2$-sheeted covering $p:S^{1}\to S^{1}$ over the
circle. One may write-down: $p:S^{1}\to{\mathbb{R}}\\!{\rm P}^{1}$, where
${\mathbb{Z}}/2\times S^{1}\to{\mathbb{R}}\\!{\rm P}^{1}$ is the standard
antipodal action. Let us consider the join of
$\frac{n-k^{\prime}+1}{2}=r$-copies of ${\mathbb{R}}\\!{\rm P}^{1}$
${\mathbb{R}}\\!{\rm P}^{1}\ast\dots\ast{\mathbb{R}}\\!{\rm
P}^{1}=S^{n-k^{\prime}}$, which is homeomorphic to the standard
$n-k^{\prime}$-dimensional sphere. Let us define the join of $r$ copies of the
mapping

$\tilde{d}:S^{1}\ast\dots\ast S^{1}\to{\mathbb{R}}\\!{\rm
P}^{1}\ast\dots\ast{\mathbb{R}}\\!{\rm P}^{1}.$

In the preimage the group ${\mathbb{Z}}/2$ by the antipodal action is
represented, this action is commuted with $\tilde{d}$. The required mapping
(15) is the composition of the projection onto the factor
$\tilde{d}/_{{\mathbb{Z}}/2}:{\mathbb{R}}\\!{\rm P}^{n-k^{\prime}}\to
S^{n-k^{\prime}}$ with the standard inclusion

$d:S^{n-k^{\prime}}\subset{\mathbb{R}}^{n}.\qed$

#### Proof of Theorem 5

Let us define $k=\frac{n-m_{\sigma}}{16}$. Assume that an element $x\in
Imm^{sf}(n-k,k)$ is represented by a skew-framed immersion
$(f,\Xi,\varkappa_{M})$, $f:M^{n-k}\looparrowright{\mathbb{R}}^{n}$. By the
assumption, there exist a compression
$\varkappa_{M}:M^{n-k}\to{\mathbb{R}}\\!{\rm P}^{n-k-q-1}$,
$q=\frac{m_{\sigma}}{2}+1$, for which the composition
$M^{n-k}\to{\mathbb{R}}\\!{\rm P}^{n-k-q-1}\subset K({\bf I}_{d},1)$ coincides
with the mapping $\varkappa_{M}:M^{n-k}\to K({\bf I}_{d},1)$.

Define $k^{\prime}$ as the biggest positive integer, which is less then
$k+q+1$, and such that $n-k^{\prime}\equiv 1\pmod{2}$. Because
$q=\frac{m_{\sigma}}{2}\geq 14$, we get $k^{\prime}\geq 7$ and the both
assumptions with the dimension $(\ref{dimdim})$ are satisfied.

By Lemma 9 there exist an (equivariant) formal mapping $d^{(2)}$, which admits
an abelian structure, see Definition 7.

Let us construct a skew-framed immersion $(f_{1},\Xi_{1},\varkappa_{1})$, for
which the self-intersection manifold is as required in Theorem 5.

An immersion $f_{1}:M^{n-k}\looparrowright{\mathbb{R}}^{n}$, which is equipped
by a skew-framing $(\varkappa_{1},\Xi_{1})$, define as a result of a small
deformation of the composition $d\circ\varkappa:M^{n-k}\to{\mathbb{R}}\\!{\rm
P}^{n-k-q-1}\subset{\mathbb{R}}\\!{\rm P}^{n-k^{\prime}}\to{\mathbb{R}}^{n}$
in the prescribed regular cobordism class of the immersion $f$. A calibre of
the deformation $d\circ\varkappa\mapsto f_{a}$ define less then a radius of a
regular neighbourhood of the polyhedron of intersection points of the mapping
$d$.

Let us denote by $N^{n-2k}$ the self-intersection manifold of the immersion
$f_{1}$. The following decomposition over a common boundary are well-defined:

$\displaystyle N^{n-2k}=N^{n-2k}_{a\times\dot{{a}},{\bf
N}(d^{(2)})}\cup_{\partial}N^{n-2k}_{reg}.$ (19)

In this formula $N^{n-2k}_{a\times\dot{{a}},{\bf N}(d^{(2)})}$ is a manifold
with boundary, which is immersed into a regular (immersed) neighbourhood
$U_{{\bf N}(d)}$ of the polyhedron ${\bf N}(d)$ (with a boundary) of self-
intersection points of the mapping $d$. The manifold $N^{n-2k}_{reg}$ with
boundary is immersed into an immersed regular neighbourhood of self-
intersection points, outside critical points; let us denote this neighbourhood
by $U_{reg}$. A common boundary of manifolds $N^{n-2k}_{a\times\dot{{a}},{\bf
N}(d^{(2)})}$, $N^{n-2k}_{reg}$ is a closed manifold of dimension $n-2k-1$,
this manifold is immersed into the boundary $\partial(U_{reg})$ of the
immersed neighbourhood $U_{reg}$.

The parametrized ${\mathbb{Z}}/2^{[2]}$–framed immersion of the immersed
submanifold $(\ref{Nregb})$ denote by $g_{a\times\dot{{a}}}$. Let us prove
that a ${\mathbb{Z}}/2^{[2]}$–framing over the component $(\ref{Nregb})$ is
reduced to a framing with the structured group ${\bf
I}_{a\times\dot{{a}}}\rtimes{\mathbb{Z}}/2$ with a control over
${\mathbb{R}}\\!{\rm P}^{2}$.

Define on the manifold $N^{n-2k}_{a\times\dot{{a}}}$ a mapping
$N^{n-2k}_{a\times\dot{{a}}}\stackrel{{\scriptstyle\eta_{a\times\dot{{a}}}}}{{\longrightarrow}}K({\bf
I}_{a\times\dot{{a}}},1)\operatorname{\rtimes}_{\chi^{[2]}}\rtimes{\mathbb{R}}\\!{\rm
P}^{2}$, as the result of a gluing of the two mappings.

Define on the component $N^{n-2k}_{a\times\dot{{a}},{\bf N}(d^{(2)})}$ in the
formula $(\ref{Nregb})$ the following mapping:

$\displaystyle\eta_{a\times\dot{{a}},{\bf
N}(d^{(2)})}:N^{n-2k}_{a\times\dot{{a}},{\bf N}(d^{(2)})}\to K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}),1),$
(20)

as the composition of the projection $N^{n-2k}_{a\times\dot{{a}},{\bf
N}(d^{(2)})}\to N(d)$ and the mapping $\eta_{a\times\dot{{a}}}:{\bf
N}(d^{(2)})\to K(({\bf
I}_{a\times\dot{{a}}})\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$,
constructed in Lemma 9.

The restrictions of the mappings $\eta_{a\times\dot{{a}},N(d^{(2)})}$,
$\eta_{a\times\dot{{a}},N_{reg}}$ on a common boundary $\partial
N^{n-2k}_{a\times\dot{{a}},N(d^{(2)})}$ ? $\partial N^{n-2k}_{reg}$ are
homotopy, because the mapping $\eta_{{\bf N}(d^{(2)})}:{\bf N}(d^{(2)})\to
K({\bf I}_{a}\times{\dot{\bf
I}}_{a}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}),1)$ satisfies on
$\partial{\bf N}(d^{(2)})$ right boundary conditions. Therefore, the required
mapping

$\displaystyle\eta_{a\times\dot{{a}}}:N^{n-2k}_{a\times\dot{{a}}}\to K({\bf
I}_{a\times\dot{{a}}},1)\rtimes_{\chi^{[2]}}{\mathbb{R}}\\!{\rm P}^{2}$ (21)

is well-defined as a result of a gluing of the mappings
$\eta_{a\times\dot{{a}},{\bf N}(d^{(2)})}$ and
$\eta_{a\times\dot{{a}},N_{reg}}$. The mapping $(\ref{etadtd})$ determines a
reduction of ${\mathbb{Z}}/2^{[2]}$–framing of the immersion
$g_{a\times\dot{{a}}}$.

Let us check Condition $C1$, see Definition 3. This is a corollary of
Condition –2 in Definition 7.

By the construction, the image of the mapping
$f_{2}:M^{n-8k}\looparrowright{\mathbb{R}}^{n}$ belongs to a regular
neighbourhood of the image of the submanifold ${\mathbb{R}}\\!{\rm
P}^{n-8k-q-1}\subset{\mathbb{R}}\\!{\rm P}^{n-7k-k^{\prime}}$ by the mapping
$d$. Because
$n-8k-q-1=n-\frac{n}{2}+\frac{m_{\sigma}}{2}-\frac{m_{\sigma}}{2}-1=\frac{n}{2}-1$,
the immersed submanifold $d({\mathbb{R}}\\!{\rm P}^{n-8k-q-1})$ is an embedded
submanifold.

Denote by $N_{2}^{n-16k}$ the manifold of self-intersection points of the
immersion $f_{2}$, this is a manifold of dimension $n-16k=m_{\sigma}$. The
following inclusion is well-defined:

$\displaystyle N_{2}^{n-16k}\subset N_{reg}^{n-2k},$ (22)

where the manifold $N_{reg}^{n-2k}$ is defined by the formula $(\ref{Nregb})$.

In particular, the following reduction of the classified mapping
$\eta_{2}:N^{n-16k}_{2}\to K({\mathbb{Z}}/2^{[2]},1)$ to a mapping

$\displaystyle\eta_{Ab,2}:N^{n-16k}_{2}\to K({\bf I}_{b\times\dot{{b}}},1)$
(23)

with the image $K({\bf I}_{b\times\dot{{b}}},1)\subset
K({\mathbb{Z}}/2^{[2]},1)$ is well-defined.

Therefore, without loss of a generality, for $\sigma\geq 5$ one may assume
that

$\displaystyle N^{n-16k}_{2}\cap N^{n-2k}_{a\times\dot{{a}},{\bf
N}(d)}=\emptyset.$ (24)

Statement 3 follows from the definition of the mapping $f_{1}$.

Theorem 5 is proved. ∎

##  3 Local coefficients and homology groups

Let us define the group $({\bf I}_{a}\times{\dot{\bf
I}}_{a})\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}$ and the epimorphism
$({\bf I}_{a}\times{\dot{\bf
I}}_{a})\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}\to{\mathbb{Z}}/2^{[2]}$.
Consider the automorphism

$\displaystyle\chi^{[2]}:{\bf I}_{a}\times{\dot{\bf I}}_{a}\to{\bf
I}_{a}\times{\dot{\bf I}}_{a}$ (25)

of the exterior conjugation of the subgroup ${\bf I}_{a}\times{\dot{\bf
I}}_{a}\subset{\bf D}$ by the element $ba\in{\bf D}$, this element is
represented by the reflection of the plane with respect to the line $Lin({\rm
e}_{2})$. Let us define the authomorphism (denotations are not changed)

$\displaystyle\chi^{[2]}:{\mathbb{Z}}/2^{[2]}\to{\mathbb{Z}}/2^{[2]},$ (26)

by permutations of the base vectors. It is not difficult to check, that the
inclusion ${\bf I}_{a}\times{\dot{\bf I}}_{a}\subset{\mathbb{Z}}/2^{[2]}$
commutes with automorphisms $(\ref{chidd})$, $(\ref{chi2})$ in the image and
the preimage.

Define the group

$\displaystyle{\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}.$ (27)

Let us consider the quotient of the group ${\bf
I}_{a\times\dot{{a}}}\ast{\mathbb{Z}}$ (the free product of the group ${\bf
I}_{a\times\dot{{a}}}$ and ${\mathbb{Z}}$) by the relation
$zxz^{-1}=\chi^{[2]}(x)$, where $z\in{\mathbb{Z}}$ is the generator, $x\in{\bf
I}_{a\times\dot{{a}}}$ is an arbitrary element.

This group is a particular example of a semi-direct product
$A\rtimes_{\phi}B$, $A={\bf I}_{a\times\dot{{a}}}$, $B={\mathbb{Z}}$, by a
homomorphism $\phi:B\to Aut(A)$; the set $A\times B$ is equipped with a binary
operation
$(a_{1},b_{1})\ast(a_{2},b_{2})\mapsto(a_{1}\phi_{b_{1}}(b_{2}),b_{1}b_{2})$.
Let us define the group $(\ref{chi2Z})$ by this construction for $A={\bf
I}_{a\times\dot{{a}}}$, $B={\mathbb{Z}}$, $\phi=\chi^{[2]}$.

The classifying space $K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$ is a
skew-product over the circle $S^{1}$ with $K({\bf I}_{a\times\dot{{a}}},1)$,
where the shift mapping in the cyclic covering $K({\bf
I}_{a\times\dot{{a}}},1)\to K({\bf I}_{a\times\dot{{a}}},1)$ over $K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$ is
induced by the automorphism $\chi^{[2]}$. The projection onto the circle is
denoted by

$\displaystyle p_{a\times\dot{{a}}}:K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)\to
S^{1}.$ (28)

Take a marked point $pt_{S^{1}}\in S^{1}$ and define the subspace

$\displaystyle K({\bf I}_{a\times\dot{{a}}},1)\subset K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$ (29)

as the inverse image of the marked point $pt_{S^{1}}$ by the mapping
$(\ref{pbb})$.

A description of the standard base of the group $H_{i}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}})$
is sufficiently complicated and is not required. The group $H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}})$ is described using the Kunneth
formula:

$\displaystyle\begin{array}[]{c}0\to\bigoplus_{i_{1}+i_{2}=i}H_{i_{1}}(K({\bf
I}_{a},1);{\mathbb{Z}})\otimes H_{i_{2}}(K({\dot{\bf
I}}_{a},1);{\mathbb{Z}})\longrightarrow\\\ \longrightarrow H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}})\longrightarrow\\\
\longrightarrow\bigoplus_{i_{1}+i_{2}=i-1}Tor^{{\mathbb{Z}}}(H_{i_{1}}(K({\bf
I}_{a},1);{\mathbb{Z}}),H_{i_{2}}(K({\dot{\bf I}}_{a},1);{\mathbb{Z}})\to
0.\\\ \end{array}$ (33)

The standard base of the group $H_{i}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1))$
contains the following elements:

$x\otimes y/(x\otimes y)-(y\otimes x),$

where $x\in H_{j}(K({\bf I}_{a},1))$, $y\in H_{i-j}(K({\dot{\bf I}}_{a},1))$
(${\mathbb{Z}}/2$–coefficients is the formulas are omitted).

In particular, for odd $i$ the group $H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}})$ contains elements, which are defined
by the fundamental classes of the following submanifolds: ${\mathbb{R}}\\!{\rm
P}^{i}\times pt\subset{\mathbb{R}}\\!{\rm P}^{i}\times{\mathbb{R}}\\!{\rm
P}^{i}\subset K({\bf I}_{a},1)\times K({\dot{\bf I}}_{a},1)=K({\bf
I}_{a\times\dot{{a}}},1)$, $pt\times{\mathbb{R}}\\!{\rm
P}^{i}\subset{\mathbb{R}}\\!{\rm P}^{i}\times{\mathbb{R}}\\!{\rm P}^{i}\subset
K({\bf I}_{a},1)\times K({\dot{\bf I}}_{a},1)=K({\bf
I}_{a\times\dot{{a}}},1)$. Let us denote the corresponding elements as
following:

$\displaystyle t_{a,i}\in H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}}),t_{\dot{{a}},i}\in H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}}).$ (34)

The following analogues of the homology groups $H_{i}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1))$,
$H_{i}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}})$
with local coefficients over the generator $\pi_{1}(S^{1})$ 333 more simple
homology groups $H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}})$
with ${\mathbb{Z}}$-local coefficient system also can be defined, in this case
on the last step of the construction difficulties with triple (non-
commutative) local coefficient system arise. is defined, the groups are
denoted by

$\displaystyle H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}/2[{\mathbb{Z}}/2]),$
(35) $\displaystyle H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2]).$
(36)

The following epimorphism

$\displaystyle p_{a\times\dot{{a}}}:{\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}\to{\mathbb{Z}}$
(37)

is well-defined by the formula: $x\ast y\mapsto y$, $x\in{\bf
I}_{a\times\dot{{a}}}$, $y\in{\mathbb{Z}}$. The following homomorphism

$\displaystyle{\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}\to{\mathbb{Z}}\to{\mathbb{Z}}/2,$
(38)

is well-defined by the formula $p_{a\times\dot{{a}}}\pmod{2}$.

Let us define the group $(\ref{Zlok})$. Let us consider the group ring
${\mathbb{Z}}[{\mathbb{Z}}/2]=\\{a+bt\\}$, $a,b\in{\mathbb{Z}}$,
$t\in{\mathbb{Z}}/2$. The generator $t\in{\mathbb{Z}}[{\mathbb{Z}}/2]$ is
represented by the involution

$\chi^{[2]}:K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)\to
K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1),$

the restriction of the involution on the subspace $K({\bf
I}_{a\times\dot{{a}}},1)\subset K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$ is
the reflection, which is induced by the automorphism ${\bf
I}_{a}\times{\dot{\bf I}}_{a}\to{\bf I}_{a}\times{\dot{\bf I}}_{a}$, which
permutes factors. Because all non-trivial homology classes of the space
$K({\bf I}_{a\times\dot{{a}}},1)$ are of the order $2$, transformation of
signs is not required. Nevertheless, even-dimensional $n-2k$-simplexes are
transformed in the case of odd $k$ by the opposition of the orientation, in
the case of even $k$ the orientation is preserved.

Let us consider the local system of the coefficient?
$\rho_{t}:{\mathbb{Z}}/2[{\mathbb{Z}}/2]\to Aut(K({\bf
I}_{a\times\dot{{a}}},1)\subset K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1))$ over
the generator of $\pi_{1}(S^{1})$, using this local system a chain
$(a+bt)\sigma$ with the support on a simplex $\sigma\subset K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$ is
transformed into a chain $(at+b)\chi^{[2]}(\sigma)$. The group $(\ref{Zlok})$
is well-defined. The group $(\ref{Z2lok})$ is defined analogously. A complete
calculation of the groups $(\ref{Z2lok})$, $(\ref{Zlok})$ is not required.

Let us define the following subgroup:

$\displaystyle D_{i}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2])\subset H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$
(39)

by the formula: $D_{i}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2])=\mathrm{Im}(H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])\to H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2]))$,
where the homomorphism

$H_{i}^{loc}(K({\bf I}_{a\times\dot{{a}}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])\to
H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$

is induced by the inclusion of the subgroup.

From the definition, there exist a natural homomorphism:

$\displaystyle H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])\to D_{i}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2]).$ (40)

The following natural homomorphism $H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}}))\otimes{\mathbb{Z}}[{\mathbb{Z}}/2]\to
H_{i}(K({\bf I}_{a\times\dot{{a}}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$ is well-
defined, this homomorphism is an isomorphism by the universal coefficients
formula. This calculation is following from the isomorphism
${\mathbb{Z}}[{\mathbb{Z}}/2]\equiv{\mathbb{Z}}\oplus{\mathbb{Z}}$ an the
additivity of the derived functor $Tor^{{\mathbb{Z}}}$. Analogously,
$H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}}/2))\otimes{\mathbb{Z}}/2[{\mathbb{Z}}/2]\equiv
H_{i}(K({\bf I}_{a\times\dot{{a}}},1);{\mathbb{Z}}/2[{\mathbb{Z}}/2])$. The
subgroup $(\ref{DZlok})$ is generated by various elements $X+Yt$, $X,Y\in
H_{i}(K({\bf I}_{a\times\dot{{a}}},1);{\mathbb{Z}})$.

A subgroup

$\displaystyle D_{i}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}/2[{\mathbb{Z}}/2])\subset
H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}/2[{\mathbb{Z}}/2])$
(41)

is defined analogously as $(\ref{DZlok})$. A description of this subgroup is
more easy, because this group is generated by elements $X+tY$, $X=x\otimes
y,Y=x^{\prime}\otimes y^{\prime}$, wherein $\chi^{[2]}_{\ast}(x\otimes
y)=y\otimes x$.

Let us define the homomorphism

$\displaystyle\Delta^{[2]}:H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])\to H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}})$ (42)

by the formula: $\Delta^{[2]}(X+Yt)=[X]+\chi^{[2]}([Y])$, where $[X],[Y]$ are
cycles which are defined as images of local cycles $X,Y$ by the inclusion
$H_{i}(K({\bf I}_{a\times\dot{{a}}},1);{\mathbb{Z}})\subset H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}})$ (note that this inclusion admits the
inverse projection $H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}})\to H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}})$ and therefore this inclusion is an
isomorphism). In Section 5 an analog of the homomorphism (42) is defined as
the following: $\Delta^{[4]}(X+Yt)=X+\chi^{4}(\omega(Y))$,
$\Delta^{[4]}:H_{i}^{loc}(K({\bf
J}_{b\times\dot{{b}}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])\to H_{i}^{loc}(K({\bf
J}_{b\times\dot{{b}}},1);{\mathbb{Z}})$ where $\omega:H_{i}^{loc}(K({\bf
J}_{b\times\dot{{b}}},1);{\mathbb{Z}})\to H_{i}^{loc}(K({\bf
J}_{b\times\dot{{b}}},1);{\mathbb{Z}})$, is a corresponding non-trivial
authomorphism of homology groups equipped with a (secondary) non-trivial local
coeffisient system.

Let us prove that the homomorphism $(\ref{delta})$ admits a factorisomorphism:

$\displaystyle\Delta^{[2]}:D_{i}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2])\to H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}}),$ (43)

which is denoted the same.

Take a local cycle $X\in H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}})$, this cycle equals to
$\chi^{[2]}([X])\in H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}})$. As well as, by the definition of
(43), the corresponding local cycle $[X]\in H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$ equals to
$t\chi^{[2]}([X])H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$. This proves that
$[X]=t[X]$. The image and the preimage in (43) are isomorphic to
$H_{i}^{loc}(K({\bf I}_{a\times\dot{{a}}},1);{\mathbb{Z}})$. Let us remarque
that an analogous statement in Section 5 for twisted secondary local
coefficients system is possible.

Analogously, the isomorphism with modulo 2 coefficients is well-defined.

$\displaystyle\Delta^{[2]}:D_{i}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}/2[{\mathbb{Z}}/2])\to H_{i}(K({\bf
I}_{a\times\dot{{a}}},1)).$ (44)

The composition of the homomorphisms

$\displaystyle H_{i}(K({\bf I}_{a\times\dot{{a}}},1);{\mathbb{Z}})\to
H_{i}(K({\bf I}_{a\times\dot{{a}}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])\to$ (45)
$D_{i}^{loc}({\bf I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2])\to
H_{i}^{loc}(K({\bf I}_{a\times\dot{{a}}},1);{\mathbb{Z}})\to H_{i}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}}),$

where the first homomorphism is the natural inclusion, the middle is
$(\ref{epilok})$, and the last two are defined by the formula
$(\ref{deltaD})$, is the identity homomorphism.

Consider the subgroup $(\ref{DZlok})$ and the epimorphism $(\ref{epilok})$.
For an arbitrary element $x\in H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$
let us define the total obstruction

$\displaystyle o(x)\in H_{i-1}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2])$ (46)

to the inclusion $x\in D_{i}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2])\subset H_{i}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$.

Consider the homomorphism $(\ref{epid})$ and consider the mapping
$(\ref{pbb})$, which is associated with this homomorphism. Let $\theta\in
H^{1}(S^{1};{\mathbb{Z}}[{\mathbb{Z}}/2])$ be the generator. Consider the
element $p_{a\times\dot{{a}}}^{\ast}(\theta)\in H^{1}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$
and define the obstruction $(\ref{ox})$ by the formula: $o(x)=x\cap
p_{a\times\dot{{a}}}^{\ast}(\theta)$.

From this construction, the obstruction $(\ref{ox})$ is calculated as follows:
intersects this singular cycle of the space $K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$ with
the subspace $(\ref{inclK})$.

In the case $i=2s$ the basis elements $y\in D_{i-1}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2])$, with the condition
$\chi^{[2]}_{\ast}(y)=y$, are the following:

1\. $y=r$, $r=t_{a,2s-1}+t_{\dot{{a}},2s-1}$, where
$t_{a,2s-1},t_{\dot{{a}},2s-1}\in H_{2s-1}(K({\bf
I}_{a\times\dot{{a}}},1);{\mathbb{Z}})$ are defined by the formula
$(\ref{tdtdd})$.

2\. $y=z(i_{1},i_{2})$, where
$z(i_{1},i_{2})=tor(r_{a,i_{1}},r_{\dot{{a}},i_{2}})+tor(r_{a,i_{2}},r_{\dot{{a}},i_{1}})$,
$i_{1},i_{2}\equiv 1\pmod{2}$, $i_{1}+i_{2}=2s-2$,
$tor(r_{a,i_{1}},r_{\dot{{a}},i_{2}})\in Tor^{{\mathbb{Z}}}(H_{i_{1}}(K({\bf
I}_{a},1);{\mathbb{Z}}),H_{i_{2}}(K({\dot{\bf I}}_{a},1);{\mathbb{Z}}))$. The
elements $\\{z(i_{1},i_{2})\\}$ belong to the kernel of the homomorphism:

$\displaystyle D_{i-1}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2])\to D_{i-1}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}/2[{\mathbb{Z}}/2]),$ (47)

which is defined as the modulo 2 reduction of the coefficients:
${\mathbb{Z}}\to{\mathbb{Z}}/2$.

The elements $R,\quad Z(i_{1},i_{2})\in H_{2s}^{lok}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$
are defined as the direct product of the corresponding $2s-1$–cycle
$f:C_{2s-1}\to K({\bf I}_{a\times\dot{{a}}},1)$ on the circle. The mapping of
the cycle into $K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$ is
defined by the composition of the Cartesian product mapping $f\times
id:C_{2s-1}\times S^{1}\to K({\bf I}_{a\times\dot{{a}}},1)\times S^{1}$ with
the standard $2$-sheeted covering $K({\bf I}_{a\times\dot{{a}}},1)\times
S^{1}\to K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$.

Using the obstruction $(\ref{ox})$, the following lemma is proved.

######  Lemma 10.

The group $H_{2s}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$
is isomorphic to the direct sum of the subgroup $D_{2s}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2])$ and the subgroup, which
is generated by elements $R,\quad\\{Z(i_{1},i_{2})\\}$. The elements from the
subgroup

$\bigoplus_{i_{1}+i_{2}=2s}H_{i_{1}}(K({\bf I}_{a},1);{\mathbb{Z}})\otimes
H_{i_{2}}(K({\dot{\bf I}}_{a},1);{\mathbb{Z}})\subset D_{2s}^{loc}({\bf
I}_{a}\times{\dot{\bf I}}_{a};{\mathbb{Z}}[{\mathbb{Z}}/2]),$

and the element $R$ generate the image $\mathrm{Im}(A)$ of the following
homomorphism:

$\displaystyle A:H_{2s}^{loc}(K({\bf
I}_{a\times\dot{{b}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])\to$
(48) $H_{2s}^{loc}(K({\bf
I}_{a\times\dot{{b}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}/2[{\mathbb{Z}}/2]),$

which is the modulo 2 reduction of the coefficient system.

#### Proof of Lemma 10

Assume that $x\in H_{2s}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$
is not in the subgroup $D^{loc}_{2s}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2])$. Then $o(x)\neq 0$ (see.
$(\ref{ox})$ for $i=2s$). The element $x$ as an element in the residue class
with respect to the considered subgroup is expressed trough elements $R,\quad
Z(i_{1},i_{2})$. A subgroup, which contains all obstructions $(\ref{ox})$ is a
direct factor in $H^{loc}_{2s}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2])$
because obstructions are realized as collection of elements
$R,\quad\\{Z(i_{1},i_{2})\\}$, and the considered collection of elements
generates a direct factor. Each element $Z(i_{1},i_{2})$ belongs to
$\mathrm{Ker}(A)$, therefore the considered collections of elements contain no
contribution of $\mathrm{Im}(A)$. Lemma 10 is proved.

Let us consider the diagonal subgroup: $i_{{\bf I}_{d},{\bf
I}_{a\times\dot{{a}}}}:{\bf I}_{d}\subset{\bf I}_{a}\times{\dot{\bf
I}}_{a}={\bf I}_{a\times\dot{{a}}}$. This subgroup coincides to the kernel of
the homomorphism

$\displaystyle\omega^{[2]}:{\bf I}_{a}\times{\dot{\bf
I}}_{a}\to{\mathbb{Z}}/2,$ (49)

which is defined by the formula $(x\times y)\mapsto xy$.

Let us define the homomorphism

$\displaystyle\Phi^{[2]}:{\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}\to{\mathbb{Z}}/2^{[2]}$
(50)

by the formula: $\Phi^{[2]}(z)=ab$, $z\in{\mathbb{Z}}$ is the generator (the
element $ab$ is represented by the inversion of the first basis vector
${\mathbb{Z}}^{[2]}\subset O(2)$),

$\Phi^{[2]}|_{{\bf I}_{a\times\dot{{a}}}\times\\{0\\}}:{\bf
I}_{a\times\dot{{a}}}\subset{\bf D}$

is the standard inclusion;

$\Phi^{[2]}|_{z^{-1}{\bf I}_{a}\times{\dot{\bf I}}_{a}z}:{\bf
I}_{a\times\dot{{a}}}\subset{\bf D}$

is the conjugated inclusion, by the composition with the exterior automorphism
in the subgroup ${\bf I}_{a\times\dot{{a}}}\subset{\mathbb{Z}}/2^{[2]}$.

Let us define

$(\Phi^{[2]})^{\ast}(\tau_{[2]})=\tau_{a\times\dot{{a}}},$

where $\tau_{a\times\dot{{a}}}\in H^{2}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1))$,
$\tau_{[2]}\in H^{2}(K({\mathbb{Z}}^{[2]},1))$.

##  4 The fundamental class of the canonical covering over the self-
intersection $D$-framed standardized immersion without a defect

Let us consider a ${\mathbb{Z}}/2^{[2]}$–framed immersion $(g,\Psi,\eta_{N})$,
$g:N^{n-2k}\looparrowright{\mathbb{R}}^{n}$, and assume that this immersion is
standardized.

The mapping $\eta_{N}$ admits a reduction, the image of this mapping belongs
to the total bundle space over ${\mathbb{R}}\\!{\rm P}^{2}$ with the fibre
$K({\bf I}_{a\times\dot{{a}}},1)$. Assume that the manifold $N^{n-2k}$ is
connected. Assume that a marked point $PROJ^{-1}({\bf x}_{\infty})\in
N^{n-2k}$ is fixed. Assume that the immersion $g$ translates a marked point
into a small neighbourhood of the central point ${\bf
x}_{\infty}\in{\mathbb{R}}\\!{\rm P}^{2}$, over this point the defect is well-
defined.

The image of the fundamental class $\eta_{N,\ast}([N^{n-2k}])$ belongs to
$H_{n-2k}({\bf D};{\mathbb{Z}}/2)$. The standardization condition of implies
an existence of a natural lift of this homology class into the group
$H_{n-2k}({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}};{\mathbb{Z}})$,
because the condition C1 implies that the submanifold $N_{sing}\subset
N^{n-2k}$, which is a fibre over the point ${\bf
x}_{\infty}\in{\mathbb{R}}\\!{\rm P}^{2}$ determines the trivial fundamental
class in $H_{n-2k-2}({\bf I}_{a\times\dot{{a}}};{\mathbb{Z}})$. If $N^{n-2k}$
is a self-intersection of the immersion
$f:M^{n-k}\looparrowright{\mathbb{R}}^{n}$, then the compression condition of
of the class $\varkappa\in H^{1}(M;{\mathbb{Z}}/2)$ in the codimension $q$,
for $q\geq 2$, implies the existence of a lift of the class
$\eta_{N,\ast}([N^{n-2k}])$ into $H_{n-2k}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}})$. Moreover, the homology Euler class
$PD(e_{\ast}(\nu(g)))$ of the normal bundle also is lifted into $H_{n-4k}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}})$.

Let us consider the self-intersection manifold $L^{n-4k}$ of the immersion $g$
and its canonical double covering $\bar{L}^{n-4k}$, which is immersed into
$N^{n-2k}$. Let us assume (this assumption gives no restriction in the proof)
that the defect $N_{sing}$ is empty and, therefore, the characteristic mapping
$\eta_{N}$ has the target space $K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$.

Let us investigate the Hurewicz image of the fundamental class
$[\bar{L}^{n-4k}]$. Define a local coefficient system and prove that the image
of the fundamental class $[\bar{L}^{n-4k},pt]$ by $\eta_{N}$ determines an
element

$\displaystyle\eta_{\ast}([\bar{L}^{n-4k},pt])\in D_{n-2k}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2])\subset
H_{n-2k}^{loc}(K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2]).$
(51)

in the homology group with this local coefficient system.

Let us consider a skeleton of the space $K({\bf I}_{a\times\dot{{a}}},1)$,
which is realized as the structured group ${\bf I}_{a\times\dot{{a}}}\subset
O(2+2)$ of a bundle over the corresponding Grasmann manifold $Gr_{{\bf
I}_{a\times\dot{{a}}}}(2+2,n)$ of non-oriented $2+2=4$-plans in $n$–space.
Denote this Grasmann manifold by

$\displaystyle KK({\bf I}_{a\times\dot{{a}}})\subset K({\bf
I}_{a\times\dot{{a}}},1).$ (52)

On the space $(\ref{KK})$ a free involution

$\displaystyle\chi^{[2]}:KK({\bf I}_{a\times\dot{{a}}})\to KK({\bf
I}_{a\times\dot{{a}}}),$ (53)

acts, this involution corresponds to the automorphism $(\ref{chidd})$ and
represents by a permutation of planes of fibres (by a block-antidiagonal
matrix).

Let us define a family of spaces (52) with the prescribed symmetry group, are
parametrized over the base $S^{1}$, the generator of $S^{1}$ transforms fibres
by the involution (100). This space is a subspace (a skeleton): $K({\bf
I}_{a\times\dot{{a}}},1)\rtimes{\mathbb{R}}\\!{\rm P}^{2}\subset K({\bf
D},1)$.

Define the required space

$\displaystyle KK({\bf
I}_{a\times\dot{{a}}}\rtimes_{\chi^{[2]}}{\mathbb{R}}\\!{\rm P}^{2})$ (54)

by a quotient of the direct product $KK({\bf I}_{a\times\dot{{a}}})\times
S^{2}$ with respect to the involution $\chi^{[2]}\times(-1)$, where
$-1:S^{2}\to S^{2}$–is the antipodal involution. The following inclusion is
well-defined:

$\displaystyle KK({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}})\subset
KK({\bf I}_{a\times\dot{{a}}})\rtimes{\mathbb{R}}\\!{\rm P}^{2}.$ (55)

The involution $(\ref{chiK})$ induces the $S^{1}$-fibrewise involution

$\displaystyle\chi^{[2]}:KK({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}})\to
KK({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}),$ (56)

which is extended to the involution

$\displaystyle\chi^{[2]}:KK({\bf
I}_{a\times\dot{{a}}}\rtimes_{\chi^{[2]}}{\mathbb{R}}\\!{\rm P}^{2})\to
KK({\bf I}_{a\times\dot{{a}}}\rtimes_{\chi^{[2]}}{\mathbb{R}}\\!{\rm P}^{2})$
(57)

on $KK({\bf I}_{a\times\dot{{a}}})\rtimes{\mathbb{R}}\\!{\rm P}^{2}$, all this
extended involutions denote the same.

The universal $4$-dimensional ${\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}/2$–bundle
over the target space in (55) $KK({\bf
I}_{a\times\dot{{a}}},1)\rtimes{\mathbb{R}}\\!{\rm P}^{2}$, which is denoted
by $\tau_{a\times\dot{{a}},\rtimes}$, is well-defined. We will need mostly the
restriction of this bundle to the subspace (55). The restriction of this
bundle on the origin subspace in $(\ref{KK})$, is the Whitney sum of the two
involutive pull-back copies of the universal $2$-bundle
$\tau_{a\times\dot{{a}}}$ over the target space of $(\ref{KK})$.

Let us define ${\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}$–reduction
of the structured mapping of $L^{n-4k}$ for the standardized immersion $g$ and
the framing $\Psi_{\bar{L}}:\nu_{g}|_{\bar{L}}\equiv
k\eta_{a\times\dot{{a}}}^{\ast}(\tau_{a\times\dot{{a}}})$. In this
construction we will use condition -2 in Definition 7, using the polyhedron
$P$ described in Remark 8. We may put $k^{\prime}=6$ when $k=7$, if we like to
do the construction for all regular cobordism classes of original skew-framed
immersions $f$. More strong retruction by Theorem 5 is possible, in the case
we do the construction for a prescribed representative.

The manifold $L^{n-4k}$ admits an open submanifold $L^{n-4k}_{reg}\subset
L^{n-4k}$, which is equipped with the control mapping $L^{n-4k}\setminus
L^{n-4k}_{reg}\to P$, $dim(P)=\frac{n-k}{2}+k^{\prime}-k$, and the structured
mapping $\eta_{P}:P\to K({\bf I}_{a\times\dot{{a}}},1)$, such that the
following mapping of pairs

$(L^{n-4k},L^{n-4k}\setminus L^{n-4k}_{reg})\to(L^{n-4k},P)\to(KK({\bf
I}_{a\times\dot{{a}}}\rtimes_{\chi^{[2]}{\mathbb{R}}\\!{\rm P}^{2}}),KK({\bf
I}_{a\times\dot{{a}}}))$

is well-defined. In particular, the Hurewicz image of the fundamental class
$[L^{n-4k}]$ by the structured mapping belongs to the subspace (55).

By the construction, the normal bundle $\nu_{g}$ of the immersion $g$ over
$\bar{L}^{n-4k}$ is the Whitney sum of $k$ isomorphic $2$-dimensional blocks.
The normal bundle over $\bar{L}^{n-4k}$ inside $N^{n-2k}$ is isomorphic is the
Whitney sum of $k$ isomorphic $2$-block, analogously. Therefore the normal
bundle over $\bar{L}^{n-4k}$ is the Whitney sum of $k$ $2+2$ blocks, each
first term in a block is orthogonal to $g(N^{n-2k})$, the second term is
parallel to the tangent space of $g(N^{n-2k})$. Each term are decomposed into
line bundles, which are classified by the classes
$\varkappa_{a},\varkappa_{\dot{{a}}}$ correspondingly, the decomposition of
the $2$-block into line subbundles is globally defined, but the denotation
$\varkappa_{a},\varkappa_{\dot{{a}}}$ are well-defined up to the permutation.

The two 2-terms in a $4$-block are permuted along a path $l\subset L^{n-4k}$,
$\bar{L}^{n-4k}\to L^{n-4k}$, which corresponds to a permutation of leafs of
the self-intersection manifold. Such a path represents also an odd element in
$H_{1}(S^{1})$ by the mapping $(\ref{pbb})$.

Let us consider an arbitrary cell $\alpha$ of a regular cell-decomposition of
the manifold $N^{n-2k}$. Assume that a prescribed path $\phi_{\alpha}$, which
connects central points of a cell $\alpha$ with a marked point $pt\in
N^{n-2k}$. The restriction of the fibre bundle $\nu_{a\times\dot{{a}}}$ on
$\alpha$ is classified by the mapping
$\eta_{a\times\dot{{a}}}(\alpha,\phi_{\alpha}):\alpha\to KK({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$ into
the space (55).

A change of the path $\phi_{\alpha}$ with a fixed boundary points in a
conjugacy class with respect to the subgroup $H_{1}(S^{1})$ determines an
alternative mapping $\eta_{a\times\dot{{a}}}(\alpha,\phi_{\alpha})$, the two
mappings are related by the composition with the involution $(\ref{chiKint})$.
In a cell complex of the space $(\ref{KKin})$ with
${\mathbb{Z}}[{\mathbb{Z}}/2]$–local coefficients this changing of a path for
a chain on $\alpha$corresponds to the multiplication of the coefficient of the
chain by the variable $t$ and, simultaneously, by the changing of the
classified map to the Grasmann manifold by the involution $(\ref{chiKint})$.
Therefore, the element $(\ref{etadd16})$ in the target group is well-defined.

Let us apply this to the canonical covering $\bar{L}^{n-4k}$ globally,
$\bar{L}^{n-4k}=\nu_{g}\oplus T(\nu_{g})$. On the $g$-normal $2$-block of
$\nu(\bar{L})$, this block is parallel to the normal bundle of $\nu_{g}$,
define this reduction as above; on the second $2$-block, this block is
parallel to the tangent bundle $T(\nu_{g})|_{N^{n-2k}}$, define the
$\chi^{[2]}$-conjugated reduction. In the case a path to the marked point is
changed into the conjugated path, the both terms are transformed by
$\chi^{[2]}$ simultaneously. Therefore, the local coefficient system on
$\bar{L}$ is well-defined.

Let us prove that the image of the element $(\ref{etadd16})$ belongs to the
group $D_{n-4k}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}/2[{\mathbb{Z}}/2])$. We will do an explicit
calculation with no reference to Lemma 10 by the following geometrical
description.

The canonical covering $\bar{L}^{n-4k}$ contains a hypersurface

$\displaystyle\bar{\Sigma}^{n-4k-1}\subset\bar{L}^{n-4k},$ (58)

which is the inverce image of a pointby the projection: $\bar{L}^{n-4k}\to
K(({\bf I}_{a}\times{\dot{\bf I}}_{a})\rtimes_{\chi}{\mathbb{Z}},1)\to
K({\mathbb{Z}},1)=S^{1}$.

Let us do a modification of the manifold $\bar{L}^{n-4k}$ along the
hypersurface $\bar{\Sigma}^{n-4k-1}$ into a manifold, which is mapped into
$K({\bf I}_{a}\times{\dot{\bf I}}_{a},1)$. This modification is an equivariant
cutting of the hypersurface and a gluing the boundary with the permutation of
the order of the preimages of the canonical covering. Singular cycles with
local coefficients over two bounrady components of the hypersurface (58) are
agree. The resulting manifold represents a homology class in
$D_{n-4k}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}/2[{\mathbb{Z}}/2])=H_{n-4k}(({\bf
I}_{a}\times{\dot{\bf I}}_{a},1);{\mathbb{Z}})$. The element (51) is well-
defined.

In the case $dim(N)=n-8k$, we get $dim(N_{a\times\dot{{a}}})=m_{\sigma}=n-16k$
(see Theorem 5); when $n=126$, $k=7$ implies $m_{\sigma}=14$.

Let us consider the image of the fundamental class of the canonical covering
$\bar{L}$, $dim(\bar{L})=m_{\sigma}$, determines an element

$\displaystyle\eta_{\ast}(\bar{L}^{m_{\sigma}}_{a\times\dot{{a}}})\in
D_{m_{\sigma}}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}[{\mathbb{Z}}/2]).$ (59)

Let us prove that the element (59) satisfies an additional property, which is
called pure immersion, Definition 17 below.

By Theorem 5 the immersion $g$ in its regular cobordism class is represented
such that the cannonical covering $\bar{L}^{m_{\sigma}}_{a\times\dot{{a}}}$ of
the self-intersection manifold admits the trivial mapping (28). The Hurewicz
image 59 by strighforeward geometrical reason, using Theorem 5 –3 (using the
isomorphism (43) and the construction (51) we get the same result) is is
simplified:

$\displaystyle\eta_{\ast}(\bar{L}^{m_{\sigma}}_{a\times\dot{{a}}})\in
H_{m_{\sigma}}({\bf I}_{a\times\dot{{a}}};{\mathbb{Z}}).$ (60)

Moreover, by the Herbert immersion Theorem, the Hurewicz image 60 coincides to
the element

$\displaystyle\eta_{\ast}(N^{m_{\sigma}}_{a\times\dot{{a}}})\in
H_{m_{\sigma}}(K({\bf I}_{a\times\dot{{a}}},1))$ (61)

by the homomorphism $(\ref{deltaD2})$.

######  Lemma 11.

–1. The element $(\ref{etadlok})$ has the image by the homomorphism
$(\ref{deltaD2})$, such as the decomposition of this image with respect to the
standard base of the group $H_{m_{\sigma}}(K({\bf I}_{a}\times{\dot{\bf
I}}_{a},1))$ contains not more then the only monomial $t_{a,i}\otimes
t_{\dot{{a}},i}$, see. $(\ref{tdtdd})$,
$i=\frac{m_{\sigma}}{2}=\frac{n-16k}{2}$. The coefficient of this monomial
coincides with the Kervaire invariant, which is calculated for a
${\mathbb{Z}}/2^{[2]}$–framed immersion $(g,\Psi,\eta_{N})$.

#### Proof of Lemma 11

Let us proof the statement for the case $m_{\sigma}=14$. Let us consider the
manifold $N_{a\times\dot{{a}}}^{14}$, which we re-denote in the proof by
$N^{14}$ for sort. This manifold is equipped with the mapping $\eta:N^{14}\to
K({\bf I}_{a\times\dot{{a}}},1)$. Let us consider all the collection of
characteristic ${\mathbb{Z}}/2$-numbers for the mapping $\eta$, which is
induced from the universal classes.

Let us consider the manifold $\bar{L}^{n-16k}$ and its fundamental class
$(\ref{etadlok})$. By arguments from Herbert theorem the following two classes
$-[N_{a\times\dot{{a}}}^{14}]$ (61) and (59) coincides in the group
$D_{m_{\sigma}}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}/2[{\mathbb{Z}}/2])$. Analogously, the
projection of the opposite class (but modulo $2$ coefficients the opposition
is not required) the classes (61) coincides to $(\ref{etadlok})$ in
$H_{m_{\sigma}}(K({\bf I}_{a\times\dot{{a}}},1))$.

Let us estimate the fundamental class $[N^{14}]$ and prove that this class is
pure (see definition 17) below. Because $N^{14}$ is oriented, among
characteristic numbers: $\varkappa_{a}\varkappa^{13}_{\dot{{a}}}$,
$\varkappa_{a}^{3}\varkappa^{11}_{\dot{{a}}}$,
$\varkappa_{a}^{5}\varkappa^{9}_{\dot{{a}}}$,
$\varkappa_{a}^{7}\varkappa^{7}_{\dot{{a}}}$,
$\varkappa_{a}^{9}\varkappa^{5}_{\dot{{a}}}$,
$\varkappa_{a}^{3}\varkappa^{11}_{\dot{{a}}}$,
$\varkappa_{a}\varkappa^{13}_{\dot{{a}}}$. the only number
$\varkappa_{a}^{7}\varkappa^{7}_{\dot{{a}}}$ could be nontrivial.

Let us prove that the characteristic number
$\varkappa_{a}\varkappa^{13}_{\dot{{a}}}$ is trivial. Let us consider the
manifold $K^{3}\subset N^{14}$, which is dual to the characteristic class
$\varkappa_{a}\varkappa_{\dot{{a}}}^{10}$. The normal bundle $\nu_{N}$ over
the manifold $N^{14}$ is isomorphic to the Whitney sum $k\varkappa_{a}\oplus
k\varkappa_{\dot{{a}}}$, where the positive integer $k$ satisfies the
equation: $k\equiv 0\pmod{8}$. The restriction of the bundle $\nu_{N}$ on the
submanifold $K^{3}$ is trivial. Therefore, the normal bundle $\nu_{K}$ of the
manifold $K^{3}$ is stably isomorphic to the bundle $\varkappa_{a}\oplus
2\varkappa_{\dot{{a}}}$.

Because the characteristic class $w_{2}(K^{3})$ is trivial we get:
$\langle\varkappa^{3}_{\dot{{a}}};[K^{3}]\rangle=0$. But,
$\langle\varkappa^{3}_{\dot{{a}}};[K^{3}]\rangle=\langle\varkappa_{a}\varkappa^{13}_{\dot{{a}}};[N^{14}]\rangle$.
This proves that the characteristic number
$\varkappa_{a}\varkappa^{13}_{\dot{{a}}}$ is trivial. Analogously, the
characteristic numbers $\varkappa_{a}^{5}\varkappa^{9}_{\dot{{a}}}$,
$\varkappa_{a}^{9}\varkappa^{5}_{\dot{{a}}}$,
$\varkappa_{a}\varkappa^{13}_{\dot{{a}}}$ are trivial.

Let us prove that the characteristic number
$\varkappa_{a}^{3}\varkappa^{11}_{\dot{{a}}}$ is trivial. Let us consider the
submanifold $K^{7}\subset N^{14}$, which is dual to the characteristic class
$\varkappa_{a}^{2}\varkappa_{\dot{{a}}}^{5}$. The normal bundle $\nu_{K}$ of
the manifold $K^{6}$ is stably isomorphic to the Whitney sum
$2\varkappa_{a}\oplus 5\varkappa_{\dot{{a}}}$. Because $w_{5}(K^{7})=0$, the
characteristic class $\varkappa_{\dot{{a}}}^{5}$ is trivial. In particular,
$\langle\varkappa_{a}\varkappa_{\dot{{a}}}^{6};[K^{7}]\rangle=0$. The
following equation is satisfied:
$\langle\varkappa_{a}\varkappa_{\dot{{a}}}^{6};[K^{7}]\rangle=\langle\varkappa_{a}^{3}\varkappa_{\dot{{a}}}^{11};[N^{14}]\rangle$.

It is proved that the characteristic number
$\varkappa_{a}^{3}\varkappa^{11}_{\dot{{a}}}$ is trivial. Analogously, the
characteristic number $\varkappa_{a}^{11}\varkappa^{3}_{\dot{{a}}}$ is
trivial.

It is sufficiently to note, that the characteristic number
$\langle\varkappa_{a}^{7}\varkappa_{\dot{{a}}}^{7};[N^{14}]\rangle$ coincides
with the characteristic number (6) in the lemma. Statement 1 is proved. Lemma
11 is proved. ∎

######  Definition 12.

Let $(g,\eta_{N},\Psi)$ be the standardized ${\bf D}$-framed immersion in the
codimension $2k$.

Let us say that this standardized immersion is negligible, if its fundamental
class $\pmod{2}$ is in the subgroup

$\displaystyle\eta_{\ast}(N^{n-2k}_{a\times\dot{{a}}})\in D_{n-2k}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}/2[{\mathbb{Z}}/2])$ (62)

and is trivial.

Let us say that this standardized immersion is pure, if its fundamental class
$\pmod{2}$ satisfies the condition of Lemma 11, i.e. the Hurewicz image of the
fundamental class is in the group $\in D_{n-2k}^{loc}({\bf
I}_{a\times\dot{{a}}};{\mathbb{Z}}/2[{\mathbb{Z}}/2])$ and contains no
monomials $t_{a,\frac{n-2k}{2}+i}\otimes t_{\dot{{a}},\frac{n-2k}{2}-i}$,
$i\in\\{\pm 1;\pm 2;\pm 3;\pm 4;\pm 5;\pm 6;\pm 7\\}$, but, probably, contains
the monomial $t_{a,\frac{n-2k}{2}}\otimes t_{\dot{{a}},\frac{n-2k}{2}}$.

We may reformulate results by Lemma 11 using Definition 17.

######  Proposition 13.

Assume an element $[(g,\eta_{N},\Psi)]\in Imm^{D}(n-2k,2k)$ is in the image of
the vertical homomorphism in the diagram (10). Then this element is
represented by a standardized immersion. Assume we have dimensions as in
Theorem 5, then the element is pure in the sence of Definition 17.

##  5 ${\bf H}_{a\times\dot{{a}}}$–structure on ${\mathbb{Z}}/2^{[3]}$–framed
immersion
${\bf J}_{b}\times{\dot{\bf J}}_{b}$–structure on
${\mathbb{Z}}/2^{[4]}$–framed immersion

The group ${\bf I}_{b}$ is defined as the cyclic subgroup of the order $4$ in
the dihedral group: ${\bf I}_{b}\subset{\mathbb{Z}}/2^{[2]}$.

Let us define an analogous subgroup

$\displaystyle i_{{\bf J}_{b}\times{\dot{\bf J}}_{b}}:{\bf
J}_{b}\times{\dot{\bf J}}_{b}\subset{\mathbb{Z}}/2^{[4]},$ (63)

which is isomorphic to the Cartesian product of the two cyclic groups of the
order $4$.

The group ${\mathbb{Z}}/2^{[4]}$ (the monodromy group for the $3$-uple
iterated self-intersection of a skew-framed immersion) is defined using the
base $({\rm e}_{1},\dots,{\rm e}_{8})$ in the Euclidean space
${\mathbb{R}}^{8}$. Let us denote the generators of the subgroup ${\bf
J}_{b}\times{\dot{\bf J}}_{b}$ by $b$, $\dot{{b}}$ correspondingly. Let us
describe transformations in ${\mathbb{Z}}/2^{[4]}$, which corresponds to each
generator.

Let us consider an orthogonal base $\\{{\bf f}_{1},\dots,{\bf f}_{8}\\}$,
which is determined by the formulas: ${\bf f}_{2i-1}=\frac{{\rm e}_{2i-1}+{\rm
e}_{2i}}{\sqrt{2}}$, ${\bf f}_{2i}=\frac{{\rm e}_{2i-1}-{\rm
e}_{2i}}{\sqrt{2}}$, $i=1,\dots,4$.

The transformation group ${\bf J}_{b}\times{\dot{\bf J}}_{b}$ have invariant
pairwise orthogonal $(2,2,2,2)$-dimensional subspaces, which we denote by
${\mathbb{R}}^{2}_{b,+}$, ${\mathbb{R}}^{2}_{b,-}$,
${\mathbb{R}}^{2}_{\dot{{b}},+}$, ${\mathbb{R}}^{2}_{\dot{{b}},-}$
correspondingly.

The subspace ${\mathbb{R}}^{2}_{b,+}$ is generated by linear combinations of
pairs of vectors: $Lin({\bf f}_{1}+{\bf f}_{5},{\bf f}_{3}+{\bf f}_{7})$. The
subspace ${\mathbb{R}}^{2}_{b,-}$ is generated by linear combinations
$Lin({\bf f}_{1}-{\bf f}_{5},{\bf f}_{3}-{\bf f}_{7})$. The subspace
${\mathbb{R}}^{2}_{\dot{{b}},+}$ is generated by linear combinations $Lin({\bf
f}_{2}+{\bf f}_{4},{\bf f}_{6}+{\bf f}_{8})$. The subspace
${\mathbb{R}}^{2}_{\dot{{b}},-}$ is generated by linear combinations $Lin({\bf
f}_{2}-{\bf f}_{4},{\bf f}_{6}-{\bf f}_{8})$.

It is convenient to pass to a new basis:

$\displaystyle\frac{{\bf f}_{1}+{\bf
f}_{5}}{\sqrt{2}}={\mathfrak{h}}_{1,+},\frac{{\bf f}_{1}-{\bf
f}_{5}}{\sqrt{2}}={\mathfrak{h}}_{1,-},\frac{{\bf f}_{3}+{\bf
f}_{7}}{\sqrt{2}}={\mathfrak{h}}_{2,+},\frac{{\bf f}_{3}-{\bf
f}_{7}}{\sqrt{2}}={\mathfrak{h}}_{2,-},$ (64) $\displaystyle\frac{{\bf
f}_{2}+{\bf f}_{4}}{\sqrt{2}}={\bf\dot{h}}_{1,+},\frac{{\bf f}_{2}-{\bf
f}_{4}}{\sqrt{2}}={\bf\dot{h}}_{1,-},\frac{{\bf f}_{6}+{\bf
f}_{8}}{\sqrt{2}}={\bf\dot{h}}_{2,+},\frac{{\bf f}_{6}-{\bf
f}_{8}}{\sqrt{2}}={\bf\dot{h}}_{2,-}.$ (65)

In the denotations above linear combinations
$Lin({\mathfrak{h}}_{1,+},{\mathfrak{h}}_{2,+})$,
$Lin({\mathfrak{h}}_{1,-},{\mathfrak{h}}_{2,-})$ determines linear subspaces
${\mathbb{R}}^{2}_{b,+}$, ${\mathbb{R}}^{2}_{b,-}$ with prescribed basis
correspondingly.

Pairs of vectors $({\bf\dot{h}}_{1,+},{\bf\dot{h}}_{2,+})$,
$({\bf\dot{h}}_{1,-},{\bf\dot{h}}_{2,-})$ determines basis in the the
subspaces $Lin({\bf\dot{h}}_{1,+},{\bf\dot{h}}_{2,+})$,
$Lin({\bf\dot{h}}_{1,-},{\bf\dot{h}}_{2,-})$, which is denoted by
${\mathbb{R}}^{2}_{\dot{{b}},+}$, ${\mathbb{R}}^{2}_{\dot{{b}},-}$
correspondingly. Linear combinations
$Lin({\bf\dot{h}}_{1,+},{\bf\dot{h}}_{2,+})$,
$Lin({\bf\dot{h}}_{1,-},{\bf\dot{h}}_{2,-})$ determines basis in the subspaces
${\mathbb{R}}^{2}_{\dot{{b}},+}$, ${\mathbb{R}}^{2}_{\dot{{b}},-}$
correspondingly. The generator $b$ is represented by the rotation on the angle
$\frac{\pi}{2}$ in each plane ${\mathbb{R}}^{2}_{b,+}$,
${\mathbb{R}}^{2}_{b,-}$ and by the central symmetry in the planes
${\mathbb{R}}^{2}_{\dot{{b}},-}$, (this symmetry is commuted with the
transformation on the element $\dot{{b}}$, which will be described below). The
generator $\dot{{b}}$ is represented by the rotation on the angle
$\frac{\pi}{2}$ in each plane ${\mathbb{R}}^{2}_{\dot{{b}},+}$
${\mathbb{R}}^{2}_{\dot{{b}},-}$ and by the central symmetry in the plane
${\mathbb{R}}^{2}_{b,-}$, which is commuted with the $b$ transformation,
described above. The subgroup (63) is defined.

Let us denote the subgroup $i_{{\bf H}_{a\times\dot{{a}}},{\bf
J}_{b}\times{\dot{\bf J}}_{b}}:{\bf H}_{a\times\dot{{a}}}\subset{\bf
J}_{b}\times{\dot{\bf J}}_{b}$, which is the product of the diagonal subgroup,
which we denote by ${\bf I}_{b}\subset{\bf J}_{b}\times{\dot{\bf J}}_{b}$ with
the elementary subgroup ${\mathbb{Z}}/2$ of the second factor, which is
denoted by ${\dot{\bf J}}_{d}\subset{\dot{\bf J}}_{b}$. The subgroup ${\bf
H}_{a\times\dot{{a}}}$ coincides with the preimage of the subgroup
${\mathbb{Z}}/2\subset{\mathbb{Z}}/4$ by the homomorphism

$\displaystyle\omega^{[4]}:{\bf J}_{b}\times{\dot{\bf
J}}_{b}\to{\mathbb{Z}}/4,$ (66)

which is defined by the formula $(x\times y)\mapsto xy$.

Define the subgroup $i_{{\bf I}_{a\times\dot{{a}}},{\bf
H}_{a\times\dot{{a}}}}:{\bf I}_{a\times\dot{{a}}}\subset{\bf
H}_{a\times\dot{{a}}}$ as the kernel of the epimorphism

$\displaystyle\omega^{[3]}:{\bf H}_{a\times\dot{{a}}}\to{\mathbb{Z}}/2,$ (67)

which is defined by the formula: $(x\times y)\mapsto x$ using generators $x,y$
of the group ${\bf J}_{b}\times{\dot{\bf J}}_{b}$.

Let us consider the diagonal subgroup
${\mathbb{Z}}/2^{[3]}\subset{\mathbb{Z}}/2^{[3]}\times{\mathbb{Z}}/2^{[3]}\subset{\mathbb{Z}}/2^{[4]}$,
which is generated by the invariant transformations in the direct sum of the
subspaces $diag({\mathbb{R}}^{2}_{b,+},{\mathbb{R}}^{2}_{\dot{{b}},+})$,
$diag({\mathbb{R}}^{2}_{b,-},{\mathbb{R}}^{2}_{\dot{{b}},-})$. This group is a
subgroup of transformations of the base vectors, which are defined by the
formulas:
${\mathfrak{h}}_{1,+}+{\bf\dot{h}}_{1,+},{\mathfrak{h}}_{2,+}+{\bf\dot{h}}_{2,+},{\mathfrak{h}}_{1,-}+{\bf\dot{h}}_{2,-},{\mathfrak{h}}_{2,-}+{\bf\dot{h}}_{2,-}$.
This collection of the vectors determines the standard base in the space
$diag({\mathbb{R}}^{2}_{b,+},{\mathbb{R}}^{2}_{\dot{{b}},+})\oplus
diag({\mathbb{R}}^{2}_{b,-},{\mathbb{R}}^{2}_{\dot{{b}},-})$. The complement
is defined by the formula:
$antidiag({\mathbb{R}}^{2}_{b,+},{\mathbb{R}}^{2}_{\dot{{b}},+})\oplus
antidiag({\mathbb{R}}^{2}_{b,-},{\mathbb{R}}^{2}_{\dot{{b}},-})$. In this
space the standard base is analogously defined.

Let us define a subgroup ${\mathbb{Z}}/2^{[2]}\subset{\mathbb{Z}}/2^{[3]}$ as
the subgroup, which is generated by transformations of unite vectors, which
are parallel to the vectors
${\mathfrak{h}}_{1,+}+{\bf\dot{h}}_{1,+}+{\mathfrak{h}}_{2,+}+{\bf\dot{h}}_{2,+}$,
${\mathfrak{h}}_{1,-}+{\bf\dot{h}}_{2,-}+{\mathfrak{h}}_{2,-}+{\bf\dot{h}}_{2,-}$.
It is easy to see, that the first vector in the collection is parallel to the
vector ${\rm e}_{1}+{\rm e}_{3}+{\rm e}_{5}+{\rm e}_{7}$. The second vetor is
parallel to the vector ${\rm e}_{2}+{\rm e}_{4}+{\rm e}_{5}+{\rm e}_{6}$.

The inclusion $i_{{\bf H}_{a\times\dot{{a}}}}:{\bf
H}_{a\times\dot{{a}}}\subset{\mathbb{Z}}/2^{[3]}$, which is corresponded to
the inclusion (63) is well defined, such that the following diagram is
commutative:

$\displaystyle\begin{array}[]{ccc}\qquad{\bf
I}_{a\times\dot{{a}}}&\stackrel{{\scriptstyle
i_{a\times\dot{{a}}}}}{{\longrightarrow}}&\qquad{\mathbb{Z}}/2^{[2]}\\\
i_{a\times\dot{{a}},{\bf
H}_{a\times\dot{{a}}}}\downarrow&&i^{[3]}\downarrow\\\ \qquad{\bf
H}_{a\times\dot{{a}}}&\stackrel{{\scriptstyle i_{{\bf
H}_{a\times\dot{{a}}}}}}{{\longrightarrow}}&\qquad{\mathbb{Z}}/2^{[3]}\\\
i_{{\bf H}_{a\times\dot{{a}}},{\bf J}_{b}\times{\dot{\bf
J}}_{b}}\downarrow&&i^{[4]}\downarrow\\\ \qquad{\bf J}_{b}\times{\dot{\bf
J}}_{b}&\stackrel{{\scriptstyle i_{{\bf J}_{b}\times{\dot{\bf
J}}_{b}}}}{{\longrightarrow}}&\qquad{\mathbb{Z}}/2^{[4]}.\\\ \end{array}$ (73)

Let us define automorphisms of the order 2:

$\displaystyle\chi^{[3]}:{\bf H}_{a\times\dot{{a}}}\to{\bf
H}_{a\times\dot{{a}}},$ (74) $\displaystyle\chi^{[4]}:{\bf
J}_{b}\times{\dot{\bf J}}_{b}\to{\bf J}_{b}\times{\dot{\bf J}}_{b},$ (75)

and

$\displaystyle\chi^{[3]}:{\mathbb{Z}}/2^{[3]}\to{\mathbb{Z}}/2^{[3]},$ (76)
$\displaystyle\chi^{[4]}:{\mathbb{Z}}/2^{[4]}\to{\mathbb{Z}}/2^{[4]},$ (77)

which are marked with a loss of the strictness.

Define the automorphism (76) by the permutation of the base vectors in each
direct factor
$diag({\mathbb{R}}^{2}_{b,+},{\mathbb{R}}^{2}_{\dot{{b}},+})\oplus
diag({\mathbb{R}}^{2}_{b,-},{\mathbb{R}}^{2}_{\dot{{b}},-})$ with indexes $b$
and $\dot{{b}}$. Define the automorphism $(\ref{chi3ZA})$ as the automorphism,
which restricted on the diagonal subgroup $diag({\bf J}_{b},{\dot{\bf
J}}_{b})={\bf I}_{b}\subset{\bf H}_{a\times\dot{{a}}}$is the identity, and the
restriction on the subgroup ${\bf I}_{a\times\dot{{a}}}\subset{\bf
H}_{a\times\dot{{a}}}$ coincides with the automorphism $\chi^{[2]}$.
Evidently, the definition is correct. Define the automorphism $(\ref{chi4ZA})$
in the standard basis in the subspaces
$diag({\mathbb{R}}^{2}_{b,+},{\mathbb{R}}^{2}_{\dot{{b}},+})\oplus
diag({\mathbb{R}}^{2}_{b,-},{\mathbb{R}}^{2}_{\dot{{b}},-})$,
$antidiag({\mathbb{R}}^{2}_{b,+},{\mathbb{R}}^{2}_{\dot{{b}},+})\oplus
antidiag({\mathbb{R}}^{2}_{b,-},{\mathbb{R}}^{2}_{\dot{{b}},-})$ as above.

The following triple of ${\mathbb{Z}}$-extensions of the group ${\bf
J}_{b}\times{\dot{\bf J}}_{b}$, defined below, is required.

The group ${\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}}$ was
defined above by the formula $(\ref{chi2Z})$. Analogously, the groups

$\displaystyle{\bf
H}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[3]}}{\mathbb{Z}},$ (78)
$\displaystyle({\bf J}_{b}\times{\dot{\bf
J}}_{b})\operatorname{\rtimes}_{\chi^{[4]}}{\mathbb{Z}},$ (79)

are defined as semi-direct products of the corresponding groups, equipped by
automorphisms, with the group ${\mathbb{Z}}$.

The classifying space $K({\bf
H}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[3]}}{\mathbb{Z}},1)$ is a
skew-product of the circle $S^{1}$ with the space $K({\bf
H}_{a\times\dot{{a}}},1)$, moreover, the mapping $K({\bf
H}_{a\times\dot{{a}}},1)\to K({\bf H}_{a\times\dot{{a}}},1)$, which
corresponds to a shift of the cyclic covering over $K({\bf
H}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[3]}}{\mathbb{Z}},1)$, is
defined by the involution, which is induced by the automorphism $\chi^{[3]}$.
The definition of the group $(\ref{chi4Z})$ is totally analogous.

Let us define another two ${\mathbb{Z}}$-extensions (Laurent extensions),
which is conjugated to each other by the automorphism $\chi^{[4]}$. Let us
denote this pair of extensions on the subgroup ${\bf J}_{b}\times{\dot{\bf
J}}_{b}$, on this subgroup the extensions are commuted. Let us denote the
extensions by

$\rtimes_{\mu_{b}^{(4)}},\quad\rtimes_{\mu_{\dot{{b}}}^{(4)}}.$

The Laurent extension on the factor ${\bf J}_{b}$ is determined by the
authomorphism $\mu_{b}^{(4)}:{\bf J}_{b}\to{\bf J}_{b}$, which inverse the
generator. The Laurent expansion on the factor ${\dot{\bf J}}_{b}$ is
determined by the automorphism $\mu_{\dot{{b}}}^{(4)}:{\dot{\bf
J}}_{b}\to{\dot{\bf J}}_{b}$, which inverse the generator. We have defined the
double Laurent extension group, which is denoted by

$\displaystyle{\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu^{(4)}_{b\times\dot{{b}}}}{\mathbb{Z}}\times{\mathbb{Z}}.$
(80)

Let us extend the extension constructed above to the extension

$\displaystyle{\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}{\mathbb{Z}}\times{\mathbb{Z}}\rtimes_{\chi^{[4]}}{\mathbb{Z}}.$
(81)

The Laurent extensions (81), and (79) (see below) are naturally represented
into ${\mathbb{Z}}/2^{[4]}$. The diagram (73) is a subdiagram in the following
diagram:

$\displaystyle\begin{array}[]{ccc}\qquad{\bf
I}_{a\times\dot{{a}}}\rtimes_{\chi^{[2]}}{\mathbb{Z}}&\stackrel{{\scriptstyle
i_{a\times\dot{{a}}}}}{{\longrightarrow}}&\qquad{\mathbb{Z}}/2^{[2]}\\\
i_{a\times\dot{{a}},{\bf
H}_{a\times\dot{{a}}}}\downarrow&&i^{[3]}\downarrow\\\ \qquad{\bf
H}_{a\times\dot{{a}}}\rtimes_{\mu^{(3)},\chi^{[3]}}{\mathbb{Z}}\rtimes{\mathbb{Z}}&\stackrel{{\scriptstyle
i_{{\bf
H}_{a\times\dot{{a}}}}}}{{\longrightarrow}}&\qquad{\mathbb{Z}}/2^{[3]}\\\
i_{{\bf H}_{a\times\dot{{a}}},{\bf J}_{b}\times{\dot{\bf
J}}_{b}}\downarrow&&i^{[4]}\downarrow\\\ \qquad{\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu^{(4)}_{b\times\dot{{b}}},\chi^{[4]}}({\mathbb{Z}}\times{\mathbb{Z}})\rtimes{\mathbb{Z}}&\stackrel{{\scriptstyle
i_{{\bf J}_{b}\times{\dot{\bf
J}}_{b}}}}{{\longrightarrow}}&\qquad{\mathbb{Z}}/2^{[4]}.\\\ \end{array}$ (87)

The middle row of the diagram contains an intermediate extension, which can be
rewrite more precisely in the following form:

$\displaystyle{\bf
H}_{a\times\dot{{a}}}\rtimes_{\mu^{(3)}}{\mathbb{Z}}\rtimes_{\chi^{[3]}}{\mathbb{Z}}.$
(88)

The extension (88) contains a sub extension (78), the both groups are
represented into ${\mathbb{Z}}/2^{[3]}$ in an agreed way, moreover, this
representation agrees to the representation (81), as a subrepresentation on a
subgroup of the index $2$ and is required into the following more precisely
form:

$\displaystyle\Phi^{[4]}:({\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}{\mathbb{Z}}\times{\mathbb{Z}})\operatorname{\rtimes}_{\chi^{[4]}}{\mathbb{Z}}\to{\mathbb{Z}}/2^{[4]},$
(89)

where the generator of the factor ${\mathbb{Z}}$ is represented in
${\mathbb{Z}}/2^{[4]}$ by the transformation $\chi^{[4]}$, which is defined by
the formula $(\ref{chi4ZA})$.

Corresponding ${\mathbb{Z}}/2$–reductions of extensions (88), (81) denote by

$\displaystyle{\bf
H}_{a\times\dot{{a}}}\rtimes_{\mu_{a}^{(3)}}{\mathbb{Z}}/2\rtimes_{\chi^{[3]}}{\mathbb{Z}},$
(90) $\displaystyle{\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}{\mathbb{Z}}/2\times{\mathbb{Z}}/2\rtimes_{\chi^{[4]}}{\mathbb{Z}}.$
(91)

The ${\mathbb{Z}}/2$–reductions are required, because integer extensions (81)
is not sufficient, a parametrization of the structured group ${\bf
J}_{b}\times{\dot{\bf J}}_{b}$ is not over $S^{1}$ (over the
M$\ddot{\rm{o}}$ebius band), but also over the projective plane
${\mathbb{R}}\\!{\rm P}^{2}$. For the reduction (91) this (minimal)
parametrization is assumed.

The following ${\mathbb{C}}\vee{\mathbb{C}}$-fibred ${\mathbb{Z}}/4$-line
bundle

$\displaystyle\beta_{b\times\dot{{b}}}.$ (92)

on the classifying space of the group (81) is denoted by (92), using the
authomorphism $\mu_{b\times\dot{{b}}}^{(4)}$.

This authomorphism inverses fibres over the components
$\rtimes_{\mu_{b}^{(4)}}$, $\rtimes_{\mu_{\dot{{b}}}^{(4)}}$ by the hermitian
conjugation. Below in Section 7 this bundle will be used to determine the
corresponding local coefficients system, with inversions of cyclic factors
(TypeII).

By the commutator with the generator $\rtimes_{\chi}$ the generator $b\in{\bf
J}_{b}$ of the subgroup (81) is transformed into the generator
$\dot{{b}}\in{\dot{\bf J}}_{b}$; the complex conjugation over $\mu_{b}$ is
transformed into the complex conjugation by $\mu_{\dot{{b}}}$. The generators
$b$, $\dot{{b}}$ correspond to the action on the first (second) factor by the
complex unite ${\bf i}$ in the covering over the classifying space.

Let us consider the homology group $H_{i}(K(({\bf J}_{b}\times{\dot{\bf
J}}_{b}),1);{\mathbb{Z}})$. In particular, for an odd $i$ the group contains
the fundamental classes of the manifolds

$S^{i}/{\bf i}\times pt\subset S^{i}/{\bf i}\times S^{i}/{\bf i}\subset K({\bf
J}_{b},1)\times K({\dot{\bf J}}_{b},1),$ $pt\times S^{i}/{\bf i}\subset
S^{i}/{\bf i}\times S^{i}/{\bf i}\subset K({\bf J}_{b},1)\times K({\dot{\bf
J}}_{b},1),$

which are denoted by

$\displaystyle t_{b,i},\in H_{i}(K({\bf J}_{b},1));\quad
t_{\dot{{b}},i^{\prime}}\in H_{i^{\prime}}(K({\dot{\bf J}}_{b},1)).$ (93)

A complete description of the group $H_{i}(K(({\bf J}_{b}\times{\dot{\bf
J}}_{b}),1);{\mathbb{Z}})$ is possible using the Kunneth formula, as in the
case (33). A description of the standard base of the group $H_{i}(K({\bf
J}_{b\times\dot{{b}}},1);{\mathbb{Z}})$ is described by:

$\displaystyle\begin{array}[]{c}0\to\bigoplus_{i_{1}+i_{2}=i}H_{i_{1}}(K({\bf
J}_{b},1);{\mathbb{Z}})\otimes H_{i_{2}}(K({\dot{\bf
J}}_{b},1);{\mathbb{Z}})\longrightarrow\\\ \longrightarrow H_{i}(K({\bf
J}_{b\times\dot{{b}}},1);{\mathbb{Z}})\longrightarrow\\\
\longrightarrow\bigoplus_{i_{1}+i_{2}=i-1}Tor^{{\mathbb{Z}}}(H_{i_{1}}(K({\bf
J}_{b},1);{\mathbb{Z}}),H_{i_{2}}(K({\dot{\bf J}}_{b},1);{\mathbb{Z}})\to
0.\\\ \end{array}$ (97)

Let us define the following local integer homology groups

$\displaystyle H_{i}^{loc}(K(({\bf J}_{b}\times{\dot{\bf
J}}_{b})\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}({\mathbb{Z}}\times{\mathbb{Z}}),1);{\mathbb{Z}}).$
(98)

which was claimed as homology with interiore system coefficients. Local cycles
in (98) are represented by oriented chains of the classifying space of the
group (80) equipped with local pathes to a marked point. An analogous more
simple system of interiore local coefficients for the single generator was
investigated in [A-P1].

Type I. The local ${\mathbb{Z}}\times{\mathbb{Z}}$-coefficients system
$\mu_{b\times\dot{{b}};I}^{(4)}$ acts along the generators $\mu_{b}$,
$\mu_{\dot{{b}}}$ by preserving orientation of local chains. In particular,
the generators of the local system acts (along paths) to global cycle as
following: the generator $t_{b,i}$ for $i\equiv 1\pmod{4}$ is changed into the
opposite along the path $\mu_{b}$ and is preserved for $i\equiv 1\pmod{4}$,
the generator $t_{b,i}$ is preserved along the path $\mu_{\dot{{b}}}$; the
generators $t_{\dot{{b}},i^{\prime}}$ for $i^{\prime}\equiv 1\pmod{4}$ is
changed into the opposite along the path $\mu_{\dot{{b}}}$ and is preserved
for $i^{\prime}\equiv 3\pmod{4}$; the generator $t_{\dot{{b}},i^{\prime}}$
along the path $\mu_{b}$ is preserved. Because the homology group is described
using generators $t_{b,i}$, $t_{\dot{{b}},i^{\prime}}$ by (97), a global
action of the local system is well-defined.

TypeIIb The local ${\mathbb{Z}}\times{\mathbb{Z}}$-coefficients system
$\mu_{b\times\dot{{b}};IIb}^{(4)}$ is determined by the same monodromy of
cycles as TypeI. The difference is the following: the orientation of chanes is
changed along the path $\mu_{b}$ and is preserved alomg the path
$\mu_{\dot{{b}}}$. The monodromy of the system is described using generators
as following: $t_{b,i}$, $i\equiv 1\pmod{4}$ keeps the sign along $\mu_{b}$
and changes the sign for $i\equiv 3\pmod{4}$ along $\mu_{b}$, the generator
keeps the sign along $\mu_{\dot{{b}}}$; the generators
$t_{\dot{{b}},i^{\prime}}$, ${}^{\prime}i\equiv 1\pmod{4}$ changes the sign
along $\mu_{\dot{{b}}}$ and keeps the sign for $i\equiv 3\pmod{4}$ along
$\mu_{\dot{{b}}}$, the generator keeps the sign along $\mu_{b}$.

TypeII$\dot{{b}}$ This local coefficient system is conjugated to the system of
the type TypeIIb.

The orientation of chanes is preserved along the path $\mu_{b}$ and is changed
alomg the path $\mu_{\dot{{b}}}$. The generators of the local system acts
(along paths) as follows: the generators $t_{b,i}$, $i\equiv 1\pmod{4}$
changes the sign along $\mu_{b}$ and keeps the sign for $i\equiv 3\pmod{4}$
along $\mu_{b}$, the generator keeps the sign along $\mu_{\dot{{b}}}$; the
generators $t_{\dot{{b}},i^{\prime}}$, ${}^{\prime}i\equiv 1\pmod{4}$ keeps
the sign along $\mu_{\dot{{b}}}$ and changes the sign for $i\equiv 3\pmod{4}$
along $\mu_{\dot{{b}}}$, the generator keeps the sign along $\mu_{b}$.

Local coefficients system TypeI, TypeII are used only.

Let us consider a skeleton of the space $B({\bf
J}_{b})\rtimes_{\mu_{b}}S^{1}\times B({\bf
J}_{\dot{{b}}})\rtimes_{\mu_{\dot{{b}}}S^{1}}$, which is realized by means of
the structured group ${\bf J}_{b\times\dot{{b}}}\subset O(8+8)$, using 73, of
a bundle over the corresponding Grasmann manifold $Gr_{{\bf
J}_{b\times\dot{{b}}}}(8+8,n)$ of non-oriented $8+8=16$-plans in $n$–space.
Denote this Grasmann manifold (analogously to (52) by

$\displaystyle KK({\bf J}_{b\times\dot{{b}}})\subset K({\bf
J}_{b\times\dot{{b}}},1)\rtimes_{\mu_{b\times\dot{{b}}}}(S^{1}\times S^{1}).$
(99)

On the space $(\ref{KK4})$ a free involution

$\displaystyle\chi^{[4]}:KK({\bf J}_{b\times\dot{{b}}})\to KK({\bf
J}_{b\times\dot{{b}}}),$ (100)

acts, this involution corresponds to the automorphism $(\ref{chi3A})$ and
represents by a permutation of planes of fibres (by a block-antidiagonal
matrix).

Take a market point $pt$ in (99) and take the image $\chi^{[4]}(pt)=pt^{t}$ of
this point by (100), take a path $\lambda$, which joins $pt$ and
$\chi^{[4]}(pt)$. Using the path $\lambda$ a conjugated pair of local systems
is defined:

$\displaystyle\chi^{[4]}:\mu_{b\times\dot{{b}}}^{loc}\mapsto\mu_{\dot{{b}}\times
b}^{loc}.$ (101)

The left system in the pair admits the corresponding homology group (98), the
right system admits the conjugated homology group:

$\displaystyle H_{i}^{loc}(K(({\dot{\bf J}}_{b}\times{\bf
J}_{b})\rtimes_{\mu_{\dot{{b}}\times
b}^{(4)}}({\mathbb{Z}}\times{\mathbb{Z}}),1);{\mathbb{Z}}).$ (102)

A pair of conjugated local system is well defined up to an equivalent
relation. Let us denote integer ${\mathbb{Z}}[{\mathbb{Z}}/2]$-homology groups
with the universal local system, when a multiplication by the generator
$t\in{\mathbb{Z}}/2$ maps a local cycle of the local system (attached to $pt$)
into the corresponding local cycle of the conjugated local system (attached to
$pt^{t}$).

Then let us define the homology groups with the $3$-uple local coefficient
system over the module ${\mathbb{Z}}[{\mathbb{Z}}/2]$, using the automorphism
(101), which changes the factors of the representation by (75) (analogously to
(36)). The local coefficients system, associated with $\chi^{[4]}$, is
referred as exterior local coefficients.

Take a skeleton

$\displaystyle KK({\bf J}_{b\times\dot{{b}}})\rtimes_{\chi^{[4]}}S^{1}\subset
K({\bf
J}_{b\times\dot{{b}}},1)\rtimes_{\mu_{b\times\dot{{b}}};\chi^{[4]}}(S^{1}\times
S^{1})\rtimes S^{1}.$ (103)

equipped with the representation into the Grasmann manifold. The path
$\lambda$ has to be a path in (103), with the projection onto the generator of
the exteriore factor of the skew-product.

The ${\mathbb{Z}}[{\mathbb{Z}}/2]$-homology groups with $3$-uple system are
defined by the formulas analogously to $(\ref{Z2lok})$, $(\ref{Zlok})$,
$(\ref{DZlok})$; for example:

$\displaystyle\begin{array}[]{c}D_{i}^{loc}({\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}({\mathbb{Z}}\times{\mathbb{Z}});{\mathbb{Z}}[{\mathbb{Z}}/2])\subset\\\
H_{i}^{loc}(K(({\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}({\mathbb{Z}}\times{\mathbb{Z}}))\operatorname{\rtimes}_{\chi^{[4]}}{\mathbb{Z}},1);{\mathbb{Z}}[{\mathbb{Z}}/2]).\end{array}$
(106)

Analogous groups are defined with
${\mathbb{Z}}/2[{\mathbb{Z}}/2]$-coefficients.

Elements in the subgroup $D_{i}^{loc}({\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}({\mathbb{Z}}\times{\mathbb{Z}});{\mathbb{Z}}[{\mathbb{Z}}/2])$
are represented by terms $a+bt$, where $a$ is a local cycle in (98), $b$ is a
local cycle in the group (102).

Analogously to $(\ref{epilok})$ the homomorphism

$\displaystyle\Delta^{[4]}_{b\times\dot{{b}}}:H_{i}^{loc}(K(({\bf
J}_{b}\times{\dot{\bf
J}}_{b})\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}({\mathbb{Z}}\times{\mathbb{Z}}),1);{\mathbb{Z}})\to
D_{i}^{loc}({\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}({\mathbb{Z}}\times{\mathbb{Z}});{\mathbb{Z}}[{\mathbb{Z}}/2])$
(107)

is well-defined. This homomorphism is an isomorphism on the subgroup generated
by elements $a+0t$, $a$ is in (98). The homomorphism

$\displaystyle\Delta^{[4]}_{\dot{{b}}\times b}:H_{i}^{loc}(K(({\dot{\bf
J}}_{b}\times{\bf J}_{b})\rtimes_{\mu_{\dot{{b}}\times
b}^{(4)}}({\mathbb{Z}}\times{\mathbb{Z}}),1);{\mathbb{Z}})\to D_{i}^{loc}({\bf
J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}({\mathbb{Z}}\times{\mathbb{Z}});{\mathbb{Z}}[{\mathbb{Z}}/2])$
(108)

is well defined. On the subgroup generated by elements $0+bt$, $b$ is in
(102). this is a conjugated isomorphism, in the origin homology with
conjugated local coefficient system (101) is.

The local systems TypeIIb, TypeII$\dot{{b}}$, which are conjugated, are
unified into a common $3$-uple system of TypeII. The local system TypeI is
self-conjugated, this determines $3$-uple system of TypeI.

The following lemma is analogous to Lemma 10.

######  Lemma 14.

–1. The following inclusion of a subgroup is well defined:

$\bigoplus_{i_{1}+i_{2}=2s}H_{i_{1}}(K({\bf J}_{b},1);{\mathbb{Z}})\otimes
H_{i_{2}}(K({\dot{\bf J}}_{b},1);{\mathbb{Z}})\subset$ $H_{2s}^{loc}(K(({\bf
J}_{b}\times{\dot{\bf
J}}_{b})\rtimes_{\mu_{b\times\dot{{b}}}^{(4)}}({\mathbb{Z}}\times{\mathbb{Z}}),1);{\mathbb{Z}}).$

This statement is claimed for a system only of TypeI for $s=1\pmod{2}$ and for
TypeII for even $s=0\pmod{2}$.

–2. The homomorphisms (107), (108) are isomorphisms restricted on subgroups
generated by elements $a+0t$, $0+bt$.

–3. For homology groups with ${\mathbb{Z}}/2$-coefficients an analogous
statement –1 is formulated for an isomorphism of the subgroup, isomorphisms in
statement –2. are well-defined.

#### Proof of Lemma 14

Statement –1. Let us assume that a non-trivial element $x\otimes y$ of the
considered subgroup is boundary in the target group. Assume that the element
$x\otimes y$ is represented by a standard closed singular manifold
$f:K^{2s}\to K({\bf J}_{b}\times{\dot{\bf J}}_{b},1)\subset K({\bf
J}_{b}\times{\dot{\bf J}}_{b},1)\rtimes_{\mu_{b\times\dot{{b}}}}(S^{1}\times
S^{1})$ with a prescribed path to the marked point, which is inside the
subcomplex $K({\bf J}_{b}\times{\dot{\bf J}}_{b},1)$. Assume there exist a
singular boundary of this local cycle $F:(W^{2s+1},\partial W=K)\mapsto(K({\bf
J}_{b}\times{\dot{\bf J}}_{b},1)\rtimes_{\mu_{b\times\dot{{b}}}}(S^{1}\times
S^{1}),K({\bf J}_{b}\times{\dot{\bf J}}_{b},1))$, equipped by the
corresponding path system.

Let us consided the closed submanifold $V^{2s-1}\subset W^{2s+1}$, which is
defined as the inverce image of a point on the torus $S^{1}\times S^{1}$ by
the projection $K({\bf J}_{b}\times{\dot{\bf
J}}_{b},1)\rtimes_{\mu_{b\times\dot{{b}}}}(S^{1}\times S^{1})\to S^{1}\times
S^{1}$ on the torus of the involutions $\mu^{(2)}_{b\times\dot{{b}}}$. The
singular manifold $f_{V}=F|_{V}:V^{2s-1}\to K({\bf J}_{b}\times{\dot{\bf
J}}_{b},1)$ represents an integer cycle in the homology group described by the
formula (97). Cycles are represented by the derivative functor in the right
term, or, by terms $x\otimes 1$, $1\otimes y$, $x\in H_{2s-1}(K({\bf
J}_{b},1);{\mathbb{Z}})$, $y\in H_{2s-1}(K({\dot{\bf
J}}_{b},1);{\mathbb{Z}})$.

Let us prove that there exist a closed manifold $F_{1}:W^{2s+1}_{1}\mapsto
K({\bf J}_{b}\times{\dot{\bf
J}}_{b},1)\rtimes_{\mu_{b\times\dot{{b}}}}(S^{1}\times S^{1})$, which
represents a local cycle, such that $F\cup F_{1}$ is a boundary of the local
cycle $f$ with the condition: $V^{2s-1}=\emptyset$.

Let us consider system of TypeI. There exist a generator $z$ in the
decomposition: denote by $V_{z}\subset V$ the corresponding closed component.
There are two analogous cases: $z=[Tor_{2s-1}^{{\mathbb{Z}}}(S^{4j-1}/{\bf
i},S^{4k-3}/{\bf i})]$, $z=[Tor^{{\mathbb{Z}}}(S^{4j-3}/{\bf i},S^{4j-1}/{\bf
i})]$; $4j+4k-4=2s-2$. The first (second) case is not possible, because the
monodromy $\mu_{\dot{{b}}}$ ($\mu_{b}$) of the cycle $f_{V}$ is nontrivial.

The case $z=1\otimes[S^{4j-3}]$ ($z=[S^{4j-3}]\otimes 1$), $4j-3=2s-1$ is not
possible, because in this case the monodromy $\mu_{\dot{{b}}}$ ($\mu_{b}$) is
nontrivial.

Let us consider system of TypeIIb. There are two different cases:
$z=[S^{4j-1}/{\bf i}\times S^{4k-1}/{\bf i}]$, $4j+4k-2=2s-2$;
$z=[S^{4j-3}/{\bf i}\times S^{4j-3}/{\bf i}]$, $4j+4k-6=2s-2$. All cases are
not possible. In the first case the monodromy $\mu_{b}$ of $f_{V}$ is non-
trivial, in the second case the monodromy $\mu_{\dot{{b}}}$ is nontrivial.

The cases $z=1\otimes[S^{4j-1}]$, $z=[S^{4j-1}]\otimes 1$ are not possible,
because in any this case the monodromy $\mu_{b}$ is nontrivial.

Then we assume that $z=2w$ is even. Let us start with the TypeI case. There is
the only cases: $w=[Tor_{2s-1}^{{\mathbb{Z}}}(S^{4j-1}/{\bf i},S^{4k-3}/{\bf
i})]$, $4j+4k-4=2s-2$. Define the cycle $V_{z}$ by
$V=[Tor^{{\mathbb{Z}}}(S^{4j-1}/{\bf i},S^{4k-3}/-1)]$. Define $W$ by the
product $W=W_{b}\times S^{1}$, where $W_{b}$ is the $\mu_{b}$-family of (even)
$V$-cycles, extended over $\mu_{\dot{{b}}}$ by the product with the circle.
The cycle $F_{1}$ is defined.

In the cases $z=2w=[S^{2s-1}/-1\times pt]$; $z=2w=[pt\times S^{2s-1}/-1]$
definition of $W$ is clear.

Let us consider TypeIIb. We have: $w=[Tor_{2s-1}^{{\mathbb{Z}}}(S^{4j-1}/{\bf
i},S^{4k-1}/{\bf i})]$, $4j+4k-2=2s-2$
$w=[Tor_{2s-1}^{{\mathbb{Z}}}(S^{4j-3}/{\bf i},S^{4k-3}/{\bf i})]$,
$4j+4k-6=2s-2$. In the first case define
$V=[Tor^{{\mathbb{Z}}}(S^{4j-1}/-1,S^{4k-1}/{\bf i})]$. The extension of
$W_{\dot{{b}}}$ is defined over $\mu_{b}$ by the trivial product. In the
second case $V=[Tor^{{\mathbb{Z}}}(S^{4j-3}/{\bf i},S^{4k-3}/-1)]$. The
extension of $W_{b}$ is defined over $\mu_{\dot{{b}}}$ by the trivial product.

In the last cases $z=2w=[S^{2s-1}/-1\times pt]$; $z=2w=[pt\times S^{2s-1}/-1]$
definition of $W$ is clear.

We assume $V^{2s-1}=\emptyset$. The manifold $W$ is cutted along the circles,
represented $\mu_{b}$ and $\mu_{\dot{{b}}}$. We have $K^{s}\mapsto
K^{s}\cup(V_{b}\cup V_{b}^{op})\cup(V_{\dot{{b}}}\cup V_{\dot{{b}}}^{op})$,
$p\circ F_{W}:W\to S^{1}\vee S^{1}\subset S^{1}\times S^{1}$. where $V_{b}$,
$V_{b}^{op}$; $V_{\dot{{b}}}$, $V_{\dot{{b}}}^{op}$ are two conjugated pairs
of local cycles, cycle in each pair cancel each other after a path to the
marked point for one cycle in a pair is changed. Statement 1 is proved.

Statement 2. Let us generalise the isomorphism (43). The only relation in the
target group in (107) isa subgroup, generated by elements
$a\equiv\chi^{[4]}(a)t$. This relation shows that the images of (107),(108)
coincide. Statement 2 is proved.

Statement 3. Because elements modulo 2 are of the order 2, the local homology
(interior local coefficient system) coincides with modulo 2 homology and the
left term in the formula (97) become trivial. Statement 3 is proved. Lemma 14
is proved.

## Standardized ${\bf J}_{b}\times{\dot{\bf J}}_{b}$-immersions

Let us formulate notions of standardized (and pre-standardized) ${\bf
J}_{b}\times{\dot{\bf J}}_{b}$-immersion.

Let us consider a ${\mathbb{Z}}/2^{[4]}$-framed immersion $(g,\Psi,\eta_{N})$
of the codimension $8k$. Assume that the image of the immersion $g$ belongs to
a regular neighbourhood of an embedding $I:S^{1}_{b}\times
S^{1}_{\dot{{b}}}\rtimes_{\chi^{[4]}}S^{1}\subset{\mathbb{R}}^{n}$.

In this formula the third factor $S^{1}$ is not a direct, the monodromy (an
involution) of a fibre $S^{1}_{b}\times S^{1}_{\dot{{b}}}$ along the exterior
circle is given by the permutation of the factors. Denote by
$U\subset{\mathbb{R}}^{n}$ a regular thin neighbourhood of the embedding $I$.

Let us consider a ${\mathbb{Z}}/2^{[4]}$–framed immersion
$g:N^{n-8k}\looparrowright U\subset{\mathbb{R}}^{n}$, for which the following
condition (Y) of a control of the structured group of the normal bundle is
satisfied.

### Condition (Y)

The immersion $g$ admits a reduction of a symmetric structured group to the
group (81). Additionally, the projection $\pi\circ g:N^{n-8k}\to
S^{1}_{b}\times S^{1}_{\dot{{b}}}\rtimes_{\chi^{[4]}}S^{1}$ of this immersion
onto the central manifold of $U$ is agreed with the projection of the
structured group $({\mathbb{Z}}\times{\mathbb{Z}})\rtimes{\mathbb{Z}}$ onto
the factors of the extension.

Formally, a weaker condition is following.

### Condition (Y1)

Assume an immersion $g$ admits a reduction of a symmetric structured group to
the group (81).

######  Definition 15.

Let us say that a ${\mathbb{Z}}/2^{[4]}$-framed immersion $(g,\Psi,\eta_{N})$
of the codimension $8k$ is standardized if Condition Y is satisfied.

Let us say that ${\mathbb{Z}}/2^{[4]}$-framed immersion $(g,\Psi,\eta_{N})$ of
the codimension $8k$ is pre-standardized if the definition above week
Condition (Y1) instead of Condition (Y) is formulated. Classification problems
of standardized and of pre-standardized immersions are equivalent.

Standardized ${\mathbb{Z}}/2^{[4]}$-framed immersions generate a cobordism
group, this group is naturally mapped into
$Imm^{{\mathbb{Z}}/4^{[4]}}(n-8k,8k)$ when a standardization is omitted.

The following definition is analogous to Definition 17. In this definition a
projection of the Hurewicz image of a fundamental class onto a corresponding
subgroup is defined using Lemma 14.

######  Definition 16.

Let $(g,\eta_{N},\Psi)$ be a standardized (a pre-standardized)
${\mathbb{Z}}/2^{[4]}$-framed immersion in the codimension $8k$.

Let us say that this standardized immersion is pure, if the Hurewicz image of
a fundamental class in $\in D_{n-8k}^{loc}({\bf J}_{b}\times{\dot{\bf
J}}_{b}\rtimes_{\mu_{b\times\dot{{b}}}}({\mathbb{Z}}\times{\mathbb{Z}});{\mathbb{Z}}/2[{\mathbb{Z}}/2])$
contains not monomial $t_{b,\frac{n-8k}{2}+i}\otimes
t_{\dot{{b}},\frac{n-8k}{2}-i}$, $i\in\\{\pm 1;\pm 2;\pm 3;\pm 4;\pm 5;\pm
6;\pm 7\\}$, but, probably, contains the only non-trivial monomial
$t_{b,\frac{n-8k}{2}}\otimes t_{\dot{{b}},\frac{n-8k}{2}}$.

######  Definition 17.

Let $(g,\eta_{N},\Psi)$ be a ${\mathbb{Z}}/2^{[4]}$-framed immersion. for
which a structured group ${\mathbb{Z}}/2^{[4]}$ of the normal bundle is
reduced to the group
${\mathbb{Z}}/2^{[3]}\times{\mathbb{Z}}/2^{[3]}\rtimes{\mathbb{Z}}$, where the
generator ${\mathbb{Z}}$ acts on
${\mathbb{Z}}/2^{[3]}\times{\mathbb{Z}}/2^{[3]}$ by the permutation of the
factors. A transfer with respect to the subgroup
${\mathbb{Z}}/2^{[2]}\subset{\mathbb{Z}}/2^{[4]}$ determines an immersion of
$4$-sheeted cover denoted by $f$. Because $g$ satisfies an extra condition
concerning its structured group, the immersion $f$ is a framed immersion:
$(f,\Xi,\eta_{N})$ and its framing admits a reduction from a generic
structured group ${\mathbb{Z}}/2^{[2]}$ to the group ${\bf
I}_{a\times\dot{{a}}}\rtimes_{\chi^{[2]}}{\mathbb{Z}}$.

Let us say that a ${\mathbb{Z}}/2^{[4]}$-framed immersion $(g,\eta_{N},\Psi)$
is negligible immersion, if its cover $(f,\Xi,\eta)$ is a negligible
standardized $D$-framed immersion in the sense of Definition 17.

### Formal regular homotopy

Let $g:N\looparrowright{\mathbb{R}}^{n}$ be an arbitrary immersion. This
immersion induces a $(T_{N},T_{{\mathbb{R}}^{n}})$\- equivariant mapping
$g^{(2)}:N\times N\looparrowright{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$,
where $T_{N}$ is a permutation of points in the origine,
$T_{{\mathbb{R}}^{n}}$ is a permutation in the target. We will call that
$F_{t}^{(2)}:N\times N\looparrowright{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$
is a formal generic regular deformation of $g^{(2)}$ $F_{0}^{(2)}=g^{(2)}$ and
the following conditions are satisfied:

–1. a support $supp(F^{(2)})\subset N\times N$ of $F^{(2)}$ is outside the
diagonal $\Delta_{N}\subset N\times N$;

–2. $F^{(2)}$ is a regular $(T_{N},T_{{\mathbb{R}}^{n}})$-equivariant homotopy
on $supp(F)$;

–3. $F^{(2)}$ is regular along $\Delta_{{\mathbb{R}}^{n}}$.

For an arbitrary formal regular homotopy of an immersion $g$ to a formal
immersion $g_{1}^{(2)}$ the self-intersection manifold $L$,
$h:L\looparrowright{\mathbb{R}}^{n}$, of $g$ is regular cobordant to the
manifold $L_{1}$ of formal self-intersection of $g^{(2)}_{1}$, which is
defined by $(g^{(2)}_{1})^{-1}(\Delta_{{\mathbb{R}}^{n}})/T_{N}$. For generic
$F^{(2)}$ the manifold $L_{1}$ is equipped with a parametrized immersion
$h_{1}:L_{1}\looparrowright{\mathbb{R}}^{n}$ and the immersions $h$ and
$h_{1}$ of manifolds $L$ and $L_{1}$ are regular cobordant. In the case $g$ is
a ${\mathbb{Z}}/2^{[s]}$-framed immersion, the immersion $h$ is
${\mathbb{Z}}/2^{[s+1]}$-framed and $L_{1}$ and the cobordism between $L$ and
$L_{1}$ are also ${\mathbb{Z}}/2^{[s+1]}$-framed.

The immersion $g$ induces a $(T^{(2)}_{N},T^{(2)}_{{\mathbb{R}}^{n}})$\-
equivariant mapping $g^{(2+2)}:(N\times N)\times(N\times
N)\looparrowright({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})\times({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$,
where $T^{(2)}_{N}$ is the involution of involutions by permutation of pairs
(in a common bracket) of points in the origine, $T^{(2)}_{{\mathbb{R}}^{n}}$
is a permutation of pairs of points in the target. We will call that
$F_{t}^{(2+2)}:N\times N\looparrowright{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$
is a formal generic regular deformation of $g^{(2+2)}$ $F_{0}^{(2+2)}=g^{(2)}$
and the following conditions are satisfied:

–1. a support $supp(F^{(2+2)})\subset(N\times N)\times(N\times N)$ of
$F^{(2+2)}$ is outside the thick diagonal $\Delta_{N}^{(2)}\subset(N\times
N)\times(N\times N)$;

–2. $F^{(2+2)}$ is a regular
$(T^{(2)}_{N},T^{(2)}_{{\mathbb{R}}^{n}})$-equivariant homotopy on $supp(F)$;

–3. $F^{(2+2)}$ is regular along
$\Delta_{{\mathbb{R}}^{n}}^{(2)}\subset({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})\times({\mathbb{R}}^{n}\times{\mathbb{R}}^{n})$.

An arbitrary generic immersion $g:N\looparrowright{\mathbb{R}}^{n}$ with a
self-intersection manifold $h:L\looparrowright{\mathbb{R}}^{n}$ admits an
iterated self-intersection manifold $f:M\looparrowright{\mathbb{R}}^{n}$ which
is a self-intersection manifold of $h$. For an arbitrary formal regular
homotopy $F^{(2+2)}$ of the extension $g^{(2+2)}$ $g$ to a formal immersion
$g_{1}^{(2+2)}$ the iterated self-intersection manifold $M$,
$f:M\looparrowright{\mathbb{R}}^{n}$, of $g$ is regular cobordant to the
manifold $M_{1}$ of formal iterated self-intersection of $g^{(2+2)}_{1}$,
which is defined by
$(g^{(2+2)}_{1})^{-1}(\Delta^{(2)}_{{\mathbb{R}}^{n}})/T^{(2)}_{N}$. For
generic $F^{(2+2)}$ the manifold $M_{1}$ is equipped with a parametrized
immersion $f_{1}:M_{1}\looparrowright{\mathbb{R}}^{n}$ and the immersions $f$
and $f_{1}$ of manifolds $M$ and $M_{1}$ are regular cobordant. In the case
$g$ is a ${\mathbb{Z}}/2^{[s]}$-framed immersion, the immersion $f$ is
${\mathbb{Z}}/2^{[s+2]}$-framed and $M_{1}$ and the cobordism between $M$ and
$M_{1}$ are also ${\mathbb{Z}}/2^{[s+2]}$-framed.

The definition of formal homotopies (with no regular condition) should be
applied for generic singular mappings.

######  Theorem 18.

Let $(g,\Psi,\eta_{N})$ be a standardized pure ${\bf D}$-framed immersion in
the codimension $2k$ (see Definitions 3, 17), $k\geq 7$. Let
$(h,\Psi_{L},\eta_{L})$ be the ${\mathbb{Z}}/2^{[3]}$-framed immersion of
self-intersection points of $g$; $(f,\Psi_{M},\zeta_{M})$ be the
${\mathbb{Z}}/2^{[4]}$-framed immersion of iterated self-intersection points
of $g$.

Then in each regular homotopy class of ${\bf D}$-framed standardized
immersions there exist $(g,\Psi,\eta)$ and a formal regular deformation
$g^{(2+2)}\mapsto\bar{g}^{(2+2)}$, the iterated self-intersection manifold of
$\bar{g}^{(2+2)}$ is a disjoin union of a pre-standardized pure
${\mathbb{Z}}/2^{[4]}$-immersion $(f,\Xi_{M},\eta_{M})$ (see Condition (Y1))
and a neglectable ${\mathbb{Z}}/2^{[4]}$-immersion.

Moreower, the cobordism class of the of ${\mathbb{Z}}/2^{[4]}$-standartized
immersion $(f,\Xi_{M},\eta_{M})$ is well-defined for $(g,\Psi,\eta_{N})$ in
its regular cobordism class.

### Proof of Proposition 2

Take a homomorphism $Imm^{sf}(n-1,1)\to Imm^{D}(n-2,2)$. By Theorem 5,
statement 1 there exist an intermediate cobordism group with a homomorphism

$Imm^{sf}(n-1,1)\to Imm^{D;stand}(n-4,4)\to Imm^{D}(n-4,4)$

(see Definition 3). By Theorem 18 there exists a homomorphism
$Imm^{D;stand}(n-4,4)\to Imm^{{\mathbb{Z}}/2^{[4]};stand}(n-8k,8k)$ into a
cobordism group of standartized immersion, in the case $k=7$, $n=62$. The
Kervaire invariant is defined as a characteristic number of the order 4 by
Lemma 14, which is well-defined homomorphism
$Imm^{{\mathbb{Z}}/2^{[4]};stand}(n-8k,8k)\to{\mathbb{Z}}/4$ . ∎

##  6 Proof of Theorem 18

Let us reformulate Definition 7 and Lemma 9, replacing the subgroup ${\bf
I}_{a}\times{\dot{\bf I}}_{a}\subset{\bf D}$ into the subgroup ${\bf
I}_{b}\subset{\bf D}$.

######  Definition 19.

Assume an equivariant (formal) mapping

$\displaystyle d^{(2)}:{\mathbb{R}}\\!{\rm
P}^{n^{\prime}-k}\times{\mathbb{R}}\\!{\rm
P}^{n^{\prime}-k}\setminus{\mathbb{R}}\\!{\rm
P}^{n^{\prime}-k}_{diag}\to{\mathbb{R}}^{n^{\prime}}\times{\mathbb{R}}^{n^{\prime}},$
(109)

which, generally speaking, is not a holonomic.

Let us say that $d^{(2)}$ admits a cyclic structure, if the following
condition is satisfied. The polyhedron of the (formal) self-intersection of
$d^{(2)}$ is divided into two components: a closed component ${\bf N}_{b}$ and
an open component (or, a component with a boundary) ${\bf
N}_{a\times\dot{{a}},\circ}$; moreover, the following conditions are
satisfied.

– 1. The fundamental class $[{\bf N}_{a\times\dot{{a}},\circ}]$ of the open
polyhedra

$\displaystyle{\bf N}_{a\times\dot{{a}},\circ}$ (110)

is well-defined. The structuring mapping $\eta:N_{a\times\dot{{a}};\circ}\to
K({\bf D},1)$ admits the transfer mapping with respect to the subgroup
$A\times\mathcal{A}\subset{\bf D}$; the image of the fundamental class
$N_{a\times\dot{{a}};\circ}]$ by the transfer is the trivial characteristic
numbers modulo $2$. In this case the component ${\bf N}_{\circ}$ is called a
negligible component.

–2. The fundamental class $[{\bf N}_{b}]$ of the closed polyhedra ${\bf
N}_{b}$ is well-defined. The restriction of the structured mapping on the
polyhedron ${\bf N}_{b}$ admits a reduction

$\displaystyle\eta_{b,{\mathbb{R}}\\!{\rm P}^{2}}:{\bf N}_{b}\to K({\bf
I}_{b},1)\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{R}}\\!{\rm P}^{2}.$ (111)

Moreover, there exist a sub-polyhedron ${\bf N}_{reg}\subset{\bf N}_{b}$ (this
polyhedron is an obstruction to a reduction of the structured mapping (13) to
a mapping $\eta_{b}:{\bf N}_{\circ}\to K({\bf
I}_{b}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)$) the dimension
${\operatorname{dim}}({\bf N}_{reg})$ equals (up to a small constant $d=2$)
satisfies the condition: $dim({\bf N}_{reg})\leq\frac{n^{\prime}-k}{2}+d$ and
which is a preimage of a marked point on ${\bf
x}_{\infty}\in{\mathbb{R}}\\!{\rm P}^{2}$ of the composition ${\bf N}_{b}\to
K({\bf
I}_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi^{[2]}}{\mathbb{Z}},1)\to{\mathbb{R}}\\!{\rm
P}^{2}$.

Additionally, the fundamental class $[{\bf N}_{b}]$ by the transfer over the
structured mapping (111) with respect to the subgroup ${\bf I}_{d}\subset{\bf
I}_{b}$ is mapped into the generator in $H_{dim({\bf N}_{b})}({\bf
I}_{d};{\mathbb{Z}}/2)$.

Let us formulate the following lemma (for the proof see Section 8).

######  Lemma 20.

Big Lemma

Assuming

$\displaystyle n^{\prime}=\frac{n}{2}=2^{l}-1,\quad k\equiv 0\pmod{2},k\geq
2\quad(n^{\prime}-k\equiv 1\pmod{2}),$ (112)

there exist a formal mapping $d^{(2)}$, which admits a cyclic structure, see
Definition 19. The formal mapping $d^{(2)}$ is a result of a non-holonomic (a
formal vertical) small deformation in the even codimension $k$ of the formal
extension of a special mapping $\bar{d}:{\mathbb{R}}\\!{\rm
P}^{n^{\prime}-k}\to S^{n^{\prime}-k}\subset{\mathbb{R}}^{n^{\prime}}$ with
the image on the sphere $S^{n^{\prime}-k}$.

### Construction of the mapping $d^{(2)}$ in Lemma 20

Consider the standard covering $p:S^{1}\to S^{1}$ of the degree $4$. It is
convenient to write-down: $p:S^{1}\to S^{1}/\\{{\bf i}\\}$, by the quotient:
${\mathbb{Z}}/4\times S^{1}\to S^{1}/\\{{\bf i}\\}$.

Consider the join of $\frac{n^{\prime}-k+1}{2}=r$-copies of the circle
$S^{1}/\\{{\bf i}\\}$ $S^{1}/\\{{\bf i}\\}\ast\dots\ast S^{1}/\\{{\bf
i}\\}=S^{n^{\prime}-k}$, which is PL-homeomorphic to the standard
$n^{\prime}-k$-dimensional sphere. Let us define the join of $r$ copies of the
mapping $p$:

$\tilde{d}:S^{1}\ast\dots\ast S^{1}\to S^{1}/\\{{\bf i}\\}\ast\dots\ast
S^{1}/\\{{\bf i}\\}.$

On the pre-image acts the group ${\mathbb{Z}}/4$ by the diagonal action, this
action is commuted with $\tilde{d}$. The mapping $\bar{d}$ is defined by the
composition of the quotient $\tilde{d}/\\{{\bf i}\\}:S^{n^{\prime}-k}/\\{{\bf
i}\\}\to S^{n^{\prime}-k}$ with the standard inclusion

$S^{n^{\prime}-k}\subset{\mathbb{R}}^{n^{\prime}}.$

A formal (non-holonomic, vertical) small deformation of the formal (holonomic)
extension of the mapping $\bar{d}$ is the required mapping $d^{(2)}$. ∎

The mapping $\bar{d}:{\mathbb{R}}\\!{\rm P}^{n^{\prime}-k}\to
S^{n^{\prime}-k}$ has to be generalized, using two-stages tower (116) of
ramified coverings.

Let us start a proof of Theorem 18 with the following construction. Let us
recall, that a positive integer $m_{\sigma}=14$. Denote by $ZZ_{{\bf
J}_{a}\times{\dot{\bf J}}_{a}}$ the Cartesian product o standard lens space
$\pmod{4}$, namely,

$\displaystyle ZZ_{{\bf J}_{a}\times{\dot{\bf
J}}_{a}}=S^{n-\frac{n-m_{\sigma}}{8}+1}/{\bf i}\times
S^{n-\frac{n-m_{\sigma}}{8}+1}/{\bf i}.$ (113)

Evidently, ${\operatorname{dim}}(ZZ_{{\bf J}_{a}\times{\dot{\bf
J}}_{a}})=\frac{7}{4}(n+m_{\sigma})+2>n$.

On the space $ZZ_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}$ a free involution
$\chi_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}:ZZ_{{\bf J}_{a}\times{\dot{\bf
J}}_{a}}\to ZZ_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}$ acts by the formula:
$\chi_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}(x\times y)=(y\times x)$.

Let us define a subpolyhedron (a manifold with singularities) $X_{{\bf
J}_{a}\times{\dot{\bf J}}_{a}}\subset ZZ_{{\bf J}_{a}\times{\dot{\bf
J}}_{a}}$. Let us consider the following family $\\{X_{j},\quad
j=0,1,\dots,j_{max}\\}$, $j_{max}\equiv 0\pmod{2}$, of submanifolds $ZZ_{{\bf
J}_{a}\times{\dot{\bf J}}_{a}}$:

$X_{0}=S^{n-\frac{n-m_{\sigma}}{8}+1}/{\bf i}\times S^{1}/{\bf i},$
$X_{1}=S^{n-\frac{n-m_{\sigma}}{8}-1}/{\bf i}\times S^{3}/{\bf i},\quad\dots$
$X_{j}=S^{n-\frac{n-m_{\sigma}}{8}+1-2j}/{\bf i}\times S^{2j+1}/{\bf i},\quad$
$X_{j_{max}}=S^{1}/{\bf i}\times S^{n-\frac{n-m_{\sigma}}{8}+1}/{\bf i},$

where

$\displaystyle j_{max}=\frac{7n+m_{\sigma}}{16}=2^{n-1},\quad m_{\sigma}=14.$
(114)

The dimension of each manifold in this family equals to
$n-\frac{n-m_{\sigma}}{8}+2$ and the codimension in $ZZ_{{\bf
J}_{a}\times{\dot{\bf J}}_{a}}$ equals to $n-\frac{n-m_{\sigma}}{8}$. Let us
define an embedding

$X_{j}\subset ZZ_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}$

by a Cartesian product of the two standard inclusions.

Let us denote by $\chi_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}:ZZ_{{\bf
J}_{a}\times{\dot{\bf J}}_{a}}\to ZZ_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}$ the
involution, which permutes coordinates. Evidently, we get: $\chi_{{\bf
J}_{a}\times{\dot{\bf J}}_{a}}(X_{j})=X_{j_{max}-j}$.

A polyhedron $X_{{\bf J}_{a}\times{\dot{\bf
J}}_{a}}=\bigcup_{j=0}^{j_{max}}X_{j}\subset ZZ_{{\bf J}_{a}\times{\dot{\bf
J}}_{a}}$ is well-defined. This polyhedron is invariant with respect to the
involution $\chi_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}$. The polyhedron
$X_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}$ can be considered as a stratified
manifolds with strata of the codimension 2.

The restriction of the involution $\chi_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}$
on the polyhedron $X_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}$ denote by
$\chi_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}$.

Write-down the sequence of the subgroups of the index $2$ of the diagram
$(\ref{a,aa})$:

$\displaystyle{\bf I}_{a}\times{\dot{\bf I}}_{a}\longrightarrow{\bf
H}_{a\times\dot{{a}}}\longrightarrow{\bf J}_{b}\times{\dot{\bf J}}_{b}.$ (115)

Define the following tower of $2$-sheeted coverings, which is associated with
the sequence $(\ref{zep})$:

$\displaystyle ZZ_{a\times\dot{{a}}}\longrightarrow ZZ_{{\bf
H}_{a}\times\dot{{a}}}\longrightarrow ZZ_{{\bf J}_{b}\times{\dot{\bf
J}}_{b}}.$ (116)

The bottom space of the tower $(\ref{z})$ coincides to a skeleton of the
Eilenberg-MacLane space: $ZZ_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}\subset
K({\bf J}_{a},1)\times K({\dot{\bf J}}_{a},1)$. This tower (117) determines
the tower $(\ref{zz})$ by means of the inclusion $ZZ_{{\bf
J}_{b}\times{\dot{\bf J}}_{b}}\subset K({\bf J}_{b},1)\times K({\dot{\bf
J}}_{b},1)$ of the bottom.

Let us define the following tower of double coverings:

$\displaystyle X_{a\times\dot{{a}}}\longrightarrow X_{{\bf
H}_{b\times\dot{{b}}}}\longrightarrow X_{{\bf J}_{a}\times{\dot{\bf J}}_{a}}.$
(117)

The bottom space of the tower $(\ref{z})$ is a subspace of the bottom space of
the tower $(\ref{zz})$ by means of an inclusion $X_{{\bf J}_{b}\times{\dot{\bf
J}}_{b}}\subset ZZ_{{\bf J}_{b}\times{\dot{\bf J}}_{b}}$. The tower
$(\ref{z})$ determines as the restriction of the tower $(\ref{zz})$ on this
subspace.

Let us describe a polyhedron $X_{a\times\dot{{a}}}\subset
ZZ_{a\times\dot{{a}}}$ explicitly. Let us define a family
$\\{X^{\prime}_{0},X^{\prime}_{1},\dots,X^{\prime}_{j_{max}}\\}$ of standard
submanifolds in the manifold $ZZ_{a\times\dot{{a}}}={\mathbb{R}}\\!{\rm
P}^{n-\frac{n-m_{\sigma}}{8}+1}\times{\mathbb{R}}\\!{\rm
P}^{n-\frac{n-m_{\sigma}}{8}+1}$ by the following formulas:

$\displaystyle X^{\prime}_{0}={\mathbb{R}}\\!{\rm
P}^{n-\frac{n-m_{\sigma}}{8}+1}\times{\mathbb{R}}\\!{\rm P}^{1}\dots$ (118)
$X^{\prime}_{j}={\mathbb{R}}\\!{\rm
P}^{n-\frac{n-m_{\sigma}}{8}+1-2j}\times{\mathbb{R}}\\!{\rm P}^{2j+1}\dots$
$X^{\prime}_{j_{max}}={\mathbb{R}}\\!{\rm P}^{1}\times{\mathbb{R}}\\!{\rm
P}^{n-\frac{n-m_{\sigma}}{8}+1}.$

In this formulas the integer index $j_{max}$ is defined by the formula
$(\ref{jmax})$. The polyhedron $X_{a\times\dot{{a}}}\subset
ZZ_{a\times\dot{{a}}}$ is defined as the union of standard submanifolds in
this family. The polyhedron $X_{{\bf H}_{a\times\dot{{a}}}}\subset ZZ_{{\bf
H}_{a\times\dot{{a}}}}$ a quotient of the double covering, which corresponds
to the tower of the groups.

The spaces $X_{{\bf H}_{a\times\dot{{a}}}}$, $X_{a\times\dot{{a}}}$ admit free
involutions, which are pullbacks of the involution $\chi_{{\bf
J}_{a}\times{\dot{\bf J}}_{a}}$ by the projection on the bottom space of the
tower.

The cylinder of the involution $\chi_{a\times\dot{{a}}}$ is well-defined,
(correspondingly, of the involution $\chi_{{\bf H}_{a\times\dot{{a}}}}$),
which is denoted by $X_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi}S^{1}$
(correspondingly, by $X_{{\bf
H}_{a\times\dot{{a}}}}\operatorname{\rtimes}_{\chi}S^{1}$). The each space is
embedded into the corresponding fibred space over ${\mathbb{R}}\\!{\rm
P}^{2}$:

$X_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi}S^{1}\subset
X_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm P}^{2},$
$X_{{\bf H}_{a\times\dot{{a}}}}\operatorname{\rtimes}_{\chi}S^{1}\subset
X_{{\bf H}_{a\times\dot{{a}}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm
P}^{2}.$

Then let us define a polyhedron $J_{b\times\dot{{b}}}$, which is a base of a
ramified covering $X_{a\times\dot{{a}}}\to J_{b\times\dot{{b}}}$.

Then let us extend the ramified covering over the bottom space of the tower to
the ramified covering:
$X_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm
P}^{2}\to J_{b\times\dot{{b}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm
P}^{2}$, and the ramified covering over the middle space of the tower of the
ramified covering $X_{{\bf
H}_{a\times\dot{{a}}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm
P}^{2}\to J_{b\times\dot{{b}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm
P}^{2}$.

Let us define a polyhedron (a manifold with singularities)
$J_{b\times\dot{{b}}}$. For an arbitrary $j=0,1,\dots,j_{max}$, where
$j_{max}$ is defined by the formula $(\ref{jmax})$, let us define the
polyhedron $J_{j}=S^{n-\frac{n-m_{\sigma}}{8}-2j+1}\times S^{2j+1}$ (the
Cartesian product). Spheres (components of this Cartesian product)
$S^{n-\frac{n-m_{\sigma}}{8}-2j+1}$, $S^{2j+1}$ are re-denoted by $J_{j,1}$,
$J_{j,2}$ correspondingly. Using this denotations, we get:

$J_{j}=J_{j,1}\times J_{j,2}.$

The standard inclusion $i_{j}:J_{j,1}\times J_{j,2}\subset
S^{\frac{n-m_{\sigma}}{8}+1}\times S^{\frac{n-m_{\sigma}}{8}+1}$ is well-
defined, each factor is included into the target sphere as the standard
subsphere.

The union $\bigcup_{j=0}^{j_{max}}Im(i_{j})$ of images of this embeddings is
denoted by

$\displaystyle J_{b\times\dot{{b}}}\subset S^{\frac{n-m_{\sigma}}{8}+1}\times
S^{\frac{n-m_{\sigma}}{8}+1}.$ (119)

The polyhedron $J_{b\times\dot{{b}}}$ is constructed.

Let us define a ramified covering

$\displaystyle\varphi_{a\times\dot{{a}}}:X_{a\times\dot{{a}}}\to
J_{b\times\dot{{b}}}.$ (120)

The covering (120) is defined as the union of the Cartesian products of the
ramified coverings, which was constructed in Lemma 20.

The covering (120) is factorized into the following ramified covering:

$\displaystyle\varphi_{{\bf H}_{a\times\dot{{a}}}}:X_{{\bf
H}_{a\times\dot{{a}}}}\to J_{b\times\dot{{b}}}.$ (121)

Because $X_{a\times\dot{{a}}}\to X_{{\bf H}_{a\times\dot{{a}}}}\to
J_{b\times\dot{{b}}}$ is a double covering, the number of sheets of the
covering (121) is greater by the factor $2^{r}$, where $r$ is the denominator
of the ramification.

The polyhedron $J_{b\times\dot{{b}}}$ is equipped by the involution $\chi$,
which is defined analogously to the involutions $\chi_{a\times\dot{{a}}}$,
$\chi_{{\bf H}_{a\times\dot{{a}}}}$.

The cylinder of the involution is well-defined, let us denote this cylinder by
$J_{b\times\dot{{b}}}\operatorname{\rtimes}_{\chi}S^{1}$. The inclusion
$J_{b\times\dot{{b}}}\operatorname{\rtimes}_{\chi}S^{1}\subset
J_{b\times\dot{{b}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm P}^{2}$.
is well-defined.

The ramified covering $(\ref{varphiX})$ commutes with the involutions
$\chi_{a\times\dot{{a}}}$, $\chi_{{\bf H}_{a\times\dot{{a}}}}$ in the origin
and the target.

Therefore the ramified covering

$\displaystyle
c_{X}:X_{a\times\dot{{a}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm
P}^{2}\to J_{b\times\dot{{b}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm
P}^{2},$ (122)

which is factorized into the ramified covering

$\displaystyle c_{Y}:X_{{\bf
H}_{a\times\dot{{a}}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm
P}^{2}\to J_{b\times\dot{{b}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm
P}^{2}.$ (123)

is well-defined.

######  Lemma 21.

There exist an inclusion

$\displaystyle
i:J_{b\times\dot{{b}}}\operatorname{\rtimes}_{\chi}{\mathbb{R}}\\!{\rm
P}^{2}\times D^{5}\rtimes D^{3}\subset{\mathbb{R}}^{n},$ (124)

where $D^{5}$ is the standard disks (of a small radius), the disk $D^{5}$ is a
fibre of the trivial $9$-bundle over ${\mathbb{R}}\\!{\rm P}^{2}$; $D^{3}$ is
the standard disks (of a small radius), the disk $D^{3}$ is a fibre of the
non-trivial twisted $3$-bundle $3\varkappa_{\chi}$ over ${\mathbb{R}}\\!{\rm
P}^{2}$, where $\varkappa_{\chi}$ is the line bundle with the generic class in
$H^{1}({\mathbb{R}}\\!{\rm P}^{2};{\mathbb{Z}}/2)$.

#### Proof of Lemma 21

Put $m_{\sigma}=14$, $k=7$, $n=126$ and prove the statement with this
restriction (this case is the most complicated). The cases $n=62$, $k=7$;
$n=2^{l}-2,l\geq 8$, $k=\frac{n-m_{\sigma}}{16},m_{\sigma}=14$ are also
possible.

Let us prove that the polyhedron $J_{b\times\dot{{b}}}$ is embeddable into the
sphere $S^{n-11}$.

The sphere $S^{n-11}$ is the unite sphere (the target of the standard
projection) of an $n-10$-dimensional fibre. Take a collection of embedding:
$S^{1}\times S^{n-13}\subset S^{n-11}$, $S^{3}\times S^{n-15}\subset
S^{n-11}$, $\dots$, $S^{1}\times S^{n-13}\subset S^{n-11}$. Take the Euclidean
coordinates $x_{1},\dots,x_{n-10}$ in ${\mathbb{R}}^{n-10}$, where the sphere
$S^{n-11}\subset{\mathbb{R}}^{n-10}$ is the unite sphere with the centre at
the origine. Each embedding $S^{2k_{1}+1}\times S^{2k_{2}+1}\subset S^{n-11}$,
$k_{1}+k_{2}=\frac{n-14}{2}$ is the hypersurface in $S^{n-11}$ given by the
equation $x_{1}^{2}+\dots+x_{2k_{1}+2}^{2}-x_{2k_{1}+3}^{2}-\dots-
x_{n-10}^{2}=0$. The polyhedron $J_{b\times\dot{{b}}}$ is $PL$-homeomorphic to
the union of the tori.

The normal bundle of the embedding $emb:{\mathbb{R}}\\!{\rm
P}^{2}\subset{\mathbb{R}}^{n}$ is the Whitney sum:

$\nu(emb)=(\frac{n-12}{2}+1)\varepsilon\oplus(\frac{n-12}{2}+1)\varkappa_{\chi}\oplus
3\varkappa_{\chi}\oplus 5\varepsilon.$

because $n-12=2\pmod{16}$. The polyhedron
$J_{b\times\dot{{b}}}\rtimes{\mathbb{R}}\\!{\rm P}^{2}$ is embedded into
$(\frac{n-12}{2}+1)\varepsilon\oplus(\frac{n-12}{2}+1)$.

Lemma 21 is proved. ∎

A reformulation of Definition 19 for ramified covering (122) over the embedded
polyhedron, which is constructed in Lemma 21 is required.

Let $\bar{d}_{X}=i\circ
c_{\bar{X}}:\bar{X}_{a\times\dot{{a}}}\rtimes_{\chi}{\mathbb{R}}\\!{\rm
P}^{2}\to{\mathbb{R}}^{n}$, $\bar{d}^{(2+2)}_{X}$ be a formal
${\mathbb{Z}}/2\times{\mathbb{Z}}/2$-equivariant holonomic extension of the
mapping $\bar{d}_{X}$ on the space of $(2+2)$-configurations: the space of
non-ordered pairs of non-ordered two-points. A general facts on such
extensions are not required, because we will considered the only following
case.

Define a formal mapping $\bar{d}^{(2+2))}_{X}$, which is not a holonomic
mapping and which is defined as a small formal deformation of the holonomic
extension $\bar{d}^{(2+2)}_{X}$ of the mapping $\bar{d}_{X}$. The deformation
is a codimension $4$ vertical with respect to the line bundles described in
Lemma 21 and is $\chi$-equivariant.

On the first stage of the construction we consider the mapping $\bar{d}_{Y}$
$\bar{d}_{Y}=i\circ c_{Y}:Y_{{\bf
H}_{a\times\dot{{a}}}}\rtimes_{\chi}{\mathbb{R}}\\!{\rm
P}^{2}\to{\mathbb{R}}^{n}$ and its formal ${\mathbb{Z}}/2$-extension
$(\bar{d}_{Y})^{(2)}$. On the second stage we consider the double formal
extension $((\bar{d}_{X})^{(2)})^{(2)}$, this extension determines an extra
self-intersections over self-intersection points of the mapping
$(\bar{d}_{Y})^{(2)}$. This resulting polyhedron is called a polyhedron of
iterated self-intersections. To define this polyhedron two steps are required
and one may changes an order of the steps, if required. By this, on the first
step self-intersection points of the upper polyhedron in the tower is
organized. On the second step self-intersection points of of the polyhedron by
the mapping $\bar{d}_{Y}$ is organized. Recall the towers of coverings are
parametrized over ${\mathbb{R}}\\!{\rm P}^{2}$ by $\chi$.

Then we define a special deformation (formal)
$((\bar{d}_{X})^{(2)})^{(2)}\mapsto d^{(2+2)}_{X}$. Properties of iterated
self-intersections are described in Definition 22 and then proved in Lemma 23.
Calculations are based on Lemma 20.

######  Definition 22.

Let us say that a formal mapping $d^{(2+2)}_{X}$ admits a bi-cyclic structure,
if the following condition is satisfied: the polyhedron of iterated self-
intersections is divided (by a formal deformation of a holonomic extension)
into two subpolyhedra: a closed subpolyhedron ${\bf N}_{b\times\dot{{b}}}$ and
a subpolyhedron ${\bf N}_{\circ}$ with a boundary. Additionally, the following
conditions, concerning a reduction of the structuring mapping are satisfied:

– 1. On the polyhedron ${\bf N}_{\circ}$ (with a boundary) the structuring
mapping admits the following reduction:

$\displaystyle\eta_{a\times\dot{{a}}}:{\bf N}_{\circ}\to K({\bf
I}_{a\times\dot{{a}}}\times\dot{{\bf
I}}_{a\times\dot{{a}}},1)\rtimes_{\mu_{a\times\dot{{a}}},\chi}({\mathbb{R}}\\!{\rm
P}^{2}\times{\mathbb{R}}\\!{\rm P}^{2}\times{\mathbb{R}}\\!{\rm P}^{2}),$
(125)

where the structured group of the mapping is clear and will not required
below.

–2. On a closed polyhedron (a component of self-intersection polyhedron is
closed and does not contain boundaries singular points) ${\bf
N}_{b\times\dot{{b}}}$, outside the critical subpolyhedron of dimension not
great then $12+d$, $d=2$, the structuring mapping admits the following
reduction:

$\displaystyle\eta_{b\times\dot{{b}}}:{\bf N}_{b\times\dot{{b}}}\to K({\bf
J}_{b\times\dot{{b}}},1)\operatorname{\rtimes}_{\mu_{b\times\dot{{b}}},\chi^{[2]}}S^{1}\times
S^{1}\times{\mathbb{R}}\\!{\rm P}^{2},$ (126)

where the structuring mapping corresponds to the group (89).

Additionally, the closed subpolyhedron ${\bf N}_{b\times\dot{{b}}}$ contains a
fundamental class $[{\bf N}_{b\times\dot{{b}}}]$. The image of the fundamental
by the structuring mapping (126) is estimated (is non-trivial) as following.
Take an arbitrary submanifold ${\mathbb{R}}\\!{\rm