#  $SU(2)$-bundles over highly connected $8$-manifolds

Samik Basu  Stat Math Unit, Indian Statistical Institute Kolkata 700108, India
<EMAIL_ADDRESS>,  Aloke Kr Ghosh  Stat Math Unit, Indian Statistical
Institute Kolkata 700108, India<EMAIL_ADDRESS>and  Subhankar Sau
Stat Math Unit, Indian Statistical Institute Kolkata 700108, India
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we analyze the possible homotopy types of the total space of a
principal $SU(2)$-bundle over a $3$-connected $8$-dimensional Poincaré duality
complex. Along the way, we also classify the $3$-connected $11$-dimensional
complexes $E$ formed from a wedge of $S^{4}$ and $S^{7}$ by attaching a
$11$-cell.

###### Key words and phrases:

Poincaré duality complexes, principal bundles, sphere fibrations, loop spaces.

###### 2020 Mathematics Subject Classification:

Primary: 55R25, 57P10; Secondary: 57R19, 55P35.

##  1\. Introduction

This paper explores $SU(2)$-bundles over $8$-manifolds, aiming for results
akin to those about circle bundles over $4$-manifolds [8, 4]. In the case of
simply connected $4$-manifolds, the results are established by leveraging the
classification of simply connected $5$-manifolds achieved by Smale [12] and
Barden [3].

A circle bundle $S^{1}\to X\to M$ over a simply connected $4$-manifold $M$ is
classified by $\alpha\in H^{2}(M)$, the total space $X(\alpha)$ is simply
connected if $\alpha$ is primitive, and there are only two possibilities of
$X(\alpha)$ via the classification of simply connected $5$-manifolds.
Explicitly, we have [8, Theorem 2]

  1. (1)

For every simply connected $4$-manifold $M$, there is a circle bundle
$\alpha$, such that $X(\alpha)$ is homotopy equivalent to a connected sum of
$S^{2}\times S^{3}$. If $M$ is spin, among primitive $\alpha$, this is the
only possibility.

  2. (2)

For a simply connected $4$-manifold $M$ which is not spin and a circle bundle
$\alpha$ over it, $X(\alpha)$ is either homotopy equivalent to a connected sum
of $S^{2}\times S^{3}$, or to a connected sum of $S^{2}\times S^{3}$ and
another manifold $B$. The manifold $B$ is (up to diffeomorphism unique) a non-
spin simply connected $5$-manifold whose homology is torsion-free, and
$\mbox{Rank}(H_{2}(B))=1$.

The results of Smale and Barden are geometric in nature, and do not generalize
easily to higher dimensions. Using homotopy theoretic methods, it was possible
to construct sphere fibrations [7] over highly connected Poincaré-duality
complexes possibly by inverting a few primes or in high enough rank. Among
these sphere fibrations, the only case where they could be principal bundles
was in dimension $8$, and the question whether they may be realized as such
was left unresolved.

In this paper, we consider principal $SU(2)$-bundles, noting that
$SU(2)=S^{3}$ is the only case apart from the circle where the sphere is a Lie
group. The base space of the $SU(2)$-bundle which is appropriate for making a
similar analysis is a highly connected $8$-manifold. More precisely, we
consider Poincaré duality complexes $M$ ($8$-dimensional) that are
$3$-connected. These are obtained by attaching a single $8$-cell to a buoquet
of $4$-spheres. We denote

${\mathcal{P}}{\mathcal{D}}_{3}^{8}=\mbox{ the collection of
}3\mbox{-connected }8\mbox{-dimensional Poincar\'{e} duality complexes. }$

The notation $M_{k}\in{\mathcal{P}}{\mathcal{D}}_{3}^{8}$ assumes that
$\mbox{Rank}(H_{4}(M_{k}))=k$. The attaching map of the $8$-cell is denoted by
$L(M_{k})$, and is of the form (once we have chosen a basis
$\\{\alpha_{1},\dots,\alpha_{k}\\}$ of $\pi_{4}(M_{k})\cong{\mathbb{Z}}^{k}$)

(1.1) ${}L(M_{k})=\sum_{1\leq i<j\leq
k}g_{i,j}[\alpha_{i},\alpha_{j}]+\sum_{i=1}^{k}g_{i,i}\nu_{i}+\sum_{i=1}^{k}l_{i}\nu^{\prime}_{i}.$

The matrix $\big{(}(g_{i,j})\big{)}$ is the matrix of the intersection form,
and hence, is invertible. The notation $\nu_{i}$ stands for
$\alpha_{i}\circ\nu$ and $\nu^{\prime}_{i}$ for $\alpha_{i}\circ\nu^{\prime}$.
Here $\nu$ is the Hopf map, and $\nu^{\prime}\in\pi_{7}(S^{4})$ is the
generator for the ${\mathbb{Z}}/(12)$ factor satisfying
$[\iota_{4},\iota_{4}]=2\nu+\nu^{\prime}$. For such complexes, we consider

${\mathcal{P}}(M_{k})=\mbox{ the set of principal }SU(2)\mbox{-bundles
}E(\psi)\stackrel{{\scriptstyle\psi}}{{\to}}M_{k}\mbox{ such that
}E(\psi)\mbox{ is }3\mbox{-connected}.$

The bundle $\psi$ is classified by a primitive element $\psi\in H^{4}(M_{k})$,
which satisfies a criterion (see Proposition 4.9). In this context, we first
encounter the question whether ${\mathcal{P}}(M_{k})$ is non-empty. We prove
(see Proposition 4.3 and Proposition 5.2)

######  Theorem A.

For $k\geq 3$, the set ${\mathcal{P}}(M_{k})$ is non-empty.

For $k=2$, there are examples where ${\mathcal{P}}(M_{k})$ is empty. This
means that for every principal $SU(2)$-bundle over such complexes, the total
space has non-trivial $\pi_{3}$. The idea here is that the existence of $\psi$
is given by a certain equation in $k$ variables, and solutions exist once $k$
is large enough.

In the case of simply connected $4$-manifolds, the first kind of
classification of circle bundles is the result of Giblin[9] which states

$\mbox{ If }X=S^{2}\times S^{2},\mbox{ then }X(\alpha)\simeq S^{2}\times
S^{3}\mbox{ for any primitive }\alpha.$

We also have an analogous result in the $8$-dimensional case

$\mbox{ If }\psi\in{\mathcal{P}}(S^{4}\times S^{4}),\mbox{ then }E(\psi)\simeq
S^{4}\times S^{7}.$

In fact, this fits into a more general framework. We call a manifold
$M_{k}\in{\mathcal{P}}{\mathcal{D}}_{3}^{8}$ stably trivial if $L(M_{k})$ is
stably null-homotopic (that is the stable homotopy class of
$L(M_{k}):S^{7}\to\big{(}S^{4}\big{)}^{\vee k}$ is $0$). In terms of (1.1),
this means for every $i$, $g_{i,i}-2l_{i}\equiv 0\pmod{24}$. We have the
following theorem (see Proposition 4.7)

######  Theorem B.

Suppose $M_{k}$ is stably trivial. Then, for every
$\psi\in{\mathcal{P}}(M_{k})$, $E(\psi)\simeq\\#^{k-1}S^{4}\times S^{7}$, a
connected sum of $k-1$ copies of $S^{4}\times S^{7}$.

This directly generalizes the result for circle bundles over simply connected
$4$-manifolds that are spin (identifying the spin manifolds as those whose
attaching map is stably null).

We proceed towards a more general classification of the homotopy type of the
space $E(\psi)$ for $\psi\in{\mathcal{P}}(M_{k})$. Let
${\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ be the class of $3$-connected
$11$-dimensional Poincaré duality complexes $E$ such that
$E\setminus\\{pt\\}\simeq$ a wedge of $S^{4}$ and $S^{7}$. We first observe
that $E(\psi)\in{\mathcal{P}}{\mathcal{D}}_{4,7}^{11}$ (see Proposition 3.2),
and we try to address the question of the classification of complexes in
${\mathcal{P}}{\mathcal{D}}_{4,7}^{11}$ up to homotopy equivalence. The
homology of such complexes $E$ is given by

$H_{m}(E)\cong\begin{cases}{\mathbb{Z}}&m=0,11\\\ {\mathbb{Z}}^{r}&m=4,7\\\
0&\mbox{otherwise}.\end{cases}$

We denote the number $r$ by $\mbox{Rank}(E)$. The classification works
differently for $r=1$, and for $r\geq 2$. Table 1 lists the various
possibilities for $r=1$. For $r\geq 2$, $E$ is a connected sum of copies of
$S^{4}\times S^{7}$, and the complexes $E_{\lambda,\epsilon,\delta}$ defined
below. Note that

$\displaystyle\pi_{10}(S^{4}\vee S^{7})$
$\displaystyle\cong\pi_{10}(S^{4})\oplus\pi_{10}(S^{7})\oplus\pi_{10}(S^{10})$
$\displaystyle\cong{\mathbb{Z}}/(24)\\{x\\}\oplus{\mathbb{Z}}/(3)\\{y\\}\oplus{\mathbb{Z}}/(24)\\{\nu_{7}\\}\oplus{\mathbb{Z}}\\{[\iota_{4},\iota_{7}]\\}.$

Here, $x=\nu\circ\nu_{7}$ and $y=\nu^{\prime}\circ\nu_{7}$. Let

$\phi_{\lambda,\epsilon,\delta}=[\iota_{4},\iota_{7}]+\lambda(\iota_{7}\circ\nu_{7})+\epsilon(\iota_{4}\circ
x)+\delta(\iota_{4}\circ y),$

and define,

$E_{\lambda,\epsilon,\delta}=(S^{4}\vee
S^{7})\cup_{\phi_{\lambda,\epsilon,\delta}}D^{11}.$

The attaching map of the top cell of $E$ takes the form

$L(E):S^{10}\to\big{(}S^{4}\vee S^{7}\big{)}^{\vee r}.$

The stable homotopy class of $L(E)$ lies in

$\pi_{10}^{s}\Big{(}\big{(}S^{4}\vee S^{7}\big{)}^{\vee
r}\Big{)}\cong\big{(}{\mathbb{Z}}/(24)\\{\nu\\}\oplus{\mathbb{Z}}/(2)\\{\nu^{2}\\}\big{)}^{\oplus
r}.$

This takes the form $\lambda_{s}\beta\circ\nu+\epsilon_{s}\alpha\circ\nu^{2}$
for some $\beta\in\pi_{7}\Big{(}\big{(}S^{4}\vee S^{7}\big{)}^{\vee r}\Big{)}$
and $\alpha\in\pi_{4}\Big{(}\big{(}S^{4}\vee S^{7}\big{)}^{\vee r}\Big{)}$. Up
to a change of basis we may assume that $\lambda_{s}\mid 24$, and if
$\lambda_{s}$ is even, $\epsilon_{s}\in{\mathbb{Z}}/(2)$. These numbers are
invariant over the homotopy equivalence class of $E$, and are denoted by
$\lambda_{s}(E)$, and $\epsilon_{s}(E)$ (defined only if $\lambda_{s}(E)$ is
even). We use these invariants to classify the homotopy types of elements in
${\mathcal{P}}{\mathcal{D}}_{4,7}^{11}$ (see Theorem 2.17)

######  Theorem C.

Let $E\in{\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$. Then the homotopy type of $E$
is determined by the following.

  1. (1)

If $\lambda_{s}(E)$ is even and $\epsilon_{s}(E)=0$, then

$E\simeq\\#^{r-1}E_{0,0,0}\\#E_{\lambda_{s},\epsilon,\delta}\quad\text{where
}\epsilon\equiv 0\pmod{2}.$

  2. (2)

If $\lambda_{s}(E)$ is even and $\epsilon_{s}(E)=1$, then

$\displaystyle
E\simeq\\#^{r-1}E_{0,0,0}\\#E_{\lambda_{s},\epsilon,\delta}\quad$
$\displaystyle\text{where }\epsilon\equiv 1\pmod{2}$ or $\displaystyle
E\simeq\\#^{r-2}E_{0,0,0}\\#E_{0,1,0}\\#E_{\lambda_{s},\epsilon,\delta}\quad$
$\displaystyle\text{where }\epsilon\equiv 0\pmod{2}.$

  3. (3)

If $\lambda_{s}(E)$ is odd, then

$E\simeq\\#^{r-1}E_{0,0,0}\\#E_{\lambda_{s},\epsilon,\delta}\quad\text{or}\quad
E\simeq\\#^{r-2}E_{0,0,0}\\#E_{0,1,0}\\#E_{\lambda_{s},\epsilon,\delta}.$

Further given $\lambda_{s}$, the choices of $\epsilon$ and $\delta$ are those
which are mentioned in Table 1.

We see that in the list given in Table 1, for certain cases the homotopy type
of $E$ is determined by $\lambda_{s}(E)$ and $\epsilon_{s}(E)$. This happens
if $\lambda_{s}(E)=$ $0$, or $12$. We also observe that the homotopy type of
$\Omega E$ depends only on the rank $r$. Now, we look at
$M_{k}\in{\mathcal{P}}{\mathcal{D}}_{3}^{8}$, and try to determine the set of
homotopy equivalence classes of $E(\psi)$ for $\psi\in{\mathcal{P}}(M_{k})$.
In this process, we determine a formula for
$\lambda(\psi):=\lambda_{s}(E(\psi))$ (Proposition 3.13), and using this we
determine the set of possible values of $\lambda_{s}(\psi)$ for
$\psi\in{\mathcal{P}}(M_{k})$. The stable homotopy class of $L(M_{k})$ lies in

$\pi_{7}^{s}\Big{(}(S^{4})^{\vee
k}\Big{)}\cong\big{(}{\mathbb{Z}}/(24)\\{\nu\\}\big{)}^{\oplus k}.$

This takes the form $\sigma_{s}\alpha\circ\nu$ for some
$\alpha\in\pi_{4}\big{(}(S^{4})^{\vee k}\big{)}$, and up to a change of basis
for $k\geq 2$, $\sigma(M_{k}):=\gcd(\sigma_{s},24)$ is an invariant of the
stable homotopy type of $M_{k}$. Other than $k$ and $\sigma(M_{k})$, the
explicit stable homotopy class of $\alpha$ above yields a linear map
$\tau:H^{4}(M_{k})\to{\mathbb{Z}}/(24)$ given by
$\tau(\psi)=\psi(\sigma_{s}\alpha)$. We use the invariants $k$,
$\sigma(M_{k})$, $\tau$, and the intersection form to completely determine the
possibilities of $\lambda(\psi)$ for $\psi\in{\mathcal{P}}(M_{k})$. (see
Theorem 3.14, Proposition 4.10, Theorem 4.11, Theorem 4.14, and Theorem 5.5)

######  Theorem D.

For any $\psi\in{\mathcal{P}}(M_{k})$, $\lambda(\psi)$ is a multiple of
$\sigma(M_{k})$ ($\pmod{24}$). Conversely, the multiples of $\sigma(M_{k})$
that may be achieved are described as follows

  1. (1)

If the intersection form of $M_{k}$ is odd and $k\geq 7$, then
$\\{\lambda(\psi)\mid\psi\in{\mathcal{P}}(M_{k})\\}$ equals the set of
multiples of $\sigma(M_{k})\pmod{24}$.

  2. (2)

If the intersection form of $M_{k}$ is even, each
$\psi\in{\mathcal{P}}(M_{k})$ satisfies $\epsilon_{s}(\psi)\equiv 0\pmod{2}$.

  3. (3)

If $k\geq 7$, there are $\psi\in{\mathcal{P}}(M_{k})$ such that
$\lambda(\psi)=\sigma(M_{k})$, and also there are
$\psi\in{\mathcal{P}}(M_{k})$ such that $\lambda(\psi)=3\sigma(M_{k})$.

  4. (4)

If $\sigma(M_{k})\equiv 2$, or $4\pmod{8}$ for $k\geq 5$, there is a
$\psi\in{\mathcal{P}}(M_{k})$ such that $\lambda(\psi)\equiv 0\pmod{8}$ if and
only if the complex satisfies hypothesis $(H_{8})$.

  5. (5)

If $\sigma(M_{k})\equiv 2\pmod{8}$ for $k\geq 5$, there is a
$\psi\in{\mathcal{P}}(M_{k})$ such that $\lambda(\psi)\equiv 4\pmod{8}$ if and
only if the complex satisfies hypothesis $(H_{4})$.

For lower values of $k$, we do not get systematic results like the above. That
is, the set $\\{\lambda(\psi)\mid\psi\in{\mathcal{P}}(M_{k})\\}$ is not
completely determined by $\sigma(M_{k})$, $k$, $\tau$, and the intersection
form. Theorem D implies that there are certain $M_{k}$ whose intersection form
is even and there is no $\psi\in{\mathcal{P}}(M_{k})$ such that
$E(\psi)\simeq\\#^{k-1}S^{4}\times S^{7}$, however if the intersection form is
odd, then for $k\geq 7$, there is a principal bundle
$SU(2)\to\\#^{k-1}(S^{4}\times S^{7})\to M_{k}$.

######  1.2.

Organization.  In §2, we prove a classification result for certain
$3$-connected $11$-dimensional complexes which proves Theorem C. In §3, we
prove formulas relating the stable homotopy type of the total space to that of
the base using the Chern character. The results for manifolds with even
intersection form are proved in §4, and those with odd intersection form in
§5.

##  2\. Homotopy classification of certain $3$-connected $11$-complexes

We study $3$-connected $11$-dimensional Poincaré duality complexes $E$ such
that $E\setminus\\{pt\\}$ is homotopic to a wedge of copies of $S^{4}$ and
$S^{7}$. We write ${\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ for the collection
of such complexes. Our target in this section is to analyze them up to
homotopy equivalence. We show that these are classified by numbers $\lambda$,
$\epsilon$ and $\delta$ which are explained in detail below.

######  2.1.

The rank one case.  Let $E\in{\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ be such
that $E\setminus\\{pt\\}\simeq S^{4}\vee S^{7}$, that is,
$\mbox{Rank}(H_{4}(E))=1$. The homotopy type of $E$ is determined by the
attaching map of the top cell, which is an element of

$\pi_{10}(S^{4}\vee
S^{7})\cong\pi_{10}(S^{4})\oplus\pi_{10}(S^{7})\oplus\pi_{10}(S^{10}).$

This must be of the form

(2.2)
${}\phi_{\lambda,\epsilon,\delta}=[\iota_{4},\iota_{7}]+\lambda(\iota_{7}\circ\nu_{7})+\epsilon(\iota_{4}\circ
x)+\delta(\iota_{4}\circ y).$

The total space associated with $\phi_{\lambda,\epsilon,\delta}$ is denoted by

$E_{\lambda,\epsilon,\delta}=(S^{4}\vee
S^{7})\cup_{\phi_{\lambda,\epsilon,\delta}}D^{11}.$

Note that as $(-\iota_{4})\circ\nu=\nu+\nu^{\prime}$, we have
$(-\iota_{4})\circ x=x+y.$ For given any $\lambda,\epsilon$ and $\delta$; we
observe the effect of the self homotopy equivalences on
$E_{\lambda,\epsilon,\delta}$ as follows

$\displaystyle\iota_{4}\mapsto-\iota_{4},\quad\iota_{7}\mapsto-\iota_{7}$
$\displaystyle\implies
E_{\lambda,\epsilon,\delta}\xrightarrow[]{\simeq}E_{-\lambda,\epsilon,\epsilon-\delta},$
$\displaystyle\iota_{4}\mapsto\iota_{4},\quad\iota_{7}\mapsto\iota_{7}+a\iota_{4}\circ\nu$
$\displaystyle\implies
E_{\lambda,\epsilon,\delta}\xrightarrow[]{\simeq}E_{\lambda,\epsilon+(\lambda+2)a,\delta},$
$\displaystyle\iota_{4}\mapsto\iota_{4},\quad\iota_{7}\mapsto\iota_{7}+b\iota_{4}\circ\nu^{\prime}$
$\displaystyle\implies
E_{\lambda,\epsilon,\delta}\xrightarrow[]{\simeq}E_{\lambda,\epsilon-4b,\delta+(\lambda+1)b}.$

This leads us to homotopy equivalences between $E_{\lambda,\epsilon,\delta}$’s
depending on the choice of $\lambda\in\pi_{10}(S^{7})\cong{\mathbb{Z}}/{24}$.
Table 1 lists the different homotopy types in
${\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ of rank $1$.

$\lambda$ |  $\\#E_{\lambda,\epsilon,\delta}$’s |  $E_{\lambda,\epsilon,\delta}$’s
---|---|---
$0$ | $2$ |  $E_{0,0,0}$, $E_{0,1,0}$
$1$ | $3$ |  $E_{1,0,0}$, $E_{1,1,0}$, $E_{1,2,0}$
$2$ | $12$ |  $E_{2,0,0}$, $E_{2,1,0}$, $E_{2,2,0}$, $E_{2,3,0}$, $E_{2,0,1}$, $E_{2,1,1}$,
|  |  $E_{2,2,1}$, $E_{2,3,1}$, $E_{2,0,2}$, $E_{2,1,2}$, $E_{2,2,2}$, $E_{2,3,2}$
$3$ | $1$ | $E_{3,0,0}$
$4$ | $6$ |  $E_{4,0,0}$, $E_{4,1,0}$, $E_{4,2,0}$, $E_{4,3,0}$, $E_{4,4,0}$, $E_{4,5,0}$
$5$ | $3$ |  $E_{5,0,0}$, $E_{5,0,1}$, $E_{5,0,2}$
$6$ | $4$ |  $E_{6,0,0}$, $E_{6,1,0}$, $E_{6,2,0}$, $E_{6,3,0}$
$7$ | $3$ |  $E_{7,0,0}$, $E_{7,1,0}$, $E_{7,2,0}$
$8$ | $6$ |  $E_{8,0,0}$, $E_{8,1,0}$, $E_{8,0,1}$, $E_{8,1,1}$, $E_{8,0,2}$, $E_{8,1,2}$
$9$ | $1$ | $E_{9,0,0}$
$10$ | $12$ |  $E_{10,0,0}$, $E_{10,1,0}$, $E_{10,2,0}$, $E_{10,3,0}$, $E_{10,4,0}$, $E_{10,5,0}$,
|  |  $E_{10,6,0}$, $E_{10,7,0}$, $E_{10,8,0}$, $E_{10,9,0}$, $E_{10,10,0}$, $E_{10,11,0}$
$11$ | $3$ |  $E_{11,0,0}$, $E_{11,0,1}$, $E_{11,0,2}$
$12$ | $2$ |  $E_{12,0,0}$, $E_{12,1,0}$
Table 1. Homotopy equivalence classes of $E_{\lambda,\epsilon,\delta}.$

######  2.4.

A simplification of the attaching map.  We simplify and reduce the attaching
map of the top cell of $E$.

######  Proposition 2.5.

Let $E\in{\mathcal{P}}{\mathcal{D}}_{4,7}^{11}$ with $\mbox{Rank}(E)=k-1$. The
attaching map $\phi$ of the top cell of $E$ as in (2.2) can be reduced, up to
homotopy, to the following form

$\phi=\sum_{i=1}^{k-1}[\iota_{4}^{i},\iota_{7}^{i}]+\sum_{i=1}^{k-1}\lambda_{i}\iota_{7}^{i}\circ\nu_{(7)}+\sum_{i=1}^{k-1}s_{i}\nu_{i}\circ\nu_{(7)}+\sum_{i=1}^{k-1}r_{i}\nu_{i}^{\prime}\circ\nu_{(7)}.$

###### Proof.

By Hilton-Milnor decomposition, we have the following equivalence

$\pi_{10}((S^{4}\vee
S^{7})^{k-1})\cong\pi_{10}(S^{4})^{\oplus(k-1)}\oplus\pi_{10}(S^{7})^{\oplus(k-1)}\oplus\pi_{10}(S^{7})^{\oplus{k-1\choose
2}}\oplus\\\
\pi_{10}(S^{10})^{\oplus{(k-1)\times(k-1)}}\oplus\pi_{10}(S^{10})^{\oplus{k-1\choose
3}}.$

We choose $\eta_{1},\dots,\eta_{k-1}\in\pi_{4}(E)$ and
$\gamma_{1},\dots,\gamma_{k-1}\in\pi_{7}(E)$ such that they correspond to the
homology generators, say $\tilde{\eta}_{1},\dots,\tilde{\eta}_{k-1}\in
H_{4}(E)$ and $\tilde{\gamma}_{1},\dots,\tilde{\gamma}_{k-1}\in H_{7}(E)$ such
that

(2.6)
${}\tilde{\eta}_{i}^{*}\cup\tilde{\gamma}_{j}^{*}=\begin{cases}1\quad\text{if
}i=j,\\\ 0\quad\text{if }i\neq j.\end{cases}$

Let $\tilde{f}\colon(S^{4}\vee S^{7})^{\vee k-1}\to E$ be the inclusion which
sends $\iota_{4}^{i}\mapsto\eta_{i}$ and $\iota_{7}^{i}\mapsto\gamma_{i}$ for
$1\leq i\leq k-1$. Then $\tilde{f}\circ\phi\in\pi_{10}(E)$ whose image under
the map $\rho\colon\pi_{10}(E)\rightarrow\pi_{9}(\Omega E)\rightarrow
H_{9}(\Omega E)$ is $0$ in the tensor algebra
$T(\tilde{\eta}_{1},\dots,\tilde{\eta}_{k-1},\tilde{\gamma}_{1},\dots,\tilde{\gamma}_{k-1})/(\sum_{i=1}^{k-1}[\tilde{\eta}_{i},\tilde{\gamma}_{i}])$.

The attaching map $\phi$ may contain triple Whitehead products as
$[\iota^{4}_{i},[\iota^{4}_{j},\iota^{4}_{\ell}]]$, the Whitehead products of
the form $[\iota_{i}^{4},\iota_{j}^{7}]$ for $i\neq j$, the terms involving
$[\iota_{i}^{4},\iota_{j}^{4}]\circ\nu_{7}$ and
$[\iota_{i}^{4},\iota_{j}^{4}]\circ\nu_{7}^{\prime}$. The triple Whitehead
product maps injectively to the loop homology and hence they can not occur in
the attaching map. The cup product formula in (2.6) implies that there is no
Whitehead product of the form $[\iota_{i}^{4},\iota_{j}^{7}]$ for $i\neq j$.
If $[\iota_{i}^{4},\iota_{j}^{4}]\circ\nu_{7}$ and
$[\iota_{i}^{4},\iota_{j}^{4}]\circ\nu_{7}^{\prime}$ appear in the attaching
map, we update the map $\tilde{f}$ by appropriately sending
$\iota_{i}^{7}\mapsto\gamma_{i}-\eta_{j}\circ\nu^{\prime}$,
$\iota_{i}^{7}\mapsto\gamma_{i}-\eta_{j}\circ\nu$ and
$\iota_{j}^{7}\mapsto\gamma_{j}-[\eta_{i},\eta_{j}]$ to get the desired form
of the attaching map. ∎

We note that the composition is given by

$S^{10}\to(S^{4}\vee S^{7})^{\vee k-1}\to(S^{7})^{\vee k-1}$

which is an element of $(\pi_{10}S^{7})^{\oplus
k-1}\cong({\mathbb{Z}}/{24}\\{\nu\\})^{\oplus k-1}$. This can be calculated
using the real $e$-invariant, see [2]. We use this to reduce Proposition 2.5
to the case $\lambda_{i}=0$ for $i\leq k-2$.

######  Proposition 2.7.

Let $E\in{\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ and $\mbox{Rank}(E)=k-1$. Then
the attaching map $\phi$ of the top cell of $E$ can be reduced to the
following form

(2.8)
${}\phi=\sum_{i=1}^{k-1}[\iota_{4}^{i},\iota_{7}^{i}]+\lambda\iota_{7}^{k-1}\circ\nu_{(7)}+\sum_{i=1}^{k-1}\epsilon_{i}\nu_{i}\circ\nu_{(7)}+\sum_{i=1}^{k-1}\delta_{i}\nu_{i}^{\prime}\circ\nu_{(7)}.$

As a consequence
$E\simeq\\#_{i=1}^{k-2}E_{0,\epsilon_{i},\delta_{i}}\\#E_{\lambda,\epsilon_{k-1},\delta_{k-1}}$.

###### Proof.

Let $\tau\colon
H^{7}(E)\cong\mathbb{Z}\\{\tilde{\gamma}_{1}^{*},\tilde{\gamma}_{2}^{*},\dots,\tilde{\gamma}_{k-1}^{*}\\}\rightarrow\mathbb{Z}/24$
be the linear map defined by

$\tau(\tilde{\gamma}_{i}^{*})=e(r_{i}\circ\phi),\quad\quad\text{for }1\leq
i\leq k-1,$

where $e$ denotes for the real $e$-invariant and $r_{i}:(S^{4}\vee
S^{7})^{\vee k-1}\to S^{7}$ is the retraction onto the $i$-th factor. Let
$\tilde{\tau}\colon H^{7}(E)\rightarrow\mathbb{Z}$ be the lift of $\tau$ and
$\lambda=\gcd(\tilde{\tau}(\tilde{\gamma}_{1}^{*}),\dots,\tilde{\tau}(\tilde{\gamma}_{k-1}^{*})$.
Then we can change the generators
$\tilde{\gamma}_{1},\dots,\tilde{\gamma}_{k-1}$ such that
$\tilde{\tau}(\tilde{\gamma}_{i})=0$ for $1\leq i<k-1$ and
$\tilde{\tau}(\tilde{\gamma}_{k-1})=\lambda.$ So, for a suitable choice of
dual bases we can have $\phi$ as in (2.8). ∎

######  2.9.

A general classification up to homotopy.  We now proceed to the classification
of elements in ${\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$. In the connected sum
$E_{\lambda_{1},\epsilon_{1},\delta_{1}}\\#E_{\lambda_{2},\epsilon_{2},\delta_{2}}$,
we transform $\alpha_{1}=\alpha_{1}^{\prime}+\alpha_{2}^{\prime}$ and
$\alpha_{2}=\alpha_{2}^{\prime}$ by $A=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}$. If $2|\lambda_{2}$, then we transform

$\beta_{1}=\beta_{1}^{\prime}-\epsilon_{1}\nu_{2}-\frac{\lambda_{2}\epsilon_{1}}{2}\nu_{2}^{\prime},\quad\beta_{2}=-\beta_{1}^{\prime}+\beta_{2}^{\prime}-\epsilon_{1}[\alpha_{1}^{\prime},\alpha_{2}^{\prime}].$

Hence by the following expression

$\displaystyle[\alpha_{1},\beta_{1}]+[\alpha_{2},\beta_{2}]+\lambda_{1}\beta_{1}\circ\nu_{(7)}+\lambda_{2}\beta_{2}\circ\nu_{7}+\epsilon_{1}x_{1}+\epsilon_{2}x_{2}+\delta_{1}y_{1}+\delta_{2}y_{2}$
$\displaystyle=$
$\displaystyle[\alpha_{1}^{\prime},\beta_{1}^{\prime}]+[\alpha_{2}^{\prime},\beta_{2}^{\prime}]+(\lambda_{1}-\lambda_{2})\beta_{1}^{\prime}\circ\nu_{(7)}+\lambda_{2}\beta_{2}^{\prime}\circ\nu_{7}+\epsilon_{1}x_{1}+(-\epsilon_{1}+\epsilon_{2}+2\lambda_{2}\epsilon_{1}-\lambda_{1}\epsilon_{1})x_{2}$
$\displaystyle\hskip
199.16928pt\delta_{1}y_{1}+(\delta_{1}+\delta_{2}+\lambda_{2}\epsilon_{1}(1+\lambda_{1}))y_{2},$

we conclude

(2.10)
${}E_{\lambda_{1},\epsilon_{1},\delta_{1}}\\#E_{\lambda_{2},\epsilon_{2},\delta_{2}}\simeq
E_{\lambda_{1}-\lambda_{2},\epsilon_{1},\delta_{1}}\\#E_{\lambda_{2},\epsilon_{2}+\epsilon_{1}(2\lambda_{2}-\lambda_{1}-1),\delta_{1}+\delta_{2}+(1+\lambda_{1})\epsilon_{1}\lambda_{2}}\quad\quad\text{when
}2|\lambda_{2}.$

######  Proposition 2.11.

For any unit $a\in{\mathbb{Z}}/24$, we have homotopy equivalence

$E_{\lambda,\epsilon,\delta}\\#E_{0,0,0}\simeq\begin{cases}E_{a\lambda,\epsilon,\delta}\\#E_{0,0,0}\quad&\text{if
}a\equiv 1\pmod{3}\\\ E_{-a\lambda,\epsilon,\delta}\\#E_{0,0,0}\quad&\text{if
}a\equiv 2\pmod{3}.\end{cases}$

###### Proof.

We transform

$\displaystyle\alpha_{1}=a\alpha_{1}^{\prime}+b\alpha_{2}^{\prime}$
$\displaystyle\beta_{1}=a\beta_{1}^{\prime}-24\beta_{2}^{\prime}-b\epsilon\nu_{2}-24b\epsilon\nu_{1}$
$\displaystyle\alpha_{2}=24\alpha_{1}^{\prime}+a\alpha_{2}^{\prime}$
$\displaystyle\beta_{2}=-b\beta_{1}^{\prime}+a\beta_{2}^{\prime}-\epsilon
b[\alpha_{1}^{\prime},\alpha_{2}^{\prime}]$

with $a^{2}-24b=1$ and calculate
$[\alpha_{1},\beta_{1}]+[\alpha_{2},\beta_{2}]+\lambda\beta\circ\nu_{(7)}+\epsilon
x_{1}+\delta y_{2}$. This gives the homotopy equivalence

(2.12) ${}E_{\lambda,\epsilon,\delta}\\#E_{0,0,0}\\\ \simeq
E_{a\lambda,a^{2}\epsilon,a\delta+{{a}\choose
2}\epsilon}\\#E_{0,-b^{2}\epsilon-b\lambda\epsilon,b\delta+{{b}\choose
2}\epsilon}.$

This proves the proposition except for when $\lambda\equiv 0\pmod{2}$ and
$\epsilon,b\equiv 1\pmod{2}$. In that case, we have $E_{0,1,0}$ instead of
$E_{0,0,0}$ in the second component of the connected sum. From (2.10), we get

(2.13) ${}E_{\lambda,\epsilon,\delta}\\#E_{0,1,0}\simeq
E_{\lambda,\epsilon,\delta}\\#E_{0,1+\epsilon(-\lambda-1),\delta}$

which we apply following the equivalence in (2.12) for $\lambda$ even and
$\epsilon$,$b$ odd. This concludes the proof. ∎

The transformations above further simplify the possibilities of
$E\in{\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ listed in Proposition 2.7.

######  Proposition 2.14.

Let $E\in{\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ and $\mbox{Rank}(E)=k-1$. Then

$E\simeq\\#^{k-3}E_{0,0,0}\\#E_{0,\hat{\epsilon},0}\\#E_{\lambda,\epsilon,\delta}$

for some $\lambda,\epsilon\in\mathbb{Z}/24$, $\delta\in{\mathbb{Z}}/3$,
$\hat{\epsilon}\in{\mathbb{Z}}/2$.

###### Proof.

From Proposition 2.7, we have the attaching map $\phi$ as in (2.8). Repeated
use of the homotopy equivalences in (2.1) and (2.13) gives the reduced form
(2.8) as follows

$\phi=\sum_{i=1}^{k-1}[\iota_{4}^{i},\iota_{7}^{i}]+\lambda\iota_{7}^{k-1}\circ\nu_{(7)}+\hat{\epsilon}x_{k-2}+\epsilon
x_{k-1}+\delta y_{k-1},$

where $\lambda,\epsilon\in\mathbb{Z}/24$, $\delta\in{\mathbb{Z}}/3$,
$\hat{\epsilon}\in{\mathbb{Z}}/2$. ∎

######  Corollary 2.15.

Let $E\in{\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ and $\mbox{Rank}(E)=k-1$. Then
stably we have

(2.16) ${}\Sigma^{\infty}E\simeq\Sigma^{\infty}(S^{4}\vee
S^{7})^{\vee{k-2}}\vee\Sigma^{\infty}Cone(\lambda_{s}(E)\nu_{(11)}+\epsilon_{s}(E)x),$

where $x$ is the generator of the stable homotopy group
$\pi_{14}(S^{8})\cong\mathbb{Z}/2$ and $\lambda_{s}(E)\in\mathbb{Z}/24$,
$\epsilon_{s}(E)\in\mathbb{Z}/2.$ Moreover, for $k-1\geq 2$ if
$\lambda_{s}(E)\equiv 1\pmod{2}$ in (2.16), then
$\epsilon_{s}(E)=0\in\mathbb{Z}/2$.

###### Proof.

From (2.8) and Proposition 2.14,

$\Sigma^{4}\phi=\lambda\iota_{11}^{k-1}\circ\nu_{(11)}+\hat{\epsilon}\iota_{8}^{k-2}\circ\nu_{(11)}+\epsilon\iota_{8}^{k-1}\circ\nu_{(11)}.$

Note that $\nu_{(11)}\in\pi^{s}_{3}(S^{0})\simeq\mathbb{Z}/24$ and
$\Sigma^{4}(\nu\circ\nu_{(7)})\in\pi^{s}_{6}(S^{0})\simeq\mathbb{Z}/2$ are the
generators. If $\hat{\epsilon}=0\in\mathbb{Z}/2$, the result readily follows.
Otherwise, if $\epsilon=1,\hat{\epsilon}=1\in\mathbb{Z}/2$, we apply the
transformation $\iota_{8}^{k-2}+\iota^{k-1}_{8}\mapsto\iota^{k-1}_{8}.$ If
$\epsilon=0,\hat{\epsilon}=1\in\mathbb{Z}/2$, we interchange $\iota_{8}^{k-2}$
and $\iota_{8}^{k-1}$ to deduce the result. ∎

We see that the stable homotopy type of
$E\in{\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ is determined by $\lambda_{s}(E)$
which is a divisor of $24$ and $\epsilon_{s}(E)\in{\mathbb{Z}}/2$. The
following theorem classifies the different homotopy types of $E$ given the
values of $\lambda_{s}$ and $\epsilon_{s}$.

######  Theorem 2.17.

Let $E\in{\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ and $\mbox{Rank}(E)=k-1$. Then
depending on $\lambda_{s}=\lambda_{s}(E)$ and $\epsilon_{s}=\epsilon_{s}(E)$,
the homotopy type of $E$ is determined by the following.

  1. (1)

If $\lambda_{s}$ is even and $\epsilon_{s}=0$, then

$E\simeq\\#^{k-2}E_{0,0,0}\\#E_{\lambda_{s},\epsilon,\delta}\quad\text{where
}\epsilon\equiv\epsilon_{s}\pmod{2}.$

  2. (2)

If $\lambda_{s}$ is even and $\epsilon_{s}=1$, then

$\displaystyle
E\simeq\\#^{k-2}E_{0,0,0}\\#E_{\lambda_{s},\epsilon,\delta}\quad$
$\displaystyle\text{where }\epsilon\equiv 1\pmod{2}$ or $\displaystyle
E\simeq\\#^{k-3}E_{0,0,0}\\#E_{0,1,0}\\#E_{\lambda_{s},\epsilon,\delta}\quad$
$\displaystyle\text{where }\epsilon\equiv 0\pmod{2}.$

  3. (3)

If $\lambda_{s}$ is odd, then

$E\simeq\\#^{k-2}E_{0,0,0}\\#E_{\lambda_{s},\epsilon,\delta}\quad\text{or}\quad
E\simeq\\#^{k-3}E_{0,0,0}\\#E_{0,1,0}\\#E_{\lambda_{s},\epsilon,\delta}.$

Further given $\lambda_{s}$, the choices of $\epsilon$ and $\delta$ are those
which are mentioned in Table 1.

###### Proof.

We write $Y_{1}=\\#^{k-2}E_{0,0,0}\\#E_{\lambda,\epsilon,\delta}$ and
$Y_{2}=\\#^{k-2}E_{0,0,0}\\#E_{\lambda,\epsilon^{\prime},\delta^{\prime}}$,
and let $Y_{1}\stackrel{{\scriptstyle f}}{{\to}}Y_{2}$ be a homotopy
equivalence with homotopy inverse $g$. We show that in this case the pair
$(\epsilon,\delta)$ is related to $(\epsilon^{\prime},\delta^{\prime})$ by the
transformations (2.1). There exists a unique (up to homotopy) factorization
$(S^{7})^{\vee k-1}\to(S^{7}\vee S^{4})^{\vee k-1}\to Y_{2}$ through cellular
approximation $\tilde{f}$ as in the following diagram.

$\textstyle{(S^{7})^{\vee
k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}}$$\scriptstyle{f_{7}}$$\textstyle{\\#^{k-2}E_{0,0,0}\\#E_{\lambda,\epsilon,\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\\#^{k-2}E_{0,0,0}\\#E_{\lambda,\epsilon^{\prime},\delta^{\prime}}}$$\textstyle{(S^{7}\vee
S^{4})^{\vee
k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{(S^{7})^{\vee
k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Now we consider the composition $f_{7}\colon=\rho\circ\tilde{f}$ where $\rho$
is the projection map. From the stable homotopy type, we get an isomorphism

$\textstyle{f_{7}\colon\underset{\displaystyle\overset{\displaystyle{\rotatebox{90.0}{$\cong$}}}{{\mathbb{Z}}\\{\beta_{1},\dots,\beta_{k-1}\\}}}{\pi_{7}((S^{7})^{\vee
k-1})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\underset{\displaystyle\overset{\displaystyle{\rotatebox{90.0}{$\cong$}}}{{\mathbb{Z}}\\{\gamma_{1},\dots,\gamma_{k-1}\\}}}{\pi_{7}((S^{7})^{\vee
k-1})}}$

where $f_{7}(\beta_{k-1})\equiv\gamma_{k-1}\pmod{24}$. Hence, the
corresponding matrix of $f_{7}$ is

$\equiv\begin{pmatrix}&&&0\\\ &A&&\vdots\\\ &&&0\\\ *&\dots&*&1\\\
\end{pmatrix}\pmod{24}$

for some $A\in GL_{k-2}({\mathbb{Z}})$. From the inverse homotopy equivalence
$g\colon Y_{2}\to Y_{1}$, we can construct a corresponding block matrix of
$g_{7}$ similar to that of $f_{7}$ for some $B\in GL_{k-2}({\mathbb{Z}})$
$\pmod{24}$. Through suitable pre-composition of $f_{7}$ and post-composition
of $g_{7}$ we may assume that $A=B=I$ where $I$ is the identity matrix in
$GL_{k-2}({\mathbb{Z}})$. Thus both $f_{7}$ and $g_{7}$ are composition of
shearing maps with $\beta_{k-1}\mapsto\gamma_{k-1}$, that is, they are
composition of maps associated to
$\beta_{i}\mapsto\gamma_{i}+c_{i}\gamma_{k-1}$ for some
$c_{i}\in{\mathbb{Z}}$.

We now consider the map $f$ on the $7$-skeleton

$\textstyle{(S^{4}\vee S^{7})^{\vee
k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{(7)}}$$\scriptstyle{\vee_{i=1}^{k-1}(\alpha_{i}\vee\beta_{i})}$$\textstyle{(S^{4}\vee
S^{7})^{\vee
k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vee_{i=1}^{k-1}(\eta_{i}\vee\gamma_{i})}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y_{2},}$

which takes the form

$\displaystyle\beta_{i}\mapsto\gamma_{i}+c_{i}\gamma_{k-1}+\sum_{j=1}^{k-1}a_{i,j}\nu_{j}+\sum_{j=1}^{k-1}a_{i,j}^{\prime}\nu_{j}^{\prime}+\sum_{\begin{subarray}{c}j=1\\\
\ell=1\\\ j\neq\ell\end{subarray}}^{\begin{subarray}{c}j=k-1\\\
\ell=k-1\end{subarray}}a_{i,j,\ell}[\eta_{j},\eta_{\ell}],$
$\displaystyle\alpha_{i}\mapsto\eta_{i},\quad\text{for }1\leq i\leq k-2,$
$\displaystyle\beta_{k-1}\mapsto\gamma_{k-1}+\sum_{j=1}^{k-1}b_{j}\nu_{j}+\sum_{j=1}^{k-1}b_{j}^{\prime}\nu_{j}^{\prime}+\sum_{\begin{subarray}{c}j=1\\\
\ell=1\\\ j\neq\ell\end{subarray}}^{\begin{subarray}{c}j=k-1\\\
\ell=k-1\end{subarray}}b_{j,\ell}[\eta_{j},\eta_{\ell}],$
$\displaystyle\alpha_{k-1}\mapsto\eta_{k-1}-\sum_{j=1}^{k-2}c_{j}\eta_{j}.$

As $f$ is a homotopy equivalence, we must have that the attaching map of the
$11$-cell of $Y_{1}$ must be carried by $f^{(7)}$ to the attaching map of
$Y_{2}$, that is, $f^{(7)}\circ L(Y_{1})\simeq L(Y_{2})$. We now look at the
coefficients of $\eta_{k-1}\circ x$ and $\eta_{k-1}\circ y$ that arises in
$f^{(7)}\circ L(Y_{1})$ and note that the only terms which contribute to these
coefficients are

$f^{(7)}([\alpha_{k-1},\beta_{k-1}]+\lambda_{s}\beta_{k-1}\circ\nu_{(7)}+\epsilon\alpha_{k-1}\circ
x+\delta\alpha_{k-1}\circ y).$

We now deduce

$\epsilon^{\prime}=(\lambda_{s}+2)b_{k-1}-4b^{\prime}_{k-1}\quad\quad\text{and}\quad\quad\delta^{\prime}=(\lambda_{s}+1)b_{k-1}^{\prime},$

which verifies the result for complexes of the type
$\\#^{k-2}E_{0,0,0}\\#E_{\lambda,\epsilon,\delta}$.

For the remaining cases, we follow the same argument with
$Y_{1}=\\#^{k-3}E_{0,0,0}\\#E_{0,1,0}\\#E_{\lambda_{s},\epsilon,\delta}$ with
$\epsilon$ even if $\lambda_{s}$ is, and
$Y_{2}=\\#^{k-3}E_{0,0,0}\\#E_{0,\hat{\epsilon},0}\\#E_{\lambda_{s},\epsilon^{\prime},\delta^{\prime}}$,
where $\hat{\epsilon}=0$ or $1$. Note that the only terms which contribute to
$\eta_{k-2}\circ x$ are

$f^{(7)}([\alpha_{k-1},\beta_{k-1}]+[\alpha_{k-2},\beta_{k-2}]+\alpha_{k-2}\circ
x+\lambda_{s}\beta_{k-1}\circ\nu_{(7)}+\epsilon\alpha_{k-1}\circ
x+\delta\alpha_{k-1}\circ y).$

A direct computation implies

(2.18) ${}\lambda_{s}b_{k-2}+\epsilon
c_{k-2}^{2}+1\equiv\hat{\epsilon}\pmod{2}.$

First let $\lambda_{s}$ be even and $\epsilon\equiv 0\pmod{2}$. This implies
$\hat{\epsilon}=1$. Finally, let $\lambda_{s}$ is odd. We look at the
coefficients of $[\eta_{k-1},[\eta_{k-2},\eta_{k-1}]]$ and
$[\eta_{k-2},\eta_{k-1}]\circ\nu_{(7)}$ in $f^{(7)}\circ L(Y_{1})$ $\pmod{2}$,
which are $b_{k-2,k-1}+c_{k-2}b_{k-1}-a_{k-2,k-1}$ and
$\lambda_{s}b_{k-2,k-1}+a_{k-2,k-1}+b_{k-2}-c_{k-2}b_{k-1}-\epsilon c_{k-2}$.
Since both these coefficients are zero, we have $b_{k-2}\equiv\epsilon
c_{k-2}\pmod{2}$. Using the relation (2.18), we observe that
$\hat{\epsilon}=1$. The conditions on $\epsilon^{\prime}$ and
$\delta^{\prime}$ are verified analogously as in the previous case. This
completes the proof of the various implications in the theorem. ∎

######  2.19.

The loopspace homotopy type.  We study the loop space homotopy type of
$E\in{\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ with $E^{(7)}\simeq(S^{4}\vee
S^{7})^{\vee k-1}$ and show that the loop space homotopy is independent of the
$\lambda,\epsilon$ and $\delta$ occurring in the attaching map $\phi$ of $E$.

######  Theorem 2.20.

The homotopy type of the loop space of $E$ is a weak product of loop spaces on
spheres, and depends only on $k-1=\mbox{Rank}(H_{4}(E))$. In particular,

$\Omega E\simeq\Omega(\\#^{k-1}(S^{4}\times S^{7})).$

###### Proof.

This follows from the arguments in [5] and [6]. More explicitly, we first
compute the homology of $\Omega E$ via cobar construction, see [1]. In this
case, $H_{*}(E)$ is a coalgebra which is quasi-isomorphic to $C_{*}(E)$, and
hence we may compute the cobar construction of $H_{*}(E)$ and deduce as in [6,
Proposition 2.2]

$H_{*}(\Omega E)\cong
T(a_{1},b_{1},\dots,a_{k-1},b_{k-1})/(\sum[a_{i},b_{i}])$

where $\rho(\alpha_{i})=a_{i}$ and $\rho(\beta_{i})=b_{i}$ with $\rho$ defined
as

$\rho\colon\pi_{r}(E)\cong\pi_{r-1}(\Omega E)\xrightarrow{Hur}H_{r-1}(\Omega
E).$

We then note that $H_{*}(\Omega E)$ is the universal enveloping algebra of the
graded Lie algebra $L(a_{1},b_{1},\dots,a_{k-1},b_{k-1})/(\sum[a_{i},b_{i}])$
where $L$ is the free Lie algebra functor. Now we apply the Poincaré-Birkhoff-
Witt theorem as in [6, Proposition 3.6] to deduce the result. ∎

##  3\. Stable homotopy type of the total space.

In this section, we examine the possible stable homotopy types of the total
space $E$ for a principal $SU(2)$-fibration over
$M_{k}\in{\mathcal{P}}{\mathcal{D}}^{8}_{3}$. We relate this to the stable
homotopy type of $M_{k}$.

Let $f\colon M_{k}\to{\mathbb{H}}P^{\infty}$ be a map such that
$\pi_{4}(f)\colon\pi_{4}(M_{k})\rightarrow\pi_{4}(\mathbb{H}P^{\infty})\cong\mathbb{Z}$
is surjective. This ensures that the homotopy fibre $E(f)$ is $3$-connected
and is a Poincaré duality complex of dimension $11$. One easily deduces

$H_{i}(E(f))=\begin{cases}\mathbb{Z}&\quad\text{}i=0,11\\\
\mathbb{Z}^{\oplus{k-1}}&\quad\text{}i=4,7\\\
0&\quad\text{otherwise}\end{cases}$

from the Serre spectral sequence associated to the fibration $S^{3}\to E(f)\to
M_{k}$. We may now consider a minimal CW-complex structure on $E:=E(f)$ with
$(k-1)$ $4$-cells, $(k-1)$ $7$-cells, and one $11$-cell, see [10, Section
2.2]. The $7$-th skeleton $E^{(7)}$ is, therefore, a pushout

(3.1)
$\textstyle{(S^{6})^{\vee{(k-1)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vee_{i=1}^{k-1}\phi_{i}}$$\textstyle{({D^{7}})^{\vee{(k-1)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(S^{4})^{\vee{(k-1)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{(7)}}$

We now observe that the $\phi_{i}$ are all $0$.

######  Proposition 3.2.

The maps $\phi_{i}\simeq 0$ for $1\leq i\leq k-1$.

###### Proof.

Note that the homotopy class of each of the attaching maps is in
$\pi_{6}(S^{4})^{\oplus{k-1}}$ which lies in the stable range. Applying the
$\Sigma^{\infty}$ functor on the diagram (3.1), we get the cofibre sequence

$\Sigma^{\infty}(S^{6})^{\vee{(k-1)}}\xrightarrow{\Sigma^{\infty}(\vee_{i=1}^{k-1}\phi_{i})}\Sigma^{\infty}(S^{4})^{\vee{(k-1)}}\rightarrow\Sigma^{\infty}E^{(7)}$

which in turn induces a long exact sequence (on the stable homotopy groups)

(3.3)
${}\dots\rightarrow\pi_{7}^{s}(E^{(7)})\rightarrow\pi_{6}^{s}(S^{6})^{\oplus
k-1}\xrightarrow{\Phi}\pi_{6}^{s}(S^{4})^{\oplus
k-1}\rightarrow\pi_{6}^{s}(E^{(7)})\rightarrow\dots$

where $\Phi$ is the induced map of $\vee_{i=1}^{k-1}\phi_{i}$. We have the
following commutative diagram

---
$\textstyle{\pi_{6}^{s}(S^{4})^{\oplus
k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{s}_{6}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{s}_{6}(E^{(7)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\underset{\displaystyle\overset{\displaystyle\shortparallel}{0}}{\pi^{s}_{6}(S^{7})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{6}^{s}(S^{4})^{\oplus
k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{6}^{s}(M_{k}).}$

The second row is a part of a long exact sequence and so, the map
$\pi_{6}^{s}(S^{4})^{\oplus k}\to\pi^{s}_{6}(M_{k})$ is injective. Hence, the
map $\pi_{6}^{s}(S^{4})^{\oplus{(k-1)}}\rightarrow\pi_{6}^{s}(E)$ is
injective, in (3.3) $\Phi$ is forced to be $0$. The result follows. ∎

Proposition 3.2 implies that $E$ fits into the pushout

$\textstyle{S^{10}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{L(E)}$$\textstyle{{D^{11}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(S^{4}\vee
S^{7})^{\vee{(k-1)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E}$

for some $[L(E)]\in\pi_{10}((S^{4}\vee S^{7})^{\vee(k-1)})$. Hence, $E$
belongs to ${\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$. We consider

$\textstyle{S^{10}\xrightarrow{L(E)}(S^{4}\vee
S^{7})^{\vee{(k-1)}}\xrightarrow{}(S^{7})^{\vee{(k-1)}}}$

which is of the form

$\sum_{i=1}^{k-1}\lambda_{i}\iota^{i}_{7}\circ\nu_{7}\in\pi_{10}(S^{7})^{\oplus{k-1}}\cong\bigoplus_{i=1}^{k-1}\mathbb{Z}/24\\{\nu_{7}\\}.$

The coefficients $\lambda_{i}$ can be computed via the $e$-invariant, see [2].
Recall that the $e$-invariant of a map $g\colon S^{11}\to S^{8}$ can be
computed using Chern character. The complex $K$-theoretic $e$-invariant
$e_{{\mathbb{C}}}$ is computed via the diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{\displaystyle\underset{\displaystyle{\rotatebox{90.0}{$\cong$}}}{{\mathbb{Z}}\\{b_{12}\\}}}{\tilde{K}(S^{12})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ch}$$\textstyle{\tilde{K}(\Sigma
C_{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ch}$$\textstyle{\overset{\displaystyle\underset{\displaystyle{\rotatebox{90.0}{$\cong$}}}{{\mathbb{Z}}\\{b_{8}\\}}}{\tilde{K}(S^{8})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ch}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\displaystyle\overset{\displaystyle{\rotatebox{90.0}{$\cong$}}}{{\mathbb{Q}}\\{a_{12}\\}}}{\tilde{H}^{ev}(S^{12};\mathbb{Q})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{H}^{ev}(\Sigma
C_{g};\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underset{\displaystyle\overset{\displaystyle{\rotatebox{90.0}{$\cong$}}}{{\mathbb{Q}}\\{a_{8}\\}}}{\tilde{H}^{ev}(S^{8};\mathbb{Q})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We obtain

$ch(b_{12})=a_{12},\quad ch(b_{8})=a_{8}+r{a}_{12}.$

If $g=\lambda\nu_{(7)}$,
$e_{{\mathbb{C}}}(g)=r=\frac{\lambda}{12}\in{\mathbb{Q}}/{\mathbb{Z}}$, see
[2]. We also have $e_{{\mathbb{C}}}=2e$, where $e$ is computed using the Chern
character of the complexification $c\colon KO\to K$. Therefore, from [2,
Proposition 7.14]

(3.4) ${}b_{8}\in Im(c)\implies e(g)=\frac{r}{2}\in{\mathbb{Q}}/{\mathbb{Z}}.$

######  3.5.

$K$-theory of $M_{k}$.  Consider the Atiyah-Hirzebruch spectral sequence

$E_{2}^{**}=H^{*}(M_{k};\pi_{*}K)\implies K^{*}(M_{k}).$

As $M$ has only even dimensional cells, this has no non-trivial differential
for degree reasons. This gives the additive structure of $K^{0}(M_{k})$. Let
$H^{4}(M_{k})\cong{\mathbb{Z}}\\{\psi_{1},\dots,\psi_{k}\\}$ and
$H^{8}(M_{k})\cong{\mathbb{Z}}\\{z\\}$. Note that if
$\alpha_{1},\dots,\alpha_{k}\in\pi_{4}(M_{k})\cong
H_{4}(M_{k})\cong{\mathbb{Z}}^{k}$ is dual to the basis
$\psi_{1},\dots,\psi_{k}$, in the expression (1.1), the matrix
$\big{(}(g_{i,j})\big{)}$ of the intersection form is related to the cup
product via the equation $\psi_{i}\cup\psi_{j}=g_{i,j}z$. Let
$\tilde{\psi_{1}},\dots,\tilde{\psi_{k}},\tilde{z}$ be classes in $K(M_{k})$
corresponding to $\psi_{1},\dots,\psi_{k},z$ respectively, in the
$E^{\infty}$-page. We have the diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{K}^{0}(S^{8})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{*}}$$\scriptstyle{ch}$$\textstyle{\tilde{K}^{0}(M_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{*}}$$\scriptstyle{ch}$$\textstyle{\tilde{K}^{0}((S^{4})^{\vee{k}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ch}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{H}^{ev}(S^{8};\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{H}^{ev}(M_{k};\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{H}^{ev}((S^{4})^{\vee{k}};\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where

(3.6) ${}ch(\tilde{z})=z,\quad ch(i^{*}(\tilde{\psi_{j}}))={\psi_{j}}(\implies
ch(\tilde{\psi_{j}})={\psi_{j}}+\tau_{j}z),1\leq j\leq k.$

Note that in terms of the formula (1.1), $\psi_{i}\psi_{j}=g_{i,j}z$. We now
use the fact that $ch$ is a ring map to get

$ch(\tilde{\psi_{i}}\tilde{\psi_{j}})=(\psi_{i}+\tau_{i}z)\cup(\psi_{j}+\tau_{j}z)=\psi_{i}\cup\psi_{j}=g_{i,j}z.$

As $ch\colon K(M_{k})\to H^{ev}(M_{k};{\mathbb{Q}})$ is injective, we deduce
$\tilde{\psi_{i}}\tilde{\psi_{j}}=g_{i,j}\tilde{z}$.

Further let $q_{i}\colon(S^{4})^{\vee k}\to S^{4}$ be the retraction onto the
$i$-th factor, and note that $q_{i}\circ L(M)$ is stably equivalent to
$(g_{i,i}-2l_{i})\alpha_{i}\circ\nu_{(7)}$. Thus the $e$-invariant

$e_{{\mathbb{C}}}(q_{i}\circ
L(M))=(g_{i,i}-2l_{i})e_{{\mathbb{C}}}(\Sigma\nu_{(7)})=\frac{g_{i,i}-2l_{i}}{12}\in\mathbb{Q}/\mathbb{Z}.$

We summarize these observations in the following proposition.

######  Proposition 3.7.

Let $M_{k}\in{\mathcal{P}}{\mathcal{D}}^{8}_{3}$ with $L(M_{k})$ as in (1.1).
Then,
${K}^{0}(M_{k})\cong{\mathbb{Z}}\\{1,\tilde{\psi}_{1},\tilde{\psi}_{2},\dots,\tilde{\psi_{k}},\tilde{z}\\}$.
The ring structure is given by

$\tilde{\psi}_{i}\tilde{\psi}_{j}=g_{i,j}\tilde{z}\quad\text{ for }1\leq
i,j\leq k.$

and

$e_{{\mathbb{C}}}(q_{i}\circ
L(M))=\frac{g_{i,i}-2l_{i}}{12}\in\mathbb{Q}/\mathbb{Z}\quad\text{ for }1\leq
i\leq k.$

######  3.8.

$K$-theory of $E(f)$.  The space $E:=E(f)$ is the total space of the sphere
bundle associated to the quaternionic line bundle classified by $f$. We note
that the quaternionic line bundle has a complex structure, and therefore, has
a $K$-orientation.

Let $\gamma_{\mathbb{H}}$ be the canonical $\mathbb{H}$-bundle over
$\mathbb{H}P^{\infty}$. The $K$-theoretic Thom class of $\gamma_{\mathbb{H}}$
is given by $\gamma_{\mathbb{H}}-2\in\tilde{K}^{0}(\mathbb{H}P^{\infty})$. As
the total space of the sphere bundle is contractible, the Thom space
$Th(\gamma_{{\mathbb{H}}})\simeq{\mathbb{H}}P^{\infty}$. Consider the map
$\pi\colon\mathbb{C}P^{\infty}\to\mathbb{H}P^{\infty}$. The pullback bundle
$\pi^{*}(\gamma_{\mathbb{H}})=\gamma_{{\mathbb{C}}}\oplus\bar{\gamma}_{{\mathbb{C}}}$,
where $\gamma_{{\mathbb{C}}}$ is the canonical line bundle over
${{\mathbb{C}}}P^{\infty}$. Therefore,

$\pi^{*}ch(\gamma_{\mathbb{H}}-2)=ch(\pi^{*}(\gamma_{\mathbb{H}}-2))=ch(\gamma_{\mathbb{C}})+ch(\bar{\gamma}_{\mathbb{C}})-2=e^{x}+e^{-x}-2,$

where $H^{\ast}({\mathbb{C}}P^{\infty};{\mathbb{Z}})\cong{\mathbb{Z}}[x]$ and
$x=c_{1}(\gamma_{\mathbb{C}})$. Since $\pi^{*}$ is injective on $H^{*}$, we
may use this formula to deduce

(3.9)
${}ch(\Phi_{K}(\gamma_{\mathbb{H}}))=\Phi_{H}(\gamma_{\mathbb{H}})(1+\frac{y}{12}+\frac{y2}{360}+\dots)$

where $H^{*}({\mathbb{H}}P^{\infty})\cong{\mathbb{Z}}[y]$ and
$\pi^{*}(y)=x^{2}$. We use the Thom isomorphism associated to
$f^{*}(\gamma_{{\mathbb{H}}})$ to deduce the following.

######  Proposition 3.10.

Assume that $f^{*}(y)=\psi_{k}$. Then,

$\tilde{K}^{0}(Th(f^{*}(\gamma_{\mathbb{H}})))\cong\tilde{K}^{0}(M)\\{\Phi_{K}(f^{*}\gamma_{\mathbb{H}})\\}$

as a $\tilde{K}^{0}(M)$-module, and

$ch(\Phi_{K}(f^{*}\gamma_{\mathbb{H}}))=\Phi_{H}(f^{*}\gamma_{\mathbb{H}})(1+\frac{\psi_{k}}{12}+\frac{\psi_{k}^{2}}{360}+\dots).$

###### Proof.

From the naturality of the Chern character as well as the Thom class, we have

$ch(\Phi_{K}(f^{*}(\gamma_{\mathbb{H}}))=ch(f^{*}(\Phi_{K}((\gamma_{\mathbb{H}}))=f^{*}ch(\Phi_{K}(\gamma_{\mathbb{H}})).$

The result follows from (3.9) and the fact $f^{*}(y)=\psi_{k}$. ∎

######  Notation 3.11.

Suppose we are in the situation of Proposition 3.10, that is,
${f^{*}(y)=\psi_{k}}$. We now assume that the basis
$\\{\psi_{1},\dots,\psi_{k}\\}$ is such that one of the following cases occur

  * Case 1

If $\psi_{k}\cup\psi_{k}=\pm z$, then $\psi_{j}\psi_{k}=0$ for $1\leq j\leq
k-1$.

  * Case 2

If $\psi_{k}\cup\psi_{k}=g_{k,k}z$ for some integer $g_{k,k}\neq\pm 1,$ then
assume $\psi_{k-1}\psi_{k}=1$ and $\psi_{j}\psi_{k}=0$ for $1\leq j\leq k-2$.

In terms of these notations we prove the following calculation. Here we
consider the cofibre sequence

$E\rightarrow M_{k}\rightarrow Th(f^{*}(\gamma_{\mathbb{H}}))\rightarrow\Sigma
E\rightarrow\Sigma M\rightarrow\dots$

which demonstrates $K(\Sigma E)$ as a submodule of
$K(Th(f^{*}\gamma_{{\mathbb{H}}}))$ because $K(\Sigma M_{k})=0$. The following
proposition identifies this submodule.

######  Proposition 3.12.

(1) Suppose we are in Case 1 of Notation 3.11, then we have

$\tilde{K}^{0}(\Sigma
E)\cong\mathbb{Z}\\{\Phi_{K}(f^{*}(\gamma_{\mathbb{H}}))\tilde{\psi}_{1},\dots,\Phi_{K}(f^{*}(\gamma_{\mathbb{H}}))\tilde{\psi}_{k-1},\Phi_{K}(f^{*}(\gamma_{\mathbb{H}}))\tilde{z}\\}.$

(2) Suppose we are in Case 2 of Notation 3.11, then

$\tilde{K}^{0}(\Sigma
E)\cong\mathbb{Z}\\{\Phi_{K}(f^{*}(\gamma_{\mathbb{H}}))\tilde{\psi}_{1},\dots,\Phi_{K}(f^{*}(\gamma_{\mathbb{H}}))\tilde{\psi}_{k-2},\Phi_{K}(f^{*}(\gamma_{\mathbb{H}}))(\tilde{\psi}_{k}-g_{k,k}\tilde{\psi}_{k-1}),\Phi_{K}(f^{*}(\gamma_{\mathbb{H}}))\tilde{z}\\}.$

###### Proof.

We have the following short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{K}^{0}(\Sigma
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{K}^{0}(Th(f^{*}(\gamma_{\mathbb{H}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{0}}$$\textstyle{\tilde{K}^{0}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

which implies that

$\tilde{K}^{0}(\Sigma
E)=\mbox{Ker}(s_{0}\colon\tilde{K}^{0}(Th(f^{*}(\gamma_{\mathbb{H}}))\rightarrow\tilde{K}^{0}(M))$

where $s_{0}$ is the restriction along the zero section. Note that $s_{0}$ is
a $\tilde{K}^{0}(M)$-module map. Hence,

$\displaystyle s_{0}(\Phi_{K}(f^{*}\gamma_{\mathbb{H}}))\tilde{z})$
$\displaystyle=e_{K}(f^{*}\gamma_{\mathbb{H}})\tilde{z}=0$ $\displaystyle
s_{0}(\Phi_{K}(f^{*}\gamma_{\mathbb{H}}))\tilde{\psi}_{i})$
$\displaystyle=e_{K}(f^{*}\gamma_{\mathbb{H}})\tilde{\psi}_{i}=g_{ik}\tilde{z},1\leq
i\leq k$

since $e_{K}(f^{*}\gamma_{\mathbb{H}}))=\tilde{\psi}_{k}+m\tilde{z}$ for some
$m$. The result follows from a direct calculation of the kernel and the
assumptions in the respective cases. ∎

We now choose the various maps $\chi_{j}\colon S^{7}\to E$ for $1\leq j\leq
k-1$ such that on $K$-theory they precisely represent the choice of the first
$(k-1)$-elements in the basis of Proposition 3.12. In these terms we calculate
the $e_{{\mathbb{C}}}$-value of the composite

$r_{j}\circ L(E)\colon S^{10}\stackrel{{\scriptstyle L(E)}}{{\to}}(S^{4}\vee
S^{7})^{\vee k-1}\stackrel{{\scriptstyle r_{j}}}{{\to}}S^{7}$

where $r_{j}$ is the restriction onto the $j$-th factor, and $L(E)$ is the
attaching map of the top cell of $E$.

######  Proposition 3.13.

(1) If we are in Case 1, then

$e_{{\mathbb{C}}}(r_{j}\circ
L(E))=\tau_{j}=\frac{g_{j,j}-2l_{j}}{12}\in\mathbb{Q}/\mathbb{Z},1\leq j\leq
k-1.$

(2) If we are in Case 2, then

$e_{{\mathbb{C}}}(r_{j}\circ
L(E))=\tau_{j}=\frac{g_{j,j}-2l_{j}}{12}\in\mathbb{Q}/\mathbb{Z},1\leq j\leq
k-2;\quad e_{{\mathbb{C}}}(r_{k-1}\circ
L(E))=\tau_{k}-g_{k,k}\tau_{k-1}\in\mathbb{Q}/\mathbb{Z}.$

###### Proof.

For the $e$-invariant, we calculate the Chern character
$ch\colon\tilde{K}^{0}(Th(f^{*}\gamma_{\mathbb{H}}))\rightarrow
H^{ev}(Th(f^{*}\gamma_{\mathbb{H}});\mathbb{Q})$ in terms of (3.6) as follows

$\displaystyle ch(\Phi_{K}((f^{*}\gamma_{\mathbb{H}}))\tilde{z})$
$\displaystyle=\Phi_{H}(f^{*}\gamma_{\mathbb{H}})(1+\frac{\psi_{k}}{12}+\dots)z=\Phi_{H}(f^{*}\gamma_{\mathbb{H}})z,$
$\displaystyle ch(\Phi_{K}((f^{*}\gamma_{\mathbb{H}}))\tilde{\psi}_{j})$
$\displaystyle=\Phi_{H}(f^{*}\gamma_{\mathbb{H}})(1+\frac{\psi_{k}}{12}+\dots)(\psi_{j}+\tau_{j}z)$
$\displaystyle=\Phi_{H}(f^{*}\gamma_{\mathbb{H}})\psi_{j}+\tau_{j}\Phi_{H}(f^{*}\gamma_{\mathbb{H}})z,$
$\displaystyle
ch((\Phi_{K}(f^{*}\gamma_{\mathbb{H}})(\tilde{\psi}_{k}-g_{k,k}\tilde{\psi}_{k-1}))$
$\displaystyle=\Phi_{H}(f^{*}\gamma_{\mathbb{H}})(1+\frac{\psi_{k}}{12}+\dots)(\psi_{k}+\tau_{k}z-g_{k,k}\psi_{k-1}-g_{k,k}\tau_{k-1}z)$
$\displaystyle=\Phi_{H}(f^{*}\gamma_{\mathbb{H}})(\psi_{k}-g_{k,k}\psi_{k-1})+\Phi_{H}(f^{*}\gamma_{\mathbb{H}})(\tau_{k}-g_{k,k}\tau_{k-1})z.$

∎

We now turn our attention to the attaching map

$S^{10}\stackrel{{\scriptstyle L(E)}}{{\to}}(S^{4}\vee S^{7})^{\vee
k-1}\to{(S^{7})}^{\vee k-1}.$

In order to identify the composite we are required to compute the
$KO$-theoretic $e$-invariant. We know that

$\displaystyle KO^{*}=\mathbb{Z}[\eta,u][\mu^{\pm 1}]/(2\eta,\eta^{3},\eta
u,u^{2}-4\mu)\quad\text{and}\quad K^{*}=\mathbb{Z}[\beta^{\pm 1}],$

with $|\eta|=-1,~{}|u|=-4,~{}|\mu|=-8$ and $|\beta|=-2$. The complexification
map $c\colon KO\rightarrow K$ induces a graded ring homomorphism $c\colon
KO^{*}(X)\rightarrow K^{*}(X)$ with

$c(\eta)=0,\quad c(u)=2\beta^{2},\quad c(\mu)=\beta^{4}.$

######  Theorem 3.14.

Let $S^{3}\rightarrow E\rightarrow M_{k}$ be the principal $SU(2)$-fibration
classified by a given map $f\colon M_{k}\rightarrow\mathbb{HP^{\infty}}$.
Suppose that $\Sigma L(M_{k})\equiv 0\pmod{\lambda}$. Then $\Sigma(r_{j}\circ
L(E))\equiv 0\pmod{\lambda},1\leq j\leq k-1$ where $r_{j}:(S^{4}\vee
S^{7})^{\vee k-1}\to S^{7}$ is the retraction onto the $j$-th factor.

###### Proof.

We consider the Atiyah-Hirzebruch spectral sequences for
${\mathbb{H}}P^{\infty}$ and $M_{k}$

$\displaystyle E^{*,*}_{2}$
$\displaystyle=H^{*}(\mathbb{H}P^{\infty};\mathbb{Z})\otimes
KO^{*}(pt)\implies KO^{*}(\mathbb{HP^{\infty}}),$ $\displaystyle E^{*,*}_{2}$
$\displaystyle=H^{*}(M_{k};\mathbb{Z})\otimes KO^{*}(pt)\implies
KO^{*}(M_{k}).$

The spectral sequences have no non-trivial differentials for degree reasons.
Thus, $KO^{*}(\mathbb{HP^{\infty}})\cong KO^{*}[\hat{y}],$ where $\hat{y}\in
KO^{4}({\mathbb{H}}P^{\infty})$. The class $\hat{y}$ serves as $KO$-theoretic
Thom class for $\gamma_{{\mathbb{H}}}$. We have

$c(\Phi_{KO}(\gamma_{\mathbb{H}}))=c(\hat{y})=\beta^{-2}\Phi_{K}(\gamma_{\mathbb{H}}).$

Let $\hat{\psi}_{j}\in\tilde{KO}^{4}(M_{k})$,
$\hat{z}\in\tilde{KO}^{8}(M_{k})$ be the class in the $E^{\infty}$-page
represented by $\psi_{j}\in H^{4}(M_{k})$ and $z\in H^{8}(M_{k})$. Then we get

$c(\hat{\psi}_{i})=\beta^{-2}\tilde{\psi}_{i},\quad
c(\hat{z})=\beta^{-4}\tilde{z}\quad\text{and}\quad\hat{\psi}_{i}\hat{\psi}_{j}=g_{i,j}\hat{z}.$

It follows that the $K$-theoretic generators
$\Phi_{K}(f^{*}\gamma_{{\mathbb{H}}})\tilde{\psi}_{j}$ and
$\Phi_{K}(f^{*}\gamma_{{\mathbb{H}}})(\tilde{\psi}_{k}-g_{k,k}\tilde{\psi}_{k-1})$
lie in the image of the map $c$. Therefore by (3.4), we get in Case 2 that

$e(r_{j}\circ
L(E))\equiv\begin{cases}\frac{g_{j,j}-2l_{j}}{24}&\pmod{\mathbb{Z}},\quad\text{for
}j<k-1,\\\
\frac{(g_{kk}-2l_{k})-g_{k,k}(g_{k-1,k-1}-2l_{k-1})}{24}&\pmod{\mathbb{Z}},\quad\text{for
}j=k-1;\end{cases}$

and in Case 1 that

$e(r_{j}\circ
L(E))\equiv\frac{g_{j,j}-2l_{j}}{24}\pmod{\mathbb{Z}},\quad\text{for }j\leq
k-1.$

The result now follows from Proposition 3.7. ∎

##  4\. $SU(2)$-bundles over even complexes

We study the homotopy type of $E\in{\mathcal{P}}{\mathcal{D}}_{4,7}^{11}$, the
total space of a stable principal $SU(2)$-bundle over
$M_{k}\in{\mathcal{P}}{\mathcal{D}}^{8}_{3}$. In this section, we consider the
case where the intersection pairing $\langle-,-\rangle\colon
H_{4}(M_{k})\times H_{4}(M_{k})\rightarrow\mathbb{Z}$ is even i.e. $\langle
x,x\rangle\in 2\mathbb{Z}$ for all $x\in H_{4}(M_{k})$. Note that in this
case, $k$ must be even. We observe the following.

  1. (1)

If $k\geq 4$, $M_{k}$ supports a principal $SU(2)$-bundle whose total space
$E$ is $3$-connected.

  2. (2)

The possible stable homotopy types of $E$ can be determined directly from the
stable homotopy type of $M_{k}$ and the intersection form. In this regard, the
formulas in Theorem 3.14 are used to demonstrate this connection.

######  4.1.

Existence of $SU(2)$-bundles.  We discuss the existence of principal
$SU(2)$-bundle over $M_{k}\in{\mathcal{P}}{\mathcal{D}}^{8}_{3}$ with an
attaching map as in (1.1) whose intersection form is even. If
$\mbox{Rank}(H_{4}(M))=2$, then up to isomorphism $\begin{pmatrix}0&1\\\
1&0\end{pmatrix}$ is the only possible matrix for the intersection form. The
attaching map $L(M)\in\pi_{7}(S^{4}\vee S^{4})$ of $M$ is of the form

(4.2)
${}L(M)=[\alpha_{1},\alpha_{2}]+l_{1}v^{\prime}_{1}+l_{2}v^{\prime}_{2},$

where $l_{1},l_{2}\in\mathbb{Z}/12$. For a principal bundle $SU(2)\to E\to
M_{k}$ where $E$ is $3$-connected, the classifying map $f_{E}\colon
M_{k}\rightarrow\mathbb{H}P^{\infty}$ is such that $\pi_{4}(f)$ is surjective.
Suppose $\alpha_{i}\mapsto n_{i}\iota_{4},i=1,2$ such that
$\gcd(n_{1},n_{2})=1$ and $f\circ L(M)\simeq*$. If both $l_{1}$ and $l_{2}$
are odd, no such $n_{1}$ and $n_{2}$ exists. However, for $k\geq 4$, one may
always construct a suitable map.

######  Proposition 4.3.

Suppose $M_{k}\in\mathcal{PD}_{3}^{8}$ such that
$\mbox{Rank}(H_{4}(M_{k}))=k\geq 4$ and the intersection form is even. Then
there exists a map $\psi\colon M_{k}\rightarrow\mathbb{H}P^{\infty}$ such that
$\mbox{hofib}(\psi)$ is $3$-connected.

###### Proof.

From [7, Theorem 4.20], if $k\geq 6$, the attaching map of $M_{k}$ can be
expressed as

$L(M_{k})=\sum_{1\leq i<j\leq
k}g_{i,j}[\alpha_{i},\alpha_{j}]+\sum_{1=1}^{k}\frac{g_{i,i}}{2}[\alpha_{i},\alpha_{i}]+\sum_{i=1}^{k}s_{i}\nu_{i}^{\prime}$

such that $s_{i}=0$ for $i\geq 2$ for a choice of basis
$\\{\alpha_{1},\dots,\alpha_{k}\\}$ of $H_{4}(M_{k})$. Then the map
$\tilde{\psi}\colon(S^{4})^{\vee
k}\xrightarrow{(0,0,\dots,0,1)}\mathbb{H}P^{\infty}$ extends to a map
$\psi\colon M_{k}\rightarrow\mathbb{H}P^{\infty}$ such that $\pi_{4}(\psi)$ is
surjective.

Now if $k=4,$ by [11], the attaching map of $M_{k}$ can be expressed as

$L(M_{k})=[\alpha_{1},\alpha_{2}]+[\alpha_{3},\alpha_{4}]+\sum_{i=1}^{4}l_{i}\nu_{i}^{\prime}$

for a choice of basis of $H_{4}(M_{k}).$ Choose two positive integers $m,n$
such that $\gcd(m,n)=1$ and $ml_{1}+nl_{3}\equiv 0\pmod{12}$. Then the map
$\tilde{\psi}\colon(S^{4})^{\vee
4}\xrightarrow{(m,0,n,0)}\mathbb{H}P^{\infty}$ extends to a map $\psi\colon
M_{k}\rightarrow\mathbb{H}P^{\infty}$ such that $\pi_{4}(\psi)$ is surjective.
∎

We now focus on the stable homotopy type of the total space $E(f)$ for
$f\colon M_{k}\to{\mathbb{H}}P^{\infty}$ such that $\pi_{4}(f)$ is injective.
From the attaching map of $M_{k}$ as in (1.1) and even intersection form, we
have

(4.4) ${}\Sigma^{\infty}M_{k}\simeq\Sigma^{\infty}(S^{4})^{\vee
k-1}\vee\Sigma^{\infty}(Cone({\sigma(M_{k})}\nu_{(7)})$

for some even $\sigma(M_{k})$. Hence the stable homotopy type of $M_{k}$ is
determined by $\sigma(M_{k})$.

######  Proposition 4.5.

Let $E(f_{\psi})$ be the total space of a principal $SU(2)$-bundle over
$M_{k}\in\mathcal{PD}^{8}_{3}$, classified by a map $f_{\psi}\colon
M_{k}\rightarrow\mathbb{H}P^{\infty}$ for $k\geq 4.$ Then

$\Sigma^{\infty}E(f_{\psi})\simeq\Sigma^{\infty}(S^{4}\vee
S^{7})^{\vee{k-2}}\vee\Sigma^{\infty}Cone({{\lambda(\psi)}}\Sigma^{4}\nu_{(7)}),$

where $\lambda(\psi):=\lambda_{s}(E(f_{\psi}))$, is even and a multiple of
$\sigma(M_{k})$.

###### Proof.

Note that $E(f_{\psi})\in\mathcal{PD}_{4,7}^{11}$ and its stable homotopy type
is given in the Corollary 2.15 where
$\epsilon=\epsilon_{s}(E(f_{\psi}))\in\mathbb{Z}/2$ and
$\lambda(\psi)\in\mathbb{Z}/24$. So, it suffices to show that
$2|\lambda(\psi)$ and $\epsilon=0$. The fact $2|\lambda(\psi)$ follows from
(4.4) and Theorem 3.14.

Now, the cofibre sequence obtained from cell structure of $E$ and $M_{k}$
induces the following commutative diagram

$\textstyle{\pi_{10}^{s}(S^{10})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{*}}$$\textstyle{\pi_{10}^{s}(S^{4}\vee
S^{7})^{\bigoplus{k-1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}\pi_{10}^{s}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}}$$\textstyle{\pi_{10}^{s}(S^{7})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{\pi_{10}^{s}(\sum^{\infty}L(M))}$$\textstyle{\pi^{s}_{10}(S^{4})^{\bigoplus
k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}\pi_{10}^{s}(M_{k})}$

of stable homotopy groups where $\phi$ is the attaching map of top cell in
$E$.

Since $\lambda(\psi)\beta_{k-1}\circ\nu_{7}+\epsilon\alpha_{k-1}\circ x=0$ in
$\pi^{s}_{10}(E)$, its image in $\pi_{10}^{s}(M)$ is $0$. Note that bottom
left map $\pi_{10}^{s}(\sum^{\infty}L(M))=0$ because $2|\sigma(M_{k})$ in
(4.4) and hence the bottom right map is injective, where
$\pi_{10}^{s}(S^{4})^{\oplus
k}\cong\mathbb{Z}/2\\{\alpha_{1}\circ\nu^{2},\dots,\alpha_{k}\circ\nu^{2}\\}$.
Since $2|\lambda(\psi)$, the middle vertical arrow sends
$\Sigma^{\infty}\phi=\lambda(\psi)\beta_{k-1}\circ\nu_{(7)}+\epsilon\alpha_{k-1}\circ\nu^{2}$
to $\epsilon\alpha_{k-1}\circ\nu^{2}$ which is in turn mapped to $0$ via the
bottom right map as $\Sigma^{\infty}\phi=0\in\pi_{10}^{s}(E).$ Hence
$\epsilon=0\in\mathbb{Z}/2$. ∎

######  4.6.

Stably trivial manifolds.  The following result states that the total space of
a principal $SU(2)$-bundle over stably trivial $M_{k}\in\mathcal{PD}^{8}_{3}$
( i.e., $\sigma(M_{k})\equiv 0\pmod{24}$), is itself a connected sum of copies
of $S^{4}\times S^{7}$.

######  Proposition 4.7.

Let $E\in{\mathcal{P}}{\mathcal{D}}^{11}_{4,7}$ be the total space of a
principal $SU(2)$-bundle over stably trivial $M_{k}\in\mathcal{PD}^{8}_{3}$.
Then $E\simeq\\#^{k-1}(S^{4}\times S^{7})$.

###### Proof.

From Theorem 2.17, we have
$E\simeq\\#^{k-2}E_{0,0,0}\\#E_{0,\epsilon,\delta}$. Then the result follows
from Proposition 4.5 for $k\geq 4$. For $k=2$, the attaching map of $M_{2}$ is
of the form (4.2). Stably trivial condition implies $M_{2}=S^{4}\times S^{4}$
which further implies $E=S^{4}\times S^{7}$. ∎

######  4.8.

Possible stable homotopy types of the total space.  Let $\psi\in
H^{4}(M_{k};\mathbb{Z})$ be a cohomology class represented by a map
$\overline{\psi}\colon M_{k}\rightarrow K(\mathbb{Z},4)$ which has a unique
lift $\tilde{\psi}\colon M_{k}\rightarrow S^{4}$ up to homotopy if
${\overline{\psi}}\big{|}_{{(S^{4})}^{\vee{k-1}}}\circ
L(M_{k})\in\pi_{7}(S^{4})$ is $0$. As the inclusion
$(S^{4})^{\vee{k}}\rightarrow M_{k}$ induces an isomorphism on $H_{4}$ and
$H^{4}$, a cohomology class $\psi\in H^{4}(M_{k})$ always induces a map
$\tilde{\psi}\colon(S^{4})^{\vee{k}}\rightarrow S^{4}.$ We consider the
following diagram

$\textstyle{(S^{4})^{\vee{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\psi}}$$\textstyle{M_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}}$$\textstyle{S^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}}$$\textstyle{\mathbb{H}P^{2}}$

and formulate when the map $M_{k}\rightarrow\mathbb{H}P^{2}$ exists. Note that
if the map exists then its homotopy fibre will be $3$-connected.

For $\psi\in H^{4}(M_{k}),$ consider the composite

$S^{7}\xrightarrow{L(M_{k})}(S^{4})^{\vee{k}}\xrightarrow{\tilde{\psi}}S^{4}$

and define $\tau(\psi)=[\tilde{\psi}\circ L(M)]\in\pi_{7}^{s}(S^{4})$. Thus

$\tau\colon H^{4}(M_{k})\rightarrow\mathbb{Z}/24$

and one can check it is a linear map.

######  Proposition 4.9.

Suppose $\psi\in H^{4}(M_{k})$ is primitive. The map
$\tilde{\psi}\colon(S^{4})^{\vee{k}}\rightarrow S^{4}$ extends to a map
$f_{\psi}:M_{k}\rightarrow\mathbb{H}P^{2}$ if and only if
$\psi\cup\psi\equiv\tau(\psi)z\pmod{24}$ for some $z\in H^{8}(M_{k})$.

###### Proof.

Consider a primitive element $\psi\in H^{4}(M_{k})$. We extend $\psi$ to a
basis of $H^{4}(M_{k})$, and use the dual basis of $\pi_{4}(M_{k})$ to write
down the attaching map of $M_{k}$ is as in $\eqref{Eq_general L(M) define}$.
In this notation, we have $\psi^{2}=g_{k,k}z$ where $z\in H^{8}(M_{k})$ is the
chosen generator and $\tau(\psi)=g_{k,k}-2l_{k}$. Thus $\tilde{\psi}\circ
L(M)$ maps to $0\in\pi_{7}(\mathbb{H}P^{2})$ if and only if $l_{k}=0$, that is
$\psi\cup\psi\equiv\tau{(\psi)}z\pmod{24}$. Hence, the result follows. ∎

Proposition 4.9 gives us criteria for constructing the maps $f_{\psi}$ out of
cohomology classes $\psi$. Using this we determine which multiples of
$\sigma(M_{k})$ may occur as $\lambda_{s}(E)$ for $E\to M_{k}$ a principal
$SU(2)$-bundle where $E$ is $3$-connected. We first show that there exist
$f_{\psi}$ such that $\lambda(\psi)=\sigma(M_{k})$ if $k$ is large enough.

######  Proposition 4.10.

Suppose the stable homotopy type of $M_{k}$ is determined by $\sigma(M_{k}).$
Then there exists $f_{\psi}\colon M_{k}\rightarrow\mathbb{H}P^{\infty}$ such
that

  1. (1)

$\lambda(\psi)\equiv\sigma(M_{k})\pmod{3}$ for $k\geq 5$,

  2. (2)

$\lambda(\psi)\equiv\sigma(M_{k})\pmod{8}$ for $k\geq 7$.

###### Proof.

We begin the proof with the first case. If $\tau\equiv 0\pmod{3}$, the proof
follows from Theorem 3.14. Let $\tau\not\equiv 0\pmod{3}$ and $\psi_{0}$ a
primitive cohomology class such that
$\tau(\psi_{0})\equiv\sigma(M_{k})\not\equiv 0\pmod{3}$. We need to choose a
$\psi\in\ker(-\cup\psi_{0})\cap\ker(\tau)$ such that
$\psi^{2}\equiv\tau(\psi)\equiv 0\pmod{3}$. This we can do for $k-2\geq 3$,
see [11, Chapter II, (3.2)-(3.4)]. By Poincaré duality, we get $\psi^{\prime}$
such that $\psi\cup\psi^{\prime}=z$. We write
$\psi^{\prime}=\psi^{\prime}_{\ker(\tau)}+t\psi_{0}$ for some $t$ where
$\psi^{\prime}_{\ker(\tau)}\in\ker(\tau)$. Since
$z=\psi^{\prime}\cup\psi=\psi^{\prime}_{\ker(\tau)}\cup\psi$, we may assume
$\psi^{\prime}=\psi^{\prime}_{\ker\tau}$. Thus by assigning
$\tau_{1}=\tau(\psi_{0})\equiv\sigma(M_{k})\pmod{3}$,
$\tau_{k-1}=\tau(\psi^{\prime})\equiv 0\pmod{3}$ and
$\tau_{k}=\tau(\psi)\equiv 0\pmod{3}$; and choosing other $\psi_{i}$’s such
that $\tau_{i}=0$ for $i=2,\dots,k-2$ we have

$\lambda(\psi)\equiv\sigma(M_{k})\pmod{3}\quad\quad\text{for }k\geq 5.$

Now we look into the second case. The proof goes similarly to that of the
above, except when we choose $\psi\in\ker(-\cup\psi_{0})\cap\ker(\tau)$ such
that $\psi^{2}\equiv\tau(\psi)\equiv 0\pmod{8}$. We can choose such $\psi$ for
$k-2\geq 5$, see [11, Chapter II, (3.2)-(3.4)]. Then following similar
arguments one can deduce

$\lambda(\psi)\equiv\sigma(M_{k})\pmod{8}\quad\quad\text{for }k\geq 7.$

∎

The following theorem constructs $f_{\psi}$ with
$\lambda(\psi)=3\sigma(M_{k})$ if $\sigma(M_{k})$ is not divisible by $3$.

######  Theorem 4.11.

Suppose $3\nmid\sigma(M_{k})$. Then for $k\geq 7$, there exists
$f_{\psi}\colon M_{k}\rightarrow\mathbb{H}P^{2}$ such that
$\lambda(\psi)=3\sigma(M_{k})$.

###### Proof.

Let

$\tau^{(3)}\colon H^{4}(M_{k},\mathbb{F}_{3})\rightarrow\mathbb{F}_{3}$

be the restriction of $\tau$ in modulo $3$. As $\tau^{3}$ is surjective, there
exists $\psi_{0}\in H^{4}(M_{k},\mathbb{F}_{3})$ such that for all cohomology
class $\psi$, $\tau^{(3)}(\psi)z\equiv\psi\cup\psi_{0}\pmod{3}$. In
particular, $\tau^{(3)}(\psi_{0})z\equiv\psi_{0}\cup\psi_{0}\pmod{3}$. We
consider two cases, $\psi_{0}^{2}=0$ or $\psi_{0}^{2}$ is unit.

First let $\psi_{0}^{2}$ is unit. Then we can choose
$\psi_{1},\dots,\psi_{k-1}$ such that $\psi_{0}\cup\psi_{i}=0$. We take the
dual basis $\alpha_{i}$ corresponding to $\psi_{i}$ and $\alpha_{k}$
corresponding to $\psi_{0}$. Thus $\psi_{0}(\alpha_{i})=0$ for $i=1,\dots,k-1$
and $\psi_{0}(\alpha_{k})=1$. Hence

$\tau_{1}\equiv\dots\equiv\tau_{k-1}\equiv
0\pmod{3},\quad\text{and}\quad\tau_{k}\equiv g_{k,k}\equiv 1\pmod{3},$

which implies $\lambda(\psi)=\gcd(\tau_{1},\dots,\tau_{k-2},\tau_{k-1})\equiv
0\pmod{3}$.

Now let $\psi_{0}^{2}=0$. Then $\tau^{(3)}(\psi_{0})z=0$. We choose
$\psi_{1},\dots,\psi_{k-1}$ such that $\psi_{k-1}\cup\psi_{0}=1$ and
$\psi_{i}\cup\psi_{0}=0$ for $i=1,\dots,k-2$. After taking the dual basis,
with similar argument we have

$\tau_{1}\equiv\dots\equiv\tau_{k-2}\equiv
0\pmod{3},\quad\tau_{k-1}\equiv\sigma(M_{k})\pmod{3},\quad\text{and}\quad\tau_{k}\equiv
g_{k,k}\equiv 0\pmod{3}.$

Hence
$\lambda(\psi)=\gcd(\tau_{1},\dots,\tau_{k-2},\tau_{k}-g_{k,k}\tau_{k-1})\equiv
0\pmod{3}$. ∎

######  Remark 4.12.

We note that Proposition 4.9, Proposition 4.10, and Theorem 4.11 does not use
the fact that the intersection form is even, and also holds in the case where
the intersection form is odd.

Now we look to prove similar results modulo $8$, which in turn provide us
desired construction $\psi$ as in Proposition 4.9 using the Chinese reminder
theorem. However, in this case certain conditions are required for obtaining
analogous $f_{\psi}$.

######  Definition 4.13.

  * •

A complex $M_{k}$ with $\sigma(M_{k})=2$ or $4$, is said to satisfy hypothesis
$(H_{8})$ if $(\ker\tau)^{\perp}=(\sigma(M_{k})\psi)\pmod{8}$ where $\psi\in
H^{4}(M_{k})$ (which is unique $\pmod{\frac{8}{\sigma(M_{k})}}$) satisfies

$\begin{cases}\psi^{2}\equiv 0\pmod{8}&\text{ if }\sigma(M_{k})=2\\\
\psi^{2}\equiv 0\pmod{4}&\text{ if }\sigma(M_{k})=4.\end{cases}$

  * •

A complex $M_{k}$ with $\sigma(M_{k})=2$ is said to satisfy hypothesis
$(H_{4})$ if $(\ker\tau)^{\perp}=(2\psi)\pmod{4}$ where $\psi\in H^{4}(M_{k})$
(which is unique $\pmod{2}$) satisfies

$\psi^{2}\equiv\tau(\psi)\equiv 0\mbox{ or }4\pmod{8}.$

Note that the hypotheses $(H_{8})$ and $(H_{4})$ depends only on the
intersection form and $\tau$ and not on the choice of $\psi$. We now prove the
existence of $f_{\psi}$ under the hypothesis defined above.

######  Theorem 4.14.

  1. (1)

Suppose $8\nmid\sigma(M_{k})$. For $k\geq 5$, there exist $f_{\psi}\colon
M_{k}\rightarrow\mathbb{H}P^{2}$ such that $\lambda(\psi)\equiv 0\pmod{8}$ if
and only if the complex satisfies hypothesis $(H_{8})$.

  2. (2)

Suppose $\sigma(M_{k})=2$. Then for $k\geq 5$, there exist $f_{\psi}\colon
M_{k}\rightarrow\mathbb{H}P^{2}$ such that $\lambda(\psi)\equiv 4\pmod{8}$ if
and only if the complex satisfies hypothesis $(H_{4})$.

###### Proof.

The condition $k\geq 5$ comes from the fact that we are required to make
certain choices modulo $3$ using Proposition 4.10. First suppose in case (1),
a $f_{\psi}$ exists such that $\lambda(\psi)=8\sigma(M_{k})$. Then there
exists a basis $\\{\psi_{1},\dots,\psi_{k-2},\psi^{\prime},\psi\\}$ satisfying
(working $\pmod{8}$)

(4.15) $\displaystyle\psi\cup\psi_{i}=0\text{ for }1\leq i\leq k-2,\quad$
$\displaystyle\psi\cup\psi^{\prime}=1,$ $\displaystyle\tau(\psi_{i})=0\text{
for }1\leq i\leq k-2,\quad$
$\displaystyle\tau(\psi^{\prime})=\sigma(M_{k})\quad\text{and}\quad\tau(\psi)=\psi^{2}=0.$

Note that
$\langle\psi\rangle^{\perp}=\langle\psi_{1},\dots,\psi_{k-2},\psi\rangle\subset\ker(\tau)=\langle\psi_{1},\dots,\psi_{k-2},\psi,\frac{8}{\sigma(M_{k})}\psi^{\prime}\rangle$.
This implies $(\ker(\tau))^{\perp}=(\sigma(M_{k})\psi)$, and thus the
hypothesis $(H_{8})$ is satisfied.

For the converse part if the complex satisfies hypothesis $(H_{8})$, one can
check that there is a choice of $\psi$ such that (4.15) is satisfied. We look
into the cases $\sigma(M_{k})=2,4$.

First, let $\sigma(M_{k})=2$. Then $(\ker(\tau))^{\perp}=(2\psi)$ and $\psi$
is well defined modulo $4$. We note that

(4.16) ${}\chi\cup(2\psi)\equiv\tau(\chi)z\pmod{8}~{}\forall\chi\in
H^{4}(M_{k}).$

This implies

$\tau(\psi)\equiv 2\psi^{2}\equiv 0\pmod{8}.$

For any $\chi\in\ker(\tau)$, we have $(2\psi)\chi=\tau(\chi)\equiv 0\pmod{8}$.
In particular $2\psi^{2}=\tau(\psi)\equiv 0\pmod{8}$. Together these two
implies $\psi^{2}\equiv 0\pmod{8}$. The equation (4.16) implies that if
$\chi\cup\psi=0$, $\tau(\chi)=0$. Now choosing a basis as in Case (2) of
Proposition 3.13, we obtain the conditions in (4.15).

Now let $\sigma(M_{k})=4$. Then $(\ker(\tau))^{\perp}=(4\psi)$ and $\psi$ is
determined modulo $2$. We proceed analogously observing that

$\chi\cup(4\psi)\equiv\tau(\chi)z\pmod{8}~{}\forall\chi\in H^{4}(M_{k}),$

which implies

$\tau(\psi)\equiv 4\psi^{2}\equiv 0\pmod{8}.$

Let $\psi^{\prime}$ be such that $\tau(\psi^{\prime})=4$, and so we have that
$\psi\cup\psi^{\prime}$ is an odd multiple of $z$. If $\psi^{2}$ is
$4\pmod{8}$ we change $\psi$ to $\psi+2\psi^{\prime}$ to ensure
$\psi^{2}=\tau(\psi)\equiv 0\pmod{8}$. Now choosing a basis as in Case (2) of
Proposition 3.13, we obtain the conditions in (4.15).

The case (2) also proceeds along analogous lines. For the existence (of
$f_{\psi}$ for some $\psi$) question, we need a basis satisfying

(4.17) $\displaystyle\psi\cup\psi_{i}=0\text{ for }1\leq i\leq
k-2,\quad\psi\cup\psi^{\prime}=1,$ $\displaystyle\tau(\psi_{i})\equiv
0\pmod{4}\text{ for }1\leq i\leq
k-2,\quad\tau(\psi^{\prime})=\sigma(M_{k}),\quad\tau(\psi)=\psi^{2}\equiv
0\mbox{ or }4\pmod{8},$ $\displaystyle\mbox{ such that at least one of
}\tau(\psi_{i})\mbox{ for }1\leq i\leq k-2,\mbox{ or }\tau(\psi)\equiv
4\pmod{8}.$

Note that
$\langle\psi\rangle^{\perp}=\langle\psi_{1},\dots,\psi_{k-2},\psi\rangle\subset\ker(\tau)=\langle\psi_{1},\dots,\psi_{k-2},\psi,2\psi^{\prime}\rangle\pmod{4}$
and $(\ker(\tau))^{\perp}=(2\psi)\pmod{4}$. Hence, the hypothesis $(H_{4})$ is
satisfied.

Conversely, if $(H_{4})$ is satisfied, we obtain a $\psi$ such that
$\psi^{2}\equiv\tau(\psi)\equiv 0\mbox{ or }4\pmod{8}$. This $\psi$ also
satisfies

$\chi\cup(2\psi)\equiv\tau(\chi)z\pmod{4}~{}\forall\chi\in H^{4}(M_{k}).$

Let $\psi^{\prime}$ be such that $\tau(\psi^{\prime})=2$, which implies that
$\psi\cup\psi^{\prime}$ is an odd multiple of $z$. Now replacing $\psi$ by
$\psi+2\psi^{\prime}$ if required we may assume that
$\psi^{2}\equiv\tau(\psi)\equiv 4\pmod{8}$. Now choosing a basis as in Case
(2) of Proposition 3.13, we obtain the conditions in (4.17). ∎

If $\mbox{Rank}(H_{4}(M_{k}))=k\geq 5$, the above results indicate a
systematic computation of possible stable homotopy types of the total space
depending on $k$, $\sigma(M_{k})$ and the intersection form. In lower rank
cases, the results depend on the explicit formula for the attachment
$L(M_{k})$, and not just on these variables. Hence the systematic description
turns out to be cumbersome. We demonstrate some observations on the
$\mbox{Rank}(H_{4}(M))=2$ case.

######  Example 4.18.

Recall that the attaching map $L(M)\in\pi_{7}(S^{4}\vee S^{4})$ of $M_{2}$ is
of the form

$L(M)=[\alpha_{1},\alpha_{2}]+l_{1}v^{\prime}_{1}+l_{2}v^{\prime}_{2}$

where $l_{1},l_{2}\in\mathbb{Z}/12$. We already have

  * •

If both $l_{1}$ and $l_{2}$ are odd, there does not exist $f\colon
M\rightarrow\mathbb{H}P^{\infty}$ such that $E=\mbox{hofib}(f)$ is
$3$-connected.

If one of $l_{1}$ and $l_{2}$ is even, or both are even, there exists $f\colon
M\rightarrow\mathbb{H}P^{\infty}$ such that $E=\mbox{hofib}(f)$ is
$3$-connected. Via an explicit calculation using Proposition 3.13, we observe
the following.

  1. (1)

If none of $l_{1}$ and $l_{2}$ are divisible by $3$, then we obtain
$\lambda(E)\equiv 0\pmod{3}$.

  2. (2)

If either of $l_{1}$ or $l_{2}$ but not both are divisible by $3$, then
$\lambda(E)\not\equiv 0\pmod{3}$.

  3. (3)

If $\sigma(M)\equiv 4\pmod{8}$ and $l_{1}l_{2}\equiv 0\pmod{8}$ where none of
$l_{1}$ and $l_{2}$ are divisible by $3$, then $\lambda(E)\equiv 0\pmod{8}$.

  4. (4)

If $\sigma(M)\equiv 2\pmod{8}$, we can never obtain $\lambda(E)\equiv
0,4\pmod{8}$.

##  5\. $SU(2)$-bundles over odd complexes

We now work out the case of $M_{k}\in{\mathcal{P}}{\mathcal{D}}_{3}^{8}$ for
which the intersection form is odd. Recall that the notation $M_{k}$ means
that $\mbox{Rank}(H_{4}(M_{k}))=k$. The intersection form being odd implies
that there are two possibilities of $\sigma(M_{k})$, namely, $1$ and $3$ among
the divisors of $24$. Here, we prove
1) For $k\geq 3$, it is possible to obtain a $SU(2)$-bundle whose total space
is $3$-connected.
2) Further if $k\geq 7$, it is possible to obtain maps
$\psi(j):M_{k}\to{\mathbb{H}}P^{\infty}$ with $\lambda(\psi(j))=j$ for every
multiple $j\pmod{24}$ of $\sigma(M_{k})$ which is also a divisor of $24$.

######  5.1.

Existence of $SU(2)$-bundles.  Through an explicit computation, we demonstrate
the existence of principal $SU(2)$-bundles over
$M_{k}\in{\mathcal{P}}{\mathcal{D}}^{8}_{3}$ if $k\geq 3$ when the
intersection form is odd.

######  Proposition 5.2.

Suppose $M_{k}\in\mathcal{PD}_{3}^{8}$ such that
$\mbox{Rank}(H_{4}(M_{k}))=k\geq 3$ and the intersection form is odd. Then
there exists a map $\psi\colon M_{k}\rightarrow\mathbb{H}P^{\infty}$ such that
$\mbox{hofib}(\psi)$ is $3$-connected.

###### Proof.

Recall that the attaching map of $M_{k}$ can be expressed as (1.1). Using
Proposition 4.9, we are required to find a primitive element $\psi$ such that
$\tau(\psi)z\equiv\psi^{2}\pmod{24}$, which is equivalent to checking that the
coefficient of $\nu^{\prime}$ in $\tilde{\psi}\circ L(M_{k})$ is $0\pmod{12}$.
It suffices to find $\psi\pmod{8}$ and $\psi\pmod{3}$ separately.

We first work out the $\pmod{3}$ case, where the base ring is a field of
characteristic $\neq 2$ so that the form is diagonalizable. Considering the
map $(S^{4})^{\vee
k}\xrightarrow[]{(0,\dots,0,n_{1},n_{2},n_{3})}{\mathbb{H}}P^{\infty}$ which
sends $L(M_{k})$ to

$\Big{(}\pm{n_{1}\choose 2}\pm{n_{2}\choose 2}\pm{n_{3}\choose
2}+n_{1}l_{k-2}+n_{2}l_{k-1}+n_{3}l_{k}\Big{)}\cdot\nu^{\prime}+\mbox{
multiple of }\nu,$

where the $\pm$ correspond to the diagonal entries $\pmod{3}$. We observe
through a direct calculation that for every fixed choice
$\epsilon_{1},\epsilon_{2},\epsilon_{3}$ of $\pm 1$s, there is a
$n_{1},n_{2},n_{3}$ with $\gcd(n_{1},n_{2},n_{3})=1$ such that

$\Big{(}\epsilon_{1}{n_{1}\choose 2}+\epsilon_{2}{n_{2}\choose
2}+\epsilon_{3}{n_{3}\choose
2}+n_{1}l_{k-2}+n_{2}l_{k-1}+n_{3}l_{k}\Big{)}\equiv 0\pmod{3}.$

This completes the argument $\pmod{3}$.

Working $\pmod{8}$, the fact that the intersection form is odd implies that we
may choose a basis such that $g_{k,k}\equiv\pm 1\pmod{8}$ and $g_{k,k-1}\equiv
0\pmod{8}$, see [11, Chapter II, (4.3)]. Also the intersection form can be
written as the block matrix $\begin{pmatrix}A&0\\\ 0&\pm 1\end{pmatrix}.$

If $A$ is an even intersection form, then the result follows from Proposition
4.3 for $k\geq 5$. Now let $A$ is not even, then the intersection form is
$\begin{pmatrix}A^{\prime}&0\\\ 0&B^{\prime}\end{pmatrix}$ where $B^{\prime}$
is a diagonal matrix of order $2$ with diagonal entries $\pm 1$. If
$A^{\prime}$ is an even intersection form, then for $k\geq 6$ the result
follows from Proposition 4.3. If $A^{\prime}$ is not even, the intersection
form updates to

$\begin{pmatrix}A^{\prime\prime}&0\\\ 0&B^{\prime\prime}\end{pmatrix}$

where $B^{\prime\prime}$ is a diagonal matrix of order $3$ with diagonal
entries $\pm 1$. For $B^{\prime\prime}=I_{3}$, the map $(S^{4})^{\vee
k}\xrightarrow[]{(0,\dots,0,n_{1},n_{2},n_{3})}{\mathbb{H}}P^{\infty}$ sends
$L(M_{k})$ to

$\Big{(}{n_{1}\choose 2}+{n_{2}\choose 2}+{n_{3}\choose
2}+n_{1}l_{k-2}+n_{2}l_{k-1}+n_{3}l_{k}\Big{)}\cdot\nu^{\prime}+\mbox{
multiple of }\nu,$

where $\gcd(n_{1},n_{2},n_{3})=1$ ensures that the corresponding $\psi$ is
primitive. We may directly compute and observe that the equations

$\Big{(}{n_{1}\choose 2}+{n_{2}\choose 2}+{n_{3}\choose
2}+n_{1}l_{k-2}+n_{2}l_{k-1}+n_{3}l_{k}\Big{)}\equiv
0\pmod{8},\gcd(n_{1},n_{2},n_{3})=1,$

have a common solution. A similar argument works for the other diagonal $\pm
1$ matrix choices for $B^{\prime\prime\prime}$. This proves the result for
$k\geq 6$.

If $3\leq k\leq 5$, we know from [11, Chapter II, (3.2)-(3.4)] that the form
is a direct sum of $\pm 1$ and the hyperbolic form. The argument in the even
case in Proposition 4.3 implies the result for a sum of two hyperbolic forms,
and the above argument implies the result for a sum of $3$ $\pm 1$s. The
remaining cases are taken care of if we show the result for the intersection
form

$\begin{pmatrix}H&0\\\ 0&\pm 1\end{pmatrix}$

where $H=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ is the hyperbolic matrix. We
consider the map $(S^{4})^{\vee
3}\xrightarrow[]{(n_{1},n_{2},n_{3})}{\mathbb{H}}P^{\infty}$, and as above we
need to find a common solution of

${n_{3}\choose 2}+n_{1}n_{2}+\sum_{i=1}^{3}n_{i}l_{i}\equiv
0\pmod{8},\gcd(n_{1},n_{2},n_{3})=1.$

Through a direct calculation, we check that such solutions always exist. This
completes the proof. ∎

The following example shows that Proposition 5.2 does not extend to the $k=2$
case.

######  Example 5.3.

Consider $M_{2}$ such that

$L(M_{2})=\nu_{1}+\nu_{2}+l_{1}\nu_{1}^{\prime}+l_{2}\nu_{2}^{\prime},\quad
l_{1}\equiv l_{2}\equiv 2\pmod{3}.$

A map $M_{2}\to{\mathbb{H}}P^{\infty}$ which restricts to $(n_{1},n_{2})$ on
the $4$-skeleton sends $L(M_{2})$ to

$\Big{(}{n_{1}\choose 2}+{n_{2}\choose
2}+n_{1}l_{1}+n_{2}l_{2}\Big{)}\cdot\nu^{\prime}+\mbox{ multiple of }\nu.$

We may check directly that

$\Big{(}{n_{1}\choose 2}+{n_{2}\choose 2}+n_{1}l_{1}+n_{2}l_{2}\Big{)}\equiv
0\pmod{3}\implies n_{1}\equiv n_{2}\equiv 0\pmod{3}.$

Therefore, there is no map $M_{2}\to{\mathbb{H}}P^{\infty}$ whose homotopy
fibre is $3$-connected.

######  5.4.

Possible stable homotopy type of the total space.  We note from §4 that
Propositions 4.9, 4.10, and Theorem 4.11 are also valid when the intersection
form is odd. We check that all stable homotopy types are achievable in the odd
case if the rank $k$ of $H_{4}(M)$ is $\geq 7$. Applying the results from §4,
it only remains to check that the different possibilities $\pmod{8}$ are
achievable. We do this in the theorem below.

######  Theorem 5.5.

Suppose $\sigma(M_{k})$ is odd. Then for $k\geq 5$ and $j\in\\{0,2,4\\}$,
there exists $\psi\colon M_{k}\to{\mathbb{H}}P^{2}$ such that
$\lambda(\psi)\equiv j\pmod{8}$.

###### Proof.

We work $\pmod{8}$, knowing that if $k\geq 5$, Proposition 4.10 allows us to
make a choice of $\psi$ so that $\lambda(\psi)$ is as required $\pmod{3}$. The
proof is very similar to the proof of Theorem 4.11. If $\sigma(M_{k})$ is odd,
the linear map $\tau\colon({\mathbb{Z}}/8)^{k}\to{\mathbb{Z}}/8$ is
represented by some primitive class $\psi$ (that is,
$\tau(\chi)=\langle\chi,\psi\rangle\pmod{8}$, and ${\mathbb{Z}}\\{\psi\\}$ is
a summand of $H^{4}(M_{k})$). In particular, $\psi^{2}=\tau(\psi)$. Now, we
have two cases.

First let $\tau(\psi)$ be odd, i.e. $\psi^{2}$ is unit in modulo $8$. Then we
can extend $\psi$ to a basis $\psi,\psi_{1},\dots,\psi_{k-1}$ such that
$\psi\cup\psi_{i}=0$ for $1\leq i\leq k-1$. We take the dual basis
$\alpha_{i}$ corresponding to $\psi_{i}$ and $\alpha_{k}$ corresponding to
$\psi$. Thus $\psi(\alpha_{i})=0$ for $i=1,\dots,k-1$ and
$\psi(\alpha_{k})=1$. Hence

$\tau_{1}\equiv\dots\equiv\tau_{k-1}\equiv
0\pmod{8},\quad\text{and}\quad\tau_{k}\equiv g_{k,k}\pmod{8},$

which implies $\lambda(\psi)=\gcd(\tau_{1},\dots,\tau_{k-2},\tau_{k-1})\equiv
0\pmod{8}$ by Case (1) of Proposition 3.13.

Now let $\psi^{2}$ be even. Extend $\psi$ to a basis
$\psi,\psi_{1},\dots,\psi_{k-1}$ such that $\psi_{k-1}\cup\psi=1$ and
$\psi_{i}\cup\psi=0$ for $i=1,\dots,k-2$. After taking the dual basis, with
similar argument we have

$\tau_{1}\equiv\dots\equiv\tau_{k-2}\equiv 0\pmod{8},\quad\tau_{k-1}\equiv
1\pmod{8},\quad\text{and}\quad\tau_{k}\equiv g_{k,k}\pmod{8}.$

Hence
$\lambda(\psi)=\gcd(\tau_{1},\dots,\tau_{k-2},\tau_{k}-g_{k,k}\tau_{k-1})\equiv
0\pmod{8}$ by Case (2) of Proposition 3.13. This proves the result for $j=0$.

For $j=2,$ or $4$, we can use $\psi(j)=\psi+j\psi_{1}$, and note that

$\tau(\psi(j))=\tau(\psi),~{}\psi(j)^{2}=\psi^{2}+j^{2}\psi_{1}^{2}\equiv\psi^{2}\pmod{8}.$

The last equivalence comes from the fact that
$\tau(\psi_{1})z=\psi\cup\psi_{1}\equiv 0\pmod{8}$, and so $\psi_{1}^{2}$ is
forced to be an even multiple of $z$. We may now compute using the formulas of
Proposition 3.13 to conclude that $\lambda(\psi(j))\equiv j\pmod{8}$. ∎

As in the even case, when the rank is high enough we have a systematic idea of
the possibilities of the total space. However, in the low rank cases ($k\leq
6$) the results are not systematic, and may depend on individual cases rather
than only on $\sigma(M_{k}),k$, and the intersection form.

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