# Transfer and the Spectrum-Level Siegel-Sullivan KO-Orientation for Singular
Spaces

Markus Banagl  Mathematisches Institut, Universität Heidelberg, Im Neuenheimer
Feld 205, 69120 Heidelberg, Germany<EMAIL_ADDRESS>
(Date: May 2023)

###### Abstract.

Integrally oriented normally nonsingular maps between singular spaces have
associated transfer homomorphisms on KO-homology at odd primes. We prove that
such transfers preserve Siegel-Sullivan orientations, defined when the
singular spaces are compact pseudomanifolds satisfying the Witt condition, for
example pure-dimensional compact complex algebraic varieties. This holds for
bundle transfers associated to block bundles with manifold fibers as well as
for Gysin restrictions associated to normally nonsingular inclusions. Our
method is based on constructing a lift of the Siegel-Sullivan orientation to a
morphism of highly structured ring spectra which factors through L-theory.

###### Key words and phrases:

Stratified Spaces, Characteristic Classes, Orientation Classes, Bundle
Transfer, Gysin maps, Intersection Homology, Bordism, KO-Homology, L-Theory

###### 2020 Mathematics Subject Classification:

55N33, 55R12, 57N80, 57R20, 55N15, 19L41, 57Q20, 57Q50

This work is funded by a research grant of the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) – Projektnummer 495696766.

###### Contents

  1. 1 Introduction
  2. 2 Multiplicative Identification of KO- and L-theory away from $2$
  3. 3 The Orientations of Sullivan and Ranicki
  4. 4 The Classical Construction of the Siegel-Sullivan Orientation
  5. 5 Ring Spectrum Level Construction of the Siegel-Sullivan Orientation
  6. 6 Multiplicativity of Orientation Classes
  7. 7 Bundle Transfer of the Siegel-Sullivan Orientation
  8. 8 Gysin Restriction of the Siegel-Sullivan Orientation

##  1\. Introduction

Let ${\operatorname{KO}}_{*}(-)$ denote topological
${\operatorname{KO}}$-homology and let $M$ be a smooth $n$-dimensional closed
oriented manifold. In his MIT notes [56], Sullivan introduced a class
$\Delta_{\operatorname{SO}}(M)\in{\operatorname{KO}}_{n}(M)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$,
which is an orientation and plays a fundamental role in studying the
${\operatorname{K}}$-theory of manifolds. For instance, Sullivan showed that
topological block bundles away from $2$ are characterized as spherical
fibrations together with a
${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$-orientation. He went on to point
out in [57] that given a class of oriented piecewise-linear (PL)
pseudomanifolds equipped with a bordism invariant signature that extends the
signature of manifolds and satisfies Novikov additivity and a product formula,
an analogous procedure (based on suitable transversality results in the
singular context) still works to determine a canonical orientation in
${\operatorname{KO}}_{*}(-)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$. Goresky
and MacPherson’s intersection homology allowed for the construction of such
signature invariants when the pseudomanifolds have only strata of even
codimension, or more generally, if they satisfy the Witt condition introduced
by Siegel in [55]. An oriented PL pseudomanifold is a Witt space, if the
middle-perversity, middle-dimensional rational intersection homology of links
of odd-codimensional strata vanishes. This class contains all complex
algebraic varieties of pure dimension. The class of Witt spaces is contained
in a yet larger class of pseudomanifolds, introduced in [3], [4], that support
a bordism invariant signature. Roughly, these are spaces that admit a
Lagrangian subsheaf in the link cohomology sheaf along strata of odd
codimension. The present paper, however, will focus only on Witt spaces.

Thus Sullivan’s general framework implies that $n$-dimensional closed Witt
spaces $X$ have a canonical orientation
$\Delta(X)\in{\operatorname{KO}}_{*}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$.
The construction of this element was described in detail by Siegel in [55],
and is recalled in the present paper. We shall refer to it as the _Siegel-
Sullivan orientation_ of a Witt space. In [17], Cappell, Shaneson and
Weinberger extended this orientation to an equivariant class for finite group
actions satisfying a weak regularity condition on the fixed point sets. Under
the Pontrjagin character, $\Delta(X)$ is a lift of the Goresky-MacPherson
$L$-class $L_{*}(X)\in H_{*}(X;\mathbb{Q})$. The latter already contains
significant global information on the singular space $X$ (see [18], [60]) and
its concrete computation is correspondingly challenging. For complex
projective varieties $X$ it is often possible to obtain information on
$L_{*}(X)$ by cutting down to transverse intersections of $X$ with smooth
subvarieties using Gysin homomorphisms. For example it is possible to reduce
$L$-class computations for singular Schubert varieties entirely to signature
computations of explicitly given _algebraic_ subvarieties. This approach,
introduced in [7] and pursued systematically in [11], requires a thorough
understanding of how characteristic and orientation classes for singular
spaces transform under Gysin restriction.

Let $g:Y\hookrightarrow X$ be an oriented normally nonsingular codimension $c$
inclusion of closed Witt spaces. Thus $Y$ has an open tubular neighborhood in
$X$ which is endowed in a stratum preserving manner with the structure of an
oriented rank $c$ vector bundle. Since ${\operatorname{SO}}$-bundles are
${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$-oriented, $g$ has an associated
Gysin homomorphism
$g^{!}:{\operatorname{KO}}_{*}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\to{\operatorname{KO}}_{*-c}(Y)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].$
We prove (Theorem 8.2):

######  Theorem.

Let $g:Y^{n-c}\hookrightarrow X^{n}$ be an oriented normally nonsingular
inclusion of closed Witt spaces. The
${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$-homology Gysin map $g^{!}$ of $g$
sends the Siegel-Sullivan orientation of $X$ to the Siegel-Sullivan
orientation of $Y$:

$g^{!}\Delta(X)=\Delta(Y).$

An important class of morphisms in algebraic geometry is given by local
complete intersection morphisms ([25]) $Y\to B$. By definition, they admit a
factorization $Y\to X\to B$, where $Y\to X$ is a closed regular algebraic
embedding and $X\to B$ is a smooth morphism. The regular embedding has an
associated algebraic normal vector bundle and the smooth morphism has an
associated relative tangent bundle, so that l.c.i. morphisms possess a virtual
tangent bundle. A parallel topological notion of normally nonsingular map
$Y\to B$ has been considered by Goresky-MacPherson [29, 5.4.3] and Fulton-
MacPherson [26]. By definition, these admit factorizations $Y\to X\to B$ into
a normally nonsingular inclusion $Y\to X$ followed by a fiber bundle
projection $X\to B$ with manifold fiber. A complete picture should therefore
include an understanding of how the Siegel-Sullivan orientation behaves under
Becker-Gottlieb type bundle transfer ([12]). We shall thus also consider
bundle transfers $\xi^{!}$ associated to block bundles $\xi$ over compact Witt
spaces $B$ ([19]). These do not require a locally trivial projection map $X\to
B$, but merely a decomposition of $X$ into blocks over cells in $B$. (A fiber
bundle is a special case of a block bundle.) Thus, let $\xi$ be an oriented PL
$F$-block bundle with closed oriented $d$-dimensional PL manifold fiber $F$
over a closed Witt base $B$. Then, since the stable vertical normal block
bundle of $\xi$ is ${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$-oriented, there
is a block bundle transfer
$\xi^{!}:{\operatorname{KO}}_{n}(B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\to{\operatorname{KO}}_{n+d}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].$
We prove (Theorem 7.4):

######  Theorem.

If $\xi$ is an oriented PL $F$-block bundle with closed oriented PL manifold
fiber $F$ over a closed Witt base $B$, then the Siegel-Sullivan orientations
of base and total space $X$ are related under block bundle transfer by

$\xi^{!}\Delta(B)=\Delta(X).$

Rather than using Siegel’s original construction of $\Delta$ directly to prove
the above transfer results, we give a new description of $\Delta$ based on a
homotopy theoretic perspective relating K- to L-theory: We provide here a lift
of the Siegel-Sullivan orientation to a ring spectrum level morphism

$\Delta:\operatorname{MWITT}\longrightarrow{\operatorname{KO}}[\mbox{$\frac{1}{2}$}],$

where $\operatorname{MWITT}$ denotes the ring spectrum representing Witt space
bordism theory, constructed as in [10] via the ad-theories of Laures and
McClure. A particularly important aspect of $\Delta$ for our present purposes
is its multiplicativity. On homotopy groups, $\Delta_{*}$ sends the bordism
class of a closed Witt space $X^{4k}$ to its signature $\sigma(X)$. In order
to obtain our ring spectrum level description of $\Delta$, we use results of
Land and Nikolaus [33] to construct in Proposition 2.1 an equivalence of
highly structured ring spectra
${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\simeq\mathbb{L}()[\mbox{$\frac{1}{2}$}]$
which maps the element in $\pi_{4}({\operatorname{KO}})[\mbox{$\frac{1}{2}$}]$
whose complexification is the square of the complex Bott element to the
signature $1$ element in $\pi_{4}(\mathbb{L}(\mathbb{Z}))$, where
$\mathbb{L}(R)$ denotes the (projective) symmetric algebraic L-theory spectrum
of a ring $R$ with involution, introduced first by Ranicki. Under this
equivalence, the Siegel-Sullivan orientation $\Delta(X)$ of a Witt space
corresponds to the $\mathbb{L}(\mathbb{Q})$-homology orientation
$[X]_{\mathbb{L}}$ of Laures, McClure and the author, which generalizes
Ranicki’s $\mathbb{L}$-homology orientation of manifolds to singular spaces.
This then enables us to use $\mathbb{L}$-theoretic transfer results
established in [7] and [8]. For a PL $F$-fiber bundle $p:X\to B$ over a PL
manifold base $B$, the transfer formula
$p^{!}[B]_{\mathbb{L}}=[X]_{\mathbb{L}}\in\mathbb{L}(\mathbb{Z})_{n+d}(X)$ was
stated by Lück and Ranicki in [37].

From the analytic viewpoint, Sullivan’s orientation
$\Delta_{\operatorname{SO}}(M)$ is for a (closed, oriented) Riemannian
manifold $M$ closely related to the class of the signature operator in
Kasparov’s model
${\operatorname{K}}_{*}(M)=\operatorname{KK}_{*}(C(M),\mathbb{C})$ of the
K-homology of $M$, see for example [52] and [34, Prop. 8.3]. Modulo $2$-power
torsion, the two classes differ by certain powers of $2$. For smoothly
stratified Witt spaces $X$ equipped with an incomplete iterated edge metric on
the regular part, Albin, Leichtnam, Mazzeo and Piazza construct in [1] a
signature operator $\eth_{\operatorname{sign}}$ and a
${\operatorname{K}}$-homology class
$[\eth_{\operatorname{sign}}]\in{\operatorname{K}}_{*}(X)$. Again, it is
possible to go well beyond Witt spaces: The topological cohomology theory of
[3] and the analytic $L^{2}$ de Rham theory have been treated from a common
perspective in [2].

In view of the algebraic results of [7] and [11], the conclusions of the
present paper are also relevant in the context of a question raised by Jörg
Schürmann in [54]: Is the Siegel-Sullivan orientation $\Delta(X)$ of a pure-
dimensional compact complex algebraic variety $X$ the image of the
intersection homology (mixed) Hodge module on $X$ under the motivic Hodge
Chern class transformation
$\operatorname{MHC}_{1}:K_{0}(\operatorname{MHM}(X))\to
K^{\operatorname{coh}}_{0}(X)$ of Brasselet-Schürmann-Yokura [14], followed by
the $K$-theoretical Riemann-Roch transformation of Baum, Fulton and
MacPherson?

The above material is developed as follows: Section 2 collects relevant
homotopy theoretic information on ${\operatorname{KO}}$ and $\mathbb{L}$, and
constructs the particular equivalence between ${\operatorname{KO}}$ and
$\mathbb{L}$ away from $2$ used throughout the rest of the paper. The
orientations of Sullivan and Ranicki for the nonsingular case are reviewed in
Section 3. Section 4 sketches the classical construction of the Siegel-
Sullivan orientation for singular Witt spaces given in [55]. Our ring spectrum
level construction of the Siegel-Sullivan orientation is the focus of Section
5. An immediate application of this construction is a proof of cartesian
multiplicativity of the Siegel-Sullivan orientation (Theorem 6.2) in Section
6. In Section 7, we proceed to apply our multiplicative spectrum level
construction in establishing the normally nonsingular block bundle transfer
result, while normally nonsingular Gysin restrictions of the Siegel-Sullivan
orientation are the subject of the final Section 8.

Acknowledgements. We thank Jörg Schürmann for interesting discussions on the
subject matter on this paper. We especially express our gratitude to Markus
Land for discussions on the relation of K- and L-theory away from $2$.

##  2\. Multiplicative Identification of KO- and L-theory away from $2$

Our method is based on finding a multiplicative equivalence
$\mathbb{L}[\mbox{$\frac{1}{2}$}]\simeq{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$
that yields the Sullivan orientation when precomposed with Ranicki’s
orientation
$\operatorname{MSPL}\to\mathbb{L}\to\mathbb{L}[\mbox{$\frac{1}{2}$}]$. We
describe such an equivalence in the present section, based on an equivalence
of highly structured ring spectra obtained by Land and Nikolaus in [33].

Let $R$ be a commutative unital ring with involution and let $\mathbb{L}(R)$
denote the (projective) symmetric algebraic L-theory spectrum of $R$,
introduced first by Ranicki. (See e.g. [47]; there is no need for our notation
to distinguish between the symmetric and the quadratic L-theory spectrum,
since the latter will not be used in the present paper.) The only instances of
$R$ used in this paper are the ring of integers and the fields of rational,
real or complex numbers. Since $\widetilde{K}_{0}(R)$ vanishes in these cases,
there is no difference between the free and the projective L-theory. The
involution on $\mathbb{Z},\mathbb{Q}$ and is taken to be the trivial
involution, while we consider $\mathbb{C}$ to be endowed with the complex
conjugation involution. If $R=\mathbb{Z},$ we shall briefly write
$\mathbb{L}=\mathbb{L}(\mathbb{Z})$. The spectrum $\mathbb{L}(R)$ is a ring
spectrum, and a morphism $R\to S$ of commutative rings with involution induces
a morphism $\mathbb{L}(R)\to\mathbb{L}(S)$. For the inclusions
$\mathbb{Z}\subset\mathbb{Q}\subset\subset\mathbb{C}$, one obtains morphisms

$\mathbb{L}(\mathbb{Z})\longrightarrow\mathbb{L}(\mathbb{Q})\longrightarrow\mathbb{L}()\longrightarrow\mathbb{L}(\mathbb{C}).$

The multiplicative symmetric Poincaré ad-theory of Laures and McClure [35]
shows that these are morphisms of ring spectra. Their lax symmetric monoidal
functor also shows that $\mathbb{L}(R)$ for a commutative ring $R$ can be
realized as a commutative symmetric ring spectrum. Thus $\mathbb{L}(R)$ is
equivalent to an $\mathbb{E}_{\infty}$-ring spectrum, that is, the
multiplication is commutative and associative not just up to homotopy, but up
to coherent systems of homotopies. If $E$ is a ring spectrum and $A$ a subring
of $\mathbb{Q}$, then the localized spectrum $E_{A}$ is endowed with a unique
(up to ring equivalence) ring structure such that the localization morphism
$E\to E_{A}$ is a ring morphism. Ring morphisms $E\to F$ localize to ring
morphisms $E_{A}\to F_{A}$. If $A=\mathbb{Z}[\mbox{$\frac{1}{2}$}],$ we will
write $E[\mbox{$\frac{1}{2}$}]$ for $E_{A}$. Thus there are canonical
morphisms of ring spectra

$\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]\longrightarrow\mathbb{L}(\mathbb{Q})[\mbox{$\frac{1}{2}$}]\longrightarrow\mathbb{L}()[\mbox{$\frac{1}{2}$}]\longrightarrow\mathbb{L}(\mathbb{C})[\mbox{$\frac{1}{2}$}].$

The first two morphisms are equivalences, the last one is not. Indeed, the
homotopy ring of $\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]$ is given by
$\mathbb{L}_{*}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]=\mathbb{Z}[\mbox{$\frac{1}{2}$}][x^{\pm
1}],$ where $x\in\mathbb{L}_{4}(\mathbb{Z})$ is the signature $1$ element.
(The degree $1$ torsion element in $\mathbb{L}_{*}(\mathbb{Z})$ is removed by
inverting $2$.) The homotopy ring of $\mathbb{L}()$ is given by
$\mathbb{L}_{*}()=\mathbb{Z}[x^{\pm 1}],$ where $x\in\mathbb{L}_{4}()$ denotes
the image of the class $x\in\mathbb{L}_{4}(\mathbb{Z})$. The homotopy groups
$\mathbb{L}_{i}(\mathbb{Q})$ vanish in degrees $i$ not divisible by $4$. For
$i=4k,$ they contain an infinitely generated amount of $2$-primary torsion,

$\mathbb{L}_{4k}(\mathbb{Q})=\mathbb{L}_{4k}()\oplus\bigoplus_{p\text{
prime}}\mathbb{L}_{4k}(\mathbb{F}_{p})=\mathbb{Z}\oplus(\mathbb{Z}/_{2})^{\infty}\oplus(\mathbb{Z}/_{4})^{\infty},$

where $\mathbb{F}_{p}$ denotes the finite field with $p$ elements. The
signature homomorphism $\mathbb{L}_{4k}(\mathbb{Q})\to\mathbb{Z}$ provides the
unique splitting for the unique ring homomorphism
$\mathbb{Z}\to\mathbb{L}_{4k}(\mathbb{Q})$. The infinitely generated amount of
$2$-primary torsion is then removed by inverting $2$. Via the above canonical
multiplicative maps, we shall identify
$\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}],$
$\mathbb{L}(\mathbb{Q})[\mbox{$\frac{1}{2}$}]$ and
$\mathbb{L}()[\mbox{$\frac{1}{2}$}]$ as $\mathbb{E}_{\infty}$-ring spectra.
The homotopy ring of $\mathbb{L}(\mathbb{C})$ (with conjugation involution on
$\mathbb{C}$) is $\mathbb{L}_{*}(\mathbb{C})=\mathbb{Z}[b^{\pm 1}],$ where $b$
has degree $2$. The spectrum $\mathbb{L}(\mathbb{C})$ is $2$-periodic. On
homotopy rings, the map $\mathbb{L}()\to\mathbb{L}(\mathbb{C})$ induces the
map $\mathbb{Z}[x^{\pm 1}]\to\mathbb{Z}[b^{\pm 1}],$ $x\mapsto b^{2}$.

Let ${\operatorname{KO}}$ denote the $8$-periodic ring spectrum representing
real K-theory and ${\operatorname{K}}$ the $2$-periodic ring spectrum
representing complex K-theory. The homotopy ring of ${\operatorname{K}}$ is
$\pi_{*}({\operatorname{K}})=\mathbb{Z}[\beta^{\pm 1}],$ where $\beta$ is the
complex Bott element in degree $2$, i.e. $\beta$ is represented by the reduced
canonical complex line bundle
$H-1\in\widetilde{{\operatorname{K}}}^{0}(S^{2})$. The complexification
$c:\operatorname{BO}\to\operatorname{BU}$ can be lifted to a morphism of
spectra $c:{\operatorname{KO}}\to{\operatorname{K}}$ ([53, p. 360, Lemma
VI.3.3]). On $\pi_{4}$, $c$ induces multiplication by $2$,
$c_{*}=2:\pi_{4}({\operatorname{KO}})=\mathbb{Z}\to\mathbb{Z}=\pi_{4}({\operatorname{K}})$.
Thus there does not exist an element in $\pi_{4}({\operatorname{KO}})$ that
maps to $\beta^{2}$. But after inverting $2$, such an element exists. Let
$a\in\pi_{4}({\operatorname{KO}})[\mbox{$\frac{1}{2}$}]$ be the element whose
complexification is $\beta^{2}$. The localization
${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$ is a $4$-periodic ring spectrum
with homotopy ring
$\pi_{*}({\operatorname{KO}})[\mbox{$\frac{1}{2}$}]=\mathbb{Z}[\mbox{$\frac{1}{2}$}][a^{\pm
1}].$

Taylor and Williams showed in [59, Theorem A] that there is an equivalence
$\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]\simeq{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$
of spectra. In [51], Rosenberg asserts that these spectra are equivalent as
homotopy ring spectra. Further arguments in this direction are supplied by
Lurie [38], who proves that these spectra are quivalent as homotopy ring
spectra. Land and Nikolaus [33, p. 550] construct an equivalence of
$\mathbb{E}_{\infty}$-ring spectra
${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\simeq\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}].$

######  Proposition 2.1.

There exists an equivalence of $\mathbb{E}_{\infty}$-ring spectra

$\kappa:{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathbb{L}()[\mbox{$\frac{1}{2}$}]$

which induces the ring isomorphism

$\mathbb{Z}[\mbox{$\frac{1}{2}$}][a^{\pm
1}]\longrightarrow\mathbb{Z}[\mbox{$\frac{1}{2}$}][x^{\pm 1}],~{}a\mapsto x$

on homotopy rings.

###### Proof.

We are indebted to Markus Land for communication on the following argument.
Let
$\tau_{:}{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\to\mathbb{L}[\mbox{$\frac{1}{2}$}]$
be the equivalence of $\mathbb{E}_{\infty}$-ring spectra given in [33, Cor.
5.4]. This equivalence is related to a complex version
$\tau_{\mathbb{C}}:{\operatorname{K}}[\mbox{$\frac{1}{2}$}]\to\mathbb{L}\mathbb{C}[\mbox{$\frac{1}{2}$}]$
by the commutative diagram

$\textstyle{{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\scriptstyle{\tau}$$\textstyle{\mathbb{L}()[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{K}}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{\mathbb{C}}}$$\textstyle{\mathbb{L}(\mathbb{C})[\mbox{$\frac{1}{2}$}].}$

(There is no integral version of $\tau_{\mathbb{C}}$ on the periodic spectra,
although there _is_ an integral version $\operatorname{k}\to\ell\mathbb{C}$ on
connective spectra, which induces $\tau_{\mathbb{C}}$ on the periodic spectra
after inverting $2$.) By [33, Lemma 4.9],
$\tau_{\mathbb{C}*}:\pi_{2}({\operatorname{K}})[\mbox{$\frac{1}{2}$}]\to\pi_{2}(\mathbb{L}\mathbb{C})[\mbox{$\frac{1}{2}$}]$
maps $\beta\mapsto 2b$. Since $\tau_{\mathbb{C}}$ is a map of
$\mathbb{E}_{\infty}$-ring spectra, it follows that it sends $\beta^{2}\mapsto
4b^{2}$. Consequently, the right hand vertical map sends the element
$\tau_{*}(a)$ to

$\tau_{\mathbb{C}*}c_{*}(a)=\tau_{\mathbb{C}*}(\beta^{2})=4b^{2}.$

Since
$\mathbb{L}()[\mbox{$\frac{1}{2}$}]\to\mathbb{L}(\mathbb{C})[\mbox{$\frac{1}{2}$}]$
maps $x\mapsto b^{2}$, we deduce that $\tau_{*}(a)=4x$.

Let

$\psi^{2}:{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\longrightarrow{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$

be the stable Adams operation, constructed as a morphism of
$\mathbb{E}_{\infty}$-ring spectra ([21], [22, p. 3], [41, p. 106]). On the
homotopy groups
$\pi_{4k}{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]=\mathbb{Z}[\mbox{$\frac{1}{2}$}],$
$\psi^{2}$ induces $a^{k}\mapsto 4^{k}a^{k}$. Thus $\psi^{2}$ induces an
isomorphism of homotopy rings, and is therefore an equivalence. Composing the
inverse of $\psi^{2}$ with the Land-Nikolaus equivalence
$\tau_{:}{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\to\mathbb{L}[\mbox{$\frac{1}{2}$}]$,
we obtain the desired equivalence $\kappa$ of $\mathbb{E}_{\infty}$-ring
spectra since the map induced by the composition on $\pi_{4}$ sends

$a\mapsto\mbox{$\frac{1}{4}$}a\stackrel{{\scriptstyle\tau_{*}}}{{\mapsto}}\mbox{$\frac{1}{4}$}(4x)=x.$

∎

##  3\. The Orientations of Sullivan and Ranicki

Let $\operatorname{MSO},\operatorname{MSPL}$ and $\operatorname{MSTOP}$ denote
the Thom spectra of oriented vector-, PL- and topological bundles. These are
ring spectra and Pontrjagin-Thom isomorphisms identify their homotopy groups
with the bordism groups
$\Omega^{\operatorname{SO}}_{*},\Omega^{\operatorname{SPL}}_{*},\Omega^{\operatorname{STOP}}_{*}$
of oriented smooth, PL or topological manifolds. (The topological spectrum
$\operatorname{MSTOP}$ will not play an essential role in what follows, but
occasional side remarks will involve it.) Sullivan obtained in [56] a morphism
of spectra

$\Delta_{\operatorname{SO}}:\operatorname{MSO}\longrightarrow{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$

such that the induced map on homotopy groups

$\Delta_{{\operatorname{SO}}*}:\Omega^{\operatorname{SO}}_{4k}=\operatorname{MSO}_{4k}\longrightarrow{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]_{4k}=\mathbb{Z}[\mbox{$\frac{1}{2}$}]\langle
a^{k}\rangle$

is

(1) $\Delta_{{\operatorname{SO}}*}[M^{4k}]=\sigma(M)\cdot a^{k},$

where $\sigma(M)$ denotes the signature of the smooth oriented closed manifold
$M$, see [39, pp. 83–85]. The Pontrjagin character
$\operatorname{ph}:{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\to
H\mathbb{Q}[t^{\pm 1}]$ of $\Delta_{\operatorname{SO}}$ is the inverse of the
universal Hirzebruch $L$-class $L\in H^{*}(\operatorname{BSO};\mathbb{Q})$ up
to multiplication with the stable Thom class $u\in
H^{0}(\operatorname{MSO};\mathbb{Z})$,

$\operatorname{ph}(\Delta_{\operatorname{SO}})=L^{-1}\cup u\in
H^{*}(\operatorname{MSO};\mathbb{Q}).$

The _Sullivan orientation_ of a smooth closed $n$-dimensional manifold $M$ is
given by the image

$\Delta_{\operatorname{SO}}(M)=\Delta_{{\operatorname{SO}}*}[\operatorname{id}_{M}]\in{\operatorname{KO}}_{n}(M)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$

of the bordism class of the identity on $M$ under the homomorphism
$\Delta_{{\operatorname{SO}}*}:\Omega^{\operatorname{SO}}_{n}(M)\to({\operatorname{KO}}[\mbox{$\frac{1}{2}$}])_{n}(M)$
induced by the spectrum level Sullivan orientation
$\Delta_{\operatorname{SO}}$. If $M$ has a boundary $\partial M$, then
$\Delta_{\operatorname{SO}}(M)$ is an element in the relative group
${\operatorname{KO}}_{n}(M,\partial
M)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$.

The orientation $\Delta_{\operatorname{SO}}$ extends canonically to a map
$\Delta_{\operatorname{SPL}}:\operatorname{MSPL}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$
with respect to the canonical map $\operatorname{MSO}\to\operatorname{MSPL}$
([39, Chapter 5.A, D]). On homotopy groups, the induced map continues to be
given by the signature, i.e.

(2) $\Delta_{{\operatorname{SPL}}*}[M^{4k}]=\sigma(M)\cdot a^{k}$

for an oriented closed PL manifold $M$. The Pontrjagin character is

(3)
$\operatorname{ph}(\Delta_{\operatorname{SPL}})=L^{-1}_{\operatorname{PL}}\cup
u_{\operatorname{PL}}\in H^{*}(\operatorname{MSPL};\mathbb{Q}),$

where $L_{\operatorname{PL}}$ is the universal PL $L$-class
$L_{\operatorname{PL}}\in H^{*}(\operatorname{BSPL};\mathbb{Q})$ and
$u_{\operatorname{PL}}$ the stable Thom class $u_{\operatorname{PL}}\in
H^{0}(\operatorname{MSPL};\mathbb{Z})=\mathbb{Z}$ ([39, Cor. 5.4, p. 102]).
Note that $L_{\operatorname{PL}}$ restricts to $L$ under the canonical map
$\operatorname{BSO}\to\operatorname{BSPL}$. We recall that this map is a
rational equivalence, so in particular induces an isomorphism
$H^{*}(\operatorname{BSPL};\mathbb{Q})\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}H^{*}(\operatorname{BSO};\mathbb{Q})$,
and this isomorphism identifies Thom’s Pontrjagin class $p_{4i}\in
H^{4i}(\operatorname{BSPL};\mathbb{Q})$ with the rational reduction $p_{4i}\in
H^{4i}(\operatorname{BSO};\mathbb{Q})$ of the integral Pontrjagin class
(though the PL Pontrjagin classes are in general not integral). For this
reason, one commonly identifies $L$ and $L_{\operatorname{PL}}$ and simply
writes $L$ for it. An $n$-dimensional closed PL manifold has a Sullivan
orientation

$\Delta_{\operatorname{SPL}}(M)=\Delta_{{\operatorname{SPL}}*}[\operatorname{id}_{M}]\in{\operatorname{KO}}_{n}(M)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}],$

where $[\operatorname{id}_{M}]\in\Omega^{\operatorname{SPL}}_{n}(M)$.

######  Remark 3.1.

Randal-Williams observes in [46, p. 9] that this map can be further
canonically extended to a map
$\Delta_{\operatorname{STOP}}:\operatorname{MSTOP}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$
with respect to the canonical forget map
$\operatorname{MSPL}\to\operatorname{MSTOP}$, since the fiber of the latter
map is $2$-local. In particular, given an element
$[N]\in\Omega^{\operatorname{STOP}}_{4k}=\operatorname{MSTOP}_{4k},$ there
exists a large integer $i$ such that $2^{i}N$ is topologically bordant to a PL
manifold $M$, for which (2) is available. Thus

$\displaystyle 2^{i}\Delta_{{\operatorname{STOP}}*}[N]$
$\displaystyle=\Delta_{{\operatorname{STOP}}*}[M]=\Delta_{{\operatorname{SPL}}*}[M]=\sigma(M)\cdot
a^{k}$ $\displaystyle=\sigma(2^{i}N)\cdot a^{k}=2^{i}\sigma(N)\cdot a^{k}.$

This shows that the map induced by $\Delta_{\operatorname{STOP}}$ on homotopy
groups is again given by the signature:

$\Delta_{{\operatorname{STOP}}*}[M^{4k}]=\sigma(M)\cdot a^{k}$

for an oriented closed topological manifold $M$.

In the present paper, we take an $\mathbb{L}$-theoretic perspective in order
to approach the orientations of Sullivan and Siegel-Sullivan. For smooth
manifolds $M$, it has been known for a long time that Ranicki’s fundamental
class $[M]_{\mathbb{L}}$ agrees with Sullivan’s class
$\Delta_{\operatorname{SO}}(M)$ under the appropriate identification of
${\operatorname{KO}}$\- and $\mathbb{L}$-homology at odd primes, see for
example Ranicki [47, p. 15], and Weinberger [60, p. 82]. Let $L^{n}(R)$ denote
Ranicki’s (projective) symmetric $L$-groups of a ring $R$ with involution. In
[48, p. 385, Prop. 15.8], Ranicki constructed a morphism of ring spectra

$\sigma^{*}:\operatorname{MSPL}\longrightarrow\mathbb{L}(\mathbb{Z})$

such that the resulting $\mathbb{L}(\mathbb{Z})$-homology fundamental class

$[M]_{\mathbb{L}}:=\sigma^{*}[\operatorname{id}_{M}]\in\mathbb{L}(\mathbb{Z})_{n}(M)$

of a closed oriented PL $n$-dimensional manifold $M$ hits the Mishchenko-
Ranicki symmetric signature

$\sigma^{*}(M)=A[M]_{\mathbb{L}}\in L^{n}(\mathbb{Z}[\pi_{1}M])$

under the assembly map

$A:\mathbb{L}(\mathbb{Z})_{n}(M)\longrightarrow L^{n}(\mathbb{Z}[\pi_{1}M]).$

(Ranicki extended $\sigma^{*}$ to a morphism of ring spectra
$\operatorname{MSTOP}\longrightarrow\mathbb{L}(\mathbb{Z})$ in [50, p. 290],
but we shall not require this extension for the purposes of the present
paper.) Technically, we will work with $\sigma^{*}$ as constructed by Laures
and McClure in [35] using ad-theories. Their incarnation of $\sigma^{*}$ is an
$\mathbb{E}_{\infty}$-ring map ([35, 1.4]). The localization morphism
$\mathbb{L}(\mathbb{Z})\to\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]$ is a
morphism of ring spectra. Thus its composition with $\sigma^{*}$ is a morphism
of ring spectra
$\operatorname{MSPL}\to\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]$, which we
shall also denote by $\sigma^{*}$. By [49, p. 243], $\sigma^{*}$ induces on
homotopy groups the map

$\sigma^{*}_{\operatorname{pt}}:\Omega^{\operatorname{SPL}}_{4k}(\operatorname{pt})=\operatorname{MSPL}_{4k}(\operatorname{pt})\longrightarrow\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]_{4k}=\mathbb{Z}[\mbox{$\frac{1}{2}$}]\langle
x^{k}\rangle$

given by

(4) $\sigma^{*}_{\operatorname{pt}}[M^{4k}]=\sigma(M)\cdot x^{k}.$

######  Remark 3.2.

Let $M$ be a closed oriented PL manifold of dimension $n$. The constant map
$c:M\to\operatorname{pt}$ induces a diagram

$\textstyle{\Omega^{\operatorname{SPL}}_{n}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{*}}$$\scriptstyle{c_{*}}$$\textstyle{\mathbb{L}(\mathbb{Z})_{n}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\scriptstyle{c_{*}}$$\textstyle{L^{n}(\mathbb{Z}[\pi_{1}M])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{*}}$$\textstyle{\Omega^{\operatorname{SPL}}_{n}(\operatorname{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{*}_{\operatorname{pt}}}$$\textstyle{\mathbb{L}(\mathbb{Z})_{n}(\operatorname{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\textstyle{L^{n}(\mathbb{Z})}$

which commutes, since the assembly map $A$ is natural. Together with Equation
(4), this diagram shows that the homomorphism $L^{n}(\mathbb{Z}[\pi_{1}M])\to
L^{n}(\mathbb{Z}),$ $n=4k,$ sends the symmetric signature of $M$ to its
ordinary signature. Indeed,

$\displaystyle c_{*}\sigma^{*}(M)$
$\displaystyle=c_{*}A[M]_{\mathbb{L}}=c_{*}A\sigma^{*}[\operatorname{id}_{M}]$
$\displaystyle=\sigma^{*}_{\operatorname{pt}}c_{*}[\operatorname{id}_{M}]=\sigma^{*}_{\operatorname{pt}}[M]=\sigma(M)x^{k}.$

The analogous fact holds also for singular pseudomanifolds that satisfy the
Witt condition and will be used later to compute the behavior of the ring-
spectrum level Siegel-Sullivan orientation defined in the present paper on
homotopy groups (Proposition 5.4).

Ranicki [48, p. 390f] introduced an $\mathbb{L}$-theoretic Thom class

$u_{\mathbb{L}}(\alpha)\in\widetilde{\mathbb{L}}^{m}(\operatorname{Th}(\alpha))$

for oriented rank $m$ PL microbundles (or PL $(^{m},0)$-bundles) $\alpha$ as
follows: The classifying map $X\to\operatorname{BSPL}_{m}$ of $\alpha$ (where
$X$ is the base space) is covered by a bundle map from $\alpha$ to the
universal oriented PL microbundle. The induced map on Thom spaces yields a
class

$u_{\operatorname{SPL}}(\alpha)\in\widetilde{\operatorname{MSPL}}^{m}(\operatorname{Th}(\alpha)),$

the _Thom class_ of $\alpha$ in oriented PL cobordism. It is indeed an
$\operatorname{MSPL}$-orientation of $\alpha$ in Dold’s sense. Ranicki then
defines

$u_{\mathbb{L}}(\alpha):=\sigma^{*}(u_{\operatorname{SPL}}(\alpha)).$

Since $\sigma^{*}:\operatorname{MSPL}\to\mathbb{L}(\mathbb{Z})$ is
multiplicative, the element $u_{\mathbb{L}}(\alpha)$ is indeed an
$\mathbb{L}$-orientation of $\alpha$. This can also be carried out for stable
PL bundles $\alpha$. For the universal stable PL bundle there is thus a
canonical $\mathbb{L}$-orientation
$u_{\mathbb{L}}\in\widetilde{\mathbb{L}}^{0}(\operatorname{MSPL})$. Since the
stable Thom class
$u_{\operatorname{SPL}}\in\widetilde{\operatorname{MSPL}}^{0}(\operatorname{MSPL})$
of the universal stable PL bundle is given by the identity
$\operatorname{MSPL}\to\operatorname{MSPL},$ we have
$u_{\mathbb{L}}=\sigma^{*}$. The morphism of ring spectra
$\mathbb{L}(\mathbb{Z})\to\mathbb{L}(\mathbb{Q})$ induces a homomorphism

$\widetilde{\mathbb{L}(\mathbb{Z})}^{m}(\operatorname{Th}(\alpha))\longrightarrow\widetilde{\mathbb{L}(\mathbb{Q})}^{m}(\operatorname{Th}(\alpha)).$

We denote the image of $u_{\mathbb{L}}(\alpha)$ under this map again by
$u_{\mathbb{L}}(\alpha).$ Furthermore, the images of these elements in the
bottom row of the localization square

$\textstyle{(\widetilde{\mathbb{L}\mathbb{Z}})^{m}(\operatorname{Th}(\alpha))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widetilde{\mathbb{L}\mathbb{Q}})^{m}(\operatorname{Th}(\alpha))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widetilde{\mathbb{L}\mathbb{Z}[\mbox{$\frac{1}{2}$}]})^{m}(\operatorname{Th}(\alpha))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widetilde{\mathbb{L}\mathbb{Q}[\mbox{$\frac{1}{2}$}]})^{m}(\operatorname{Th}(\alpha))}$

will also be written as $u_{\mathbb{L}}(\alpha).$ Since all maps in the square
are induced by morphisms of ring spectra, all these image elements are again
orientations of $\alpha$.

Comparing Equations (2) and (4), we find that the diagram

$\textstyle{\operatorname{MSPL}_{4k}(\operatorname{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta_{{\operatorname{SPL}}*}}$$\scriptstyle{\sigma^{*}_{\operatorname{pt}}}$$\textstyle{\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]_{4k}}$$\textstyle{{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]_{4k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\kappa}$

commutes. More is true:

######  Proposition 3.3.

The composition

$\operatorname{MSPL}\stackrel{{\scriptstyle\Delta_{\operatorname{SPL}}}}{{\longrightarrow}}{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\stackrel{{\scriptstyle\kappa}}{{\longrightarrow}}\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]$

of Sullivan’s orientation $\Delta_{\operatorname{SPL}}$ with the ring
equivalence $\kappa$ from Proposition 2.1 is homotopic to Ranicki’s
orientation $\sigma^{*}$.

###### Proof.

We start _at_ the prime $2$ with the cohomology class $L\in
H^{4*}(\mathbb{L};\mathbb{Z}_{(2)})$ constructed by Taylor and Williams in
[59]. This yields a specific homotopy class

$L:\mathbb{L}(\mathbb{Z})_{(2)}\longrightarrow\bigoplus_{i\in\mathbb{Z}}H\mathbb{Z}_{(2)}[4i].$

The pullback of this class under Ranicki’s orientation $\sigma^{*}$
corresponds under the Thom isomorphism to the Morgan-Sullivan class
$\mathcal{L}\in H^{*}(\operatorname{BSPL};\mathbb{Z}_{(2)})$ of [43].
Rationally, $\mathcal{L}$ becomes the inverse $L^{-1}\in
H^{*}(\operatorname{BSPL};\mathbb{Q})$ of the Thom-Hirzebruch $L$-class. The
composition

$\mathbb{L}()_{(2)}\longrightarrow\mathbb{L}(\mathbb{Z})_{(2)}\stackrel{{\scriptstyle
L}}{{\longrightarrow}}\bigoplus_{i\in\mathbb{Z}}H\mathbb{Z}_{(2)}[4i]$

is an equivalence. (The individual arrows are not — the discrepancy is the de
Rham invariant.) Rationally (i.e. inverting $2$), this gives an equivalence

$\mathbb{L}()_{(0)}=\mathbb{L}(\mathbb{Z})_{(0)}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\bigoplus_{i\in\mathbb{Z}}H\mathbb{Q}[4i].$

The map

$\mathbb{L}^{*}(\operatorname{MSPL})\longrightarrow
H^{*}(\operatorname{MSPL};\mathbb{Q})$

induced by the composition

$\mathbb{L}(\mathbb{Z})\stackrel{{\scriptstyle\operatorname{loc}}}{{\longrightarrow}}\mathbb{L}(\mathbb{Z})_{(0)}=\mathbb{L}()_{(0)}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\bigoplus_{i\in\mathbb{Z}}H\mathbb{Q}[4i]$

thus sends the universal stable $\mathbb{L}$-orientation
$u_{\mathbb{L}}=\sigma^{*}\in\widetilde{\mathbb{L}}^{0}(\operatorname{MSPL})$
to

$L^{-1}_{\operatorname{PL}}\cup u_{\operatorname{PL}}\in
H^{*}(\operatorname{MSPL};\mathbb{Q}),$

see also [47, Remark 16.2, p. 176]. It sends the signature $1$ element
$x^{k}\in\mathbb{L}_{4k}(\mathbb{Z})$ to
$1\in\pi_{4k}(\bigoplus_{i\in\mathbb{Z}}H\mathbb{Q}[4i])$. Consider the
diagram

(5)
$\textstyle{{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\scriptstyle{\simeq}$$\scriptstyle{\operatorname{loc}}$$\textstyle{{\operatorname{KO}}_{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{(0)}}$$\scriptstyle{\simeq}$$\scriptstyle{\operatorname{ph}}$$\scriptstyle{\simeq}$$\textstyle{\bigoplus_{i\in\mathbb{Z}}H\mathbb{Q}[4i]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{L}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{loc}}$$\textstyle{\mathbb{L}_{(0)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\bigoplus_{i\in\mathbb{Z}}H\mathbb{Q}[4i].}$

The left hand localization square commutes for general reasons. The right hand
square commutes up to homotopy as well: Since the involved spectra are graded
Eilenberg-MacLane spectra of graded $\mathbb{Q}$-vector spaces, it suffices to
check that the induced square on homotopy rings commutes. The vertical
isomorphism induced by $\kappa_{(0)}$ sends, according to its very
construction, the generator $a^{k}\in\pi_{4k}({\operatorname{KO}}_{(0)})$ to
$x^{k}\in\pi_{4k}(\mathbb{L}_{(0)})$, which in turn maps to
$1\in\pi_{4k}(\bigoplus_{i\in\mathbb{Z}}H\mathbb{Q}[4i])$. As for the
Pontrjagin character,

$\operatorname{ph}(a^{k})=(\operatorname{ch}\circ
c)(a^{k})=\operatorname{ch}(\beta^{2k})=\operatorname{ch}(\beta)^{2k}=1.$

Hence the right hand square commutes up to homotopy. During the course of the
above argument, we have drawn upon several relevant remarks made by Randal-
Williams in [46]. Now evaluate the above diagram on $\operatorname{MSPL}$:

$\textstyle{({\operatorname{KO}}[\mbox{$\frac{1}{2}$}])^{0}(\operatorname{MSPL})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\scriptstyle{\simeq}$$\scriptstyle{\operatorname{loc}}$$\textstyle{({\operatorname{KO}}_{(0)})^{0}(\operatorname{MSPL})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{(0)}}$$\scriptstyle{\simeq}$$\scriptstyle{\operatorname{ph}}$$\scriptstyle{\simeq}$$\textstyle{\bigoplus_{i\in\mathbb{Z}}H^{4i}(\operatorname{MSPL};\mathbb{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{L}[\mbox{$\frac{1}{2}$}])^{0}(\operatorname{MSPL})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{loc}}$$\textstyle{(\mathbb{L}_{(0)})^{0}(\operatorname{MSPL})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\bigoplus_{i\in\mathbb{Z}}H^{4i}(\operatorname{MSPL};\mathbb{Q}).}$

We shall show that the elements

$\Delta_{\operatorname{SPL}},~{}\kappa^{-1}\circ\sigma^{*}\in({\operatorname{KO}}[\mbox{$\frac{1}{2}$}])^{0}(\operatorname{MSPL})$

are equal. According to (3),
$\operatorname{ph}(\Delta_{\operatorname{SPL}})=L^{-1}_{\operatorname{PL}}\cup
u_{\operatorname{PL}}$. By the commutativity of the diagram, the Pontrjagin
character $\operatorname{ph}(\kappa^{-1}\circ\sigma^{*})$, given by mapping
the element horizontally, can alternatively be calculated by first mapping
down vertically, and then mapping to the right horizontally. Mapping down via
$\kappa$ yields $\sigma^{*}$, which is then mapped to
$L^{-1}_{\operatorname{PL}}\cup u_{\operatorname{PL}}$ as discussed above. It
follows that the two elements have the same Pontrjagin character,

$\operatorname{ph}(\Delta_{\operatorname{SPL}})=\operatorname{ph}(\kappa^{-1}\circ\sigma^{*}).$

Now the element $\Delta_{\operatorname{SPL}}$ is characterized by its
Pontrjagin character, [39, p. 115]. (Madsen and Milgram show that the
Pontrjagin character
$\operatorname{ph}:\widetilde{{\operatorname{KO}}}(\operatorname{MSPL}_{(p)})\to
H^{*}(\operatorname{MSPL};\mathbb{Q})$ is injective at every odd prime $p$,
[39, Cor. 5.25].) It follows that $\kappa^{-1}\sigma^{*}$ is homotopic to
$\Delta_{\operatorname{SPL}}$. ∎

######  Corollary 3.4.

The Sullivan orientation
$\Delta_{\operatorname{SPL}}:\operatorname{MSPL}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$
is homotopic to a morphism of (homotopy) ring spectra.

In view of Proposition 3.3, we may thus adopt the following convention for the
spectrum level Sullivan orientation:

######  Definition 3.5.

Let

$\Delta:\operatorname{MSPL}\longrightarrow{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$

be the morphism of ring spectra given by the composition

$\operatorname{MSPL}\stackrel{{\scriptstyle\sigma^{*}}}{{\longrightarrow}}\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]=\mathbb{L}()[\mbox{$\frac{1}{2}$}]\stackrel{{\scriptstyle\kappa^{-1}}}{{\simeq}}{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$

of Ranicki’s orientation with a ring equivalence $\kappa^{-1}$ inverse to the
ring equivalence $\kappa$ of Proposition 2.1.

######  Definition 3.6.

For an oriented rank $m$ PL microbundle (or PL $(^{m},0)$-bundle) $\alpha$,
let

$\Delta(\alpha)\in\widetilde{{\operatorname{KO}}}^{m}(\operatorname{Th}(\alpha))[\mbox{$\frac{1}{2}$}]$

be the image

$\Delta(\alpha)=\Delta_{*}(u_{\operatorname{SPL}}(\alpha))$

of the $\operatorname{MSPL}$-orientation under

$\widetilde{\operatorname{MSPL}}^{m}(\operatorname{Th}(\alpha))\stackrel{{\scriptstyle\Delta_{*}}}{{\longrightarrow}}\widetilde{{\operatorname{KO}}}^{m}(\operatorname{Th}(\alpha))[\mbox{$\frac{1}{2}$}].$

This is indeed a ${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$-orientation of
$\alpha$, since $\Delta$ is multiplicative ([53, p. 305, Prop. V.1.6]).

From this perspective, it is immediate that $\kappa$ aligns $\Delta(\alpha)$
and Ranicki’s Thom class $u_{\mathbb{L}}(\alpha)$:

######  Lemma 3.7.

Let $\alpha:X\to\operatorname{BSPL}(m)$ be an oriented PL microbundle of rank
$m$. Then the isomorphism

$\kappa_{*}:\widetilde{{\operatorname{KO}}}^{m}(\operatorname{Th}(\alpha))\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\widetilde{\mathbb{L}}^{m}(\operatorname{Th}(\alpha))\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$

maps $\Delta(\alpha)$ to $u_{\mathbb{L}}(\alpha)$.

###### Proof.

According to Ranicki’s definition,
$u_{\mathbb{L}}(\alpha)=\sigma^{*}(u_{\operatorname{SPL}}(\alpha)).$
Consequently,

$\kappa_{*}\Delta(\alpha)=\kappa_{*}\Delta_{*}(u_{\operatorname{SPL}}(\alpha))=\kappa_{*}\kappa^{-1}_{*}\sigma^{*}(u_{\operatorname{SPL}}(\alpha))=\sigma^{*}u_{\operatorname{SPL}}(\alpha)=u_{\mathbb{L}}(\alpha).$

∎

Given an oriented vector or PL bundle $\alpha$ of rank $m$, let
$u_{\mathbb{Z}}(\alpha)\in\widetilde{H}^{m}(\operatorname{Th}(\alpha);\mathbb{Z})$
denote its integral Thom class and
$u_{\mathbb{Q}}(\alpha)\in\widetilde{H}^{m}(\operatorname{Th}(\alpha);\mathbb{Q})$
the image of $u_{\mathbb{Z}}(\alpha)$ under
$\widetilde{H}*(-;\mathbb{Z})\to\widetilde{H}^{*}(-;\mathbb{Q})$. The methods
used to prove Proposition 3.3 imply readily:

######  Lemma 3.8.

Let $\alpha$ be an oriented PL microbundle over $X$. Rationally,
$\Delta(\alpha)$ is given by

$\operatorname{ph}\Delta(\alpha)=L^{-1}(\alpha)\cup u_{\mathbb{Q}}(\alpha)\in
H^{*}(X;\mathbb{Q}).$

###### Proof.

One evaluates the commutative diagram (5) on the base space of $\alpha$ and
notes that by Lemma 3.7,
$\kappa_{*}\Delta(\alpha)=u_{\mathbb{L}}(\alpha_{\operatorname{STOP}}).$ By
commutativity, $\operatorname{ph}\Delta(\alpha)$ may be computed by mapping
$\kappa_{*}\Delta(\alpha)$ along the lower horizontal composition. As observed
in the proof of the proposition, the $\mathbb{L}$-cohomology Thom class is
given rationally (i.e. along the lower horizontal composition) by the product
of the inverse $L$-class with the $H\mathbb{Q}$-cohomology Thom class
$u_{\mathbb{Q}}$, $L^{-1}(\alpha)\cup u_{\mathbb{Q}}(\alpha)$. (See [47,
Remark 16.2]; Ranicki writes $-\otimes\mathbb{Q}$ instead of
$\operatorname{ch}_{\mathbb{L}}\operatorname{loc}$ and omits cupping with
$u_{\mathbb{Q}}$ in his notation.) ∎

For the sake of completeness, we also record the case of the trivial bundle:

######  Lemma 3.9.

If $\alpha$ is the trivial rank $4k=m$ PL bundle over a point, then
$\Delta(\alpha)=a^{k}\in\widetilde{{\operatorname{KO}}}^{4k}(S^{4k})[\mbox{$\frac{1}{2}$}]$.

###### Proof.

The Chern character
$\operatorname{ch}:{\operatorname{K}}^{0}(S^{4k})\to\bigoplus_{i=0}^{\infty}H^{2i}(S^{4k};\mathbb{Q})$
is injective and has image $H^{*}(S^{4k};\mathbb{Z})\subset
H^{*}(S^{4k};\mathbb{Q})$. Thus it is an isomorphism

$\operatorname{ch}:{\operatorname{K}}^{0}(S^{4k})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H^{*}(S^{4k};\mathbb{Z})=\mathbb{Z}\oplus\mathbb{Z}.$

It restricts to an isomorphism

$\operatorname{ch}:\widetilde{{\operatorname{K}}}^{0}(S^{4k})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\widetilde{H}^{*}(S^{4k};\mathbb{Z})=\mathbb{Z}$

between reduced groups. The localization at odd primes is an isomorphism

$\operatorname{ch}[\mbox{$\frac{1}{2}$}]:\widetilde{{\operatorname{K}}}^{0}(S^{4k})[\mbox{$\frac{1}{2}$}]\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}H^{4k}(S^{4k})[\mbox{$\frac{1}{2}$}]=\mathbb{Z}[\mbox{$\frac{1}{2}$}].$

We turn next to complexification. This is a morphism
$c:{\operatorname{KO}}\to{\operatorname{K}}$ of spectra which induces a ring
homomorphism
$c_{*}:\pi_{*}({\operatorname{KO}})\to\pi_{*}({\operatorname{K}})$ on homotopy
rings ([58, p. 304]). In degree $4$,
$c_{*}=2:\pi_{4}({\operatorname{KO}})=\mathbb{Z}\to\mathbb{Z}=\pi_{4}({\operatorname{K}})$,
while in degree $8$,
$c_{*}:\pi_{8}({\operatorname{KO}})=\mathbb{Z}\to\mathbb{Z}=\pi_{8}({\operatorname{K}})$
is an isomorphism. The localized ring homomorphism

$c_{*}[\mbox{$\frac{1}{2}$}]:\pi_{*}({\operatorname{KO}})[\mbox{$\frac{1}{2}$}]=\mathbb{Z}[\mbox{$\frac{1}{2}$}][a^{\pm
1}]\longrightarrow\mathbb{Z}[\mbox{$\frac{1}{2}$}][\beta^{\pm
1}]=\pi_{*}({\operatorname{K}})[\mbox{$\frac{1}{2}$}]$

sends $a\in\pi_{4}({\operatorname{KO}})[\mbox{$\frac{1}{2}$}]$ to
$\beta^{2}\in\pi_{4}({\operatorname{K}})[\mbox{$\frac{1}{2}$}]$. In
particular,
$c_{*}[\mbox{$\frac{1}{2}$}]:\pi_{4k}({\operatorname{KO}})[\mbox{$\frac{1}{2}$}]\to\pi_{4k}({\operatorname{K}})[\mbox{$\frac{1}{2}$}]$
is an isomorphism, mapping $a^{k}\mapsto\beta^{2k}$. The Chern character of
$\beta^{k}$ is given by $v^{k}$, where $v\in H^{2}(S^{2};\mathbb{Z})$ is the
canonical generator, i.e. the first Chern class $c_{1}(H)$ of the canonical
(hyperplane) line bundle $H$ over $S^{2}=\mathbb{C}P^{1}$. The Pontrjagin
character $\operatorname{ph}=\operatorname{ch}\circ c$ localizes to
$\operatorname{ph}[\mbox{$\frac{1}{2}$}]$ given by the composition

$\textstyle{\widetilde{{\operatorname{KO}}}^{0}(S^{4k})[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ph}[\mbox{$\frac{1}{2}$}]}$$\scriptstyle{c_{*}[\mbox{$\frac{1}{2}$}]}$$\scriptstyle{\cong}$$\textstyle{\widetilde{{\operatorname{K}}}^{0}(S^{4k})[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ch}[\mbox{$\frac{1}{2}$}]}$$\scriptstyle{\cong}$$\textstyle{H^{4k}(S^{4k};\mathbb{Z})[\mbox{$\frac{1}{2}$}].}$

Here we have used suspension isomorphisms to identify

$\textstyle{\widetilde{{\operatorname{KO}}}^{0}(S^{4k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{*}}$$\scriptstyle{\sim}$$\textstyle{\widetilde{{\operatorname{KO}}}^{-4k}(S^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{KO}}^{-4k}(\operatorname{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{4k}({\operatorname{KO}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{*}}$$\textstyle{\widetilde{{\operatorname{K}}}^{0}(S^{4k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\widetilde{{\operatorname{K}}}^{-4k}(S^{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{K}}^{-4k}(\operatorname{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{4k}({\operatorname{K}}).}$

Therefore,

$\operatorname{ph}[\mbox{$\frac{1}{2}$}](a^{k})=\operatorname{ch}[\mbox{$\frac{1}{2}$}](c_{*}[\mbox{$\frac{1}{2}$}](a^{k}))=\operatorname{ch}[\mbox{$\frac{1}{2}$}](\beta^{2k})=v^{2k}.$

The generator $v^{2k}\in H^{4k}(S^{4k};\mathbb{Z})$ agrees with the Thom class
$u_{\mathbb{Z}}(\alpha)\in H^{4k}(^{4k}\cup\\{\infty\\};\mathbb{Z})$ of the
trivial rank $4k$-bundle $\alpha$ over a point. Let
$\iota:H^{*}(S^{4k};\mathbb{Z}[\mbox{$\frac{1}{2}$}])\hookrightarrow
H^{*}(S^{4k};\mathbb{Q})$ be the injection induced by
$\mathbb{Z}[\mbox{$\frac{1}{2}$}]\subset\mathbb{Q}$. By Lemma 3.8,

$\displaystyle\iota\operatorname{ph}[\mbox{$\frac{1}{2}$}](\Delta(\alpha))$
$\displaystyle=L^{-1}(\alpha)\cup u_{\mathbb{Q}}(\alpha)=1\cup
u_{\mathbb{Q}}(\alpha)=u_{\mathbb{Q}}(\alpha)$
$\displaystyle=\iota(u_{\mathbb{Z}}(\alpha))=\iota(v^{2k})=\iota\operatorname{ph}[\mbox{$\frac{1}{2}$}](a^{k}).$

Since $\iota\operatorname{ph}[\mbox{$\frac{1}{2}$}]$ is injective,

$\Delta(\alpha)=a^{k}.$

∎

##  4\. The Classical Construction of the Siegel-Sullivan Orientation

Using Goresky-MacPherson’s intersection homology, Witt spaces have been
introduced by P. Siegel in [55] as a geometric cycle theory representing
${\operatorname{KO}}$-homology at odd primes. Sources on intersection homology
include [28], [29], [31], [23], [36], [13], [5].

######  Definition 4.1.

A _Witt space_ is an oriented PL pseudomanifold such that the links $L^{2k}$
of odd codimensional PL intrinsic strata have vanishing lower middle-
perversity degree $k$ rational intersection homology,
$IH^{\bar{m}}_{k}(L^{2k};\mathbb{Q})=0$.

For example, pure-dimensional complex algebraic varieties are Witt spaces,
since they are oriented pseudomanifolds and possess a Whitney stratification
whose strata all have even codimension. The vanishing condition on the
intersection homology of links $L^{2k}$ is equivalent to requiring the
canonical morphism from lower middle to upper middle perversity intersection
chain sheaves to be an isomorphism in the derived category of sheaf complexes.
Consequently, these middle perversity intersection chain sheaves are Verdier
self-dual, and this induces global Poincaré duality for the middle perversity
intersection homology groups of a compact Witt space. In particular, compact
Witt spaces $X$ have a well-defined bordism invariant signature $\sigma(X)$
and $L$-classes $L_{*}(X)\in H_{*}(X;\mathbb{Q})$ which agree with the
Poincaré duals of Hirzebruch’s tangential $L$-classes when $X$ is smooth. The
notion of _Witt spaces with boundary_ can be introduced as pairs $(X,\partial
X),$ where $X$ is a PL space and $\partial X$ a stratum preservingly collared
PL subspace of $X$ such that $X-\partial X$ and $\partial X$ are both
compatibly oriented Witt spaces. Let $\Omega^{\operatorname{Witt}}_{*}(-)$
denote the bordism theory based on Witt cycles. Elements of
$\Omega^{\operatorname{Witt}}_{n}(Y)$ are Witt bordism classes of continuous
maps $f:X^{n}\to Y$ defined on $n$-dimensional closed Witt spaces $X$. The
theory $\Omega^{\operatorname{Witt}}_{*}(-)$ is a homology theory, whose
coefficients have been computed by Siegel. They are nontrivial only in
nonnegative degrees divisible by $4$, where they are given by
$\mathbb{L}_{4k}(\mathbb{Q}),$ $4k>0,$ and by $\mathbb{Z}$ in degree $0$.

Let $X$ be a closed Witt space of dimension $n$. Drawing on Sullivan’s methods
as laid out in [56] and [57], Siegel constructs in [55] a canonical
orientation class

$\mu_{X}\in{\operatorname{KO}}_{n}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].$

(In fact, the class lives in connective ${\operatorname{KO}}$-homology.) We
shall refer to $\mu_{X}$ as the _Siegel-Sullivan orientation class_ of $X$.
Let us briefly outline Siegel’s construction, which rests on two fundamental
facts due to Sullivan: First, there is an exact sequence

$0\to{\operatorname{KO}}^{i}(Y,B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\longrightarrow{\operatorname{KO}}^{i}(Y,B)^{\wedge}\oplus{\operatorname{KO}}^{i}(Y,B)\otimes\mathbb{Q}\longrightarrow{\operatorname{KO}}^{i}(Y,B)^{\wedge}\otimes\mathbb{Q}\to
0,$

where ${\operatorname{KO}}^{i}(Y,B)^{\wedge}$ denotes the profinite completion
of ${\operatorname{KO}}^{i}(Y,B)$ with respect to groups of odd order. Second,
the natural transformation
$\Delta_{{\operatorname{SO}}*}:\Omega^{\operatorname{SO}}_{i}(Y,B)\to({\operatorname{KO}}[\mbox{$\frac{1}{2}$}])_{i}(Y,B)$
induces a Conner-Floyd type isomorphism

$\Omega^{\operatorname{SO}}_{i+4*}(Y,B)\otimes_{\Omega^{\operatorname{SO}}_{*}(\operatorname{pt})}\mathbb{Z}[\mbox{$\frac{1}{2}$}]\cong{\operatorname{KO}}_{i}(Y,B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$

of $\mathbb{Z}/4\mathbb{Z}$-periodic theories for compact PL pairs $(Y,B)$,
[56], [39, p. 85]. Together with universal coefficient considerations, these
two facts imply that elements of
${\operatorname{KO}}^{i}(Y,B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$ are
pairs $(\sigma_{0},\tau_{0})$ of homomorphisms
$\sigma_{0}:\Omega^{\operatorname{SO}}_{i+4*}(Y,B)\otimes\mathbb{Q}\to\mathbb{Q}$
and
$\tau_{0}:\Omega^{\operatorname{SO}}_{i+4*}(Y,B;\mathbb{Q}/\mathbb{Z}[\mbox{$\frac{1}{2}$}])\to\mathbb{Q}/\mathbb{Z}[\mbox{$\frac{1}{2}$}]$
such that the _periodicity relations_

(6)
$\sigma_{0}([f][M\to\operatorname{pt}])=\sigma(M)\cdot\sigma_{0}[f],~{}\tau_{0}([f][M\to\operatorname{pt}])=\sigma(M)\cdot\tau_{0}[f]$

with respect to multiplication by a closed manifold $M$ hold and the diagram

$\textstyle{\Omega^{\operatorname{SO}}_{i+4*}(Y,B)\otimes\mathbb{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{0}}$$\textstyle{\mathbb{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{\operatorname{SO}}_{i+4*}(Y,B;\mathbb{Q}/\mathbb{Z}[\mbox{$\frac{1}{2}$}])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{0}}$$\textstyle{\mathbb{Q}/\mathbb{Z}[\mbox{$\frac{1}{2}$}]}$

commutes. To define $\mu_{X}$ for a closed Witt space $X^{n}$, choose a PL
embedding $X\subset^{m}$, $m$ large, of codimension $4k$. Let $(N,\partial N)$
be a regular neighborhood of $X$. We will describe an element in
${\operatorname{KO}}^{4k}(N,\partial
N)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}],$ which corresponds under Alexander-
Spanier-Whitehead duality to
$\mu_{X}\in{\operatorname{KO}}_{n}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$.
Therefore, we need to specify homomorphisms $(\sigma_{X},\tau_{X})$ satisfying
the above periodicity relations and the integrality condition, i.e.
commutativity of

$\textstyle{\Omega^{\operatorname{SO}}_{4(k+*)}(N,\partial
N)\otimes\mathbb{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{X}}$$\textstyle{\mathbb{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{\operatorname{SO}}_{4(k+*)}(N,\partial
N;\mathbb{Q}/\mathbb{Z}[\mbox{$\frac{1}{2}$}])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{X}}$$\textstyle{\mathbb{Q}/\mathbb{Z}[\mbox{$\frac{1}{2}$}].}$

The homomorphism $\sigma_{X}$ is

$\sigma_{X}([(M,\partial M)\stackrel{{\scriptstyle
f}}{{\longrightarrow}}(N,\partial N)]\otimes
r):=\sigma(\widetilde{f}^{-1}(X))\otimes r,~{}r\in\mathbb{Q},$

where one uses the block-transversality results of [15], [40] to make $f$
transverse to $X$ in the PL manifold $N$. The preimage
$\widetilde{f}^{-1}(X)\subset M$ under the transverse map $\widetilde{f}$ has
the same local structure as $X$ and thus is again a Witt space with a well-
defined signature $\sigma(\widetilde{f}^{-1}(X))\in\mathbb{Z}$. The
homomorphism $\tau_{X}$ is obtained by specifying a sequence of homomorphisms

$\tau_{X,k}:\Omega^{\operatorname{SO}}_{*}(N,\partial
N;\mathbb{Z}/_{k})\longrightarrow\mathbb{Z}/_{k},~{}k\text{ odd},$

compatible with respect to divisibility, which are defined in much the same
way as $\sigma_{X}$, but using oriented $\mathbb{Z}/_{k}$-manifolds to
represent elements of $\Omega^{\operatorname{SO}}_{*}(N,\partial
N;\mathbb{Z}/_{k})$. By Novikov additivity for the signature of compact Witt
spaces with boundary, the transverse inverse image of $X$ in the
$\mathbb{Z}/_{k}$-manifold has a well-defined (and bordism invariant)
signature in $\mathbb{Z}/_{k}$, which defines $\tau_{X,k}$. The periodicity
and integrality conditions are satisfied and thus an element $\mu_{X}$ is
obtained.

The homomorphism
$c_{*}:{\operatorname{KO}}_{n}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\to{\operatorname{KO}}_{n}(\operatorname{pt})\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$
induced by the constant map $c:X\to\operatorname{pt}$ sends $\mu_{X}$ to the
signature of $X$. Using the orientation class $\mu_{X}$, Siegel obtains a
natural transformation

$\mu^{\operatorname{Witt}}:\Omega^{\operatorname{Witt}}_{*}(-)\longrightarrow{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]_{*}(-)$

of homology theories by setting

$\mu^{\operatorname{Witt}}([X\stackrel{{\scriptstyle
f}}{{\longrightarrow}}Y])=f_{*}(\mu_{X}).$

This transformation then reduces to the signature homomorphism on coefficient
groups. In terms of the transformation, the orientation class can of course be
recovered as

$\mu_{X}=\mu^{\operatorname{Witt}}([\operatorname{id}_{X}]).$

Siegel’s transformation factors through the homomorphism induced by the
connective cover
${\operatorname{ko}}[\mbox{$\frac{1}{2}$}]\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}],$
since $\Omega^{\operatorname{Witt}}_{*}(-)$ is connective.

######  Theorem 4.2.

(Siegel.) The natural transformation

$\mu^{\operatorname{Witt}}[\mbox{$\frac{1}{2}$}]:\Omega^{\operatorname{Witt}}_{*}(-)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\longrightarrow{\operatorname{ko}}[\mbox{$\frac{1}{2}$}]_{*}(-)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$

is an equivalence of homology theories.

######  Proposition 4.3.

(Siegel.) If $X=M$ is a smooth compact manifold, then $\mu_{M}$ agrees with
the Sullivan orientation $\Delta_{\operatorname{SO}}(M)$,

$\mu_{M}=\Delta_{\operatorname{SO}}(M).$

###### Proof.

As pointed out by Siegel [55, p. 1069], the statement follows directly from
the above construction, since it has been used by Sullivan [56] in the case of
a manifold to construct his canonical
${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$-orientation. (In fact, as pointed
out in [57] and by Siegel, the construction applies in general to any bordism
theory based on a class $\mathcal{F}$ of PL spaces which is closed under
taking cartesian product with a PL manifold and intersecting transversely with
a closed manifold in Euclidean space, carries a bordism invariant signature
which satisfies Novikov additivity and is multiplicative with respect to
taking products with closed manifolds. Beyond PL manifolds and Witt spaces
there exist much larger classes of singular spaces $\mathcal{F}$ that satisfy
this, in particular the class of stratified pseudomanifolds that admits
Lagrangian structures along strata of odd codimension, considered in [3],
[4].) ∎

##  5\. Ring Spectrum Level Construction of the Siegel-Sullivan Orientation

Let $\operatorname{MWITT}$ be the Quinn spectrum associated to the ad-theory
of Witt spaces, representing Witt bordism, see Banagl-Laures-McClure [10]. A
weakly equivalent spectrum had been considered first by Curran in [20]. He
verified that this spectrum is an $\operatorname{MSO}$-module ([20, Thm. 3.6,
p. 117]). The product of two Witt spaces is again a Witt space. This implies
essentially that $\operatorname{MWITT}$ is a ring spectrum; for more details
see [10]. (There, we focused on the spectrum $\operatorname{MIP}$ representing
bordism of integral intersection homology Poincaré spaces studied by Goresky
and Siegel in [30] and by Pardon in [45], but everything works in an
analogous, indeed simpler, manner for $\mathbb{Q}$-Witt spaces.) Every
oriented PL manifold is a Witt space. Hence there is a map

$\phi_{W}:\operatorname{MSPL}\longrightarrow\operatorname{MWITT},$

which, using the methods of ad-theories and Quinn spectra employed in [10],
can be constructed to be multiplicative. In [10], we constructed a map

$\tau:\operatorname{MWITT}\longrightarrow\mathbb{L}(\mathbb{Q}).$

(We even constructed an integral map $\operatorname{MIP}\to\mathbb{L}$.) This
map is multiplicative, i.e. a ring map, as shown in [10, Section 12], and the
diagram

(7)
$\textstyle{\operatorname{MSPL}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{*}}$$\scriptstyle{\phi_{W}}$$\textstyle{\mathbb{L}(\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{MWITT}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\mathbb{L}(\mathbb{Q})}$

homotopy commutes, since it comes from a commutative diagram of ad-theories
under applying the symmetric spectrum functor $\mathbf{M}$ of Laures and
McClure [35]. The localization morphism
$\mathbb{L}(\mathbb{Q})\to\mathbb{L}(\mathbb{Q})[\mbox{$\frac{1}{2}$}]$ is a
morphism of ring spectra. Thus the composition of $\tau$ with the localization
morphism is a morphism of ring spectra
$\operatorname{MWITT}\to\mathbb{L}(\mathbb{Q})[\mbox{$\frac{1}{2}$}]=\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]$,
which we shall also denote by $\tau$. It was known to the experts early on
that carrying out Mishchenko’s method [42] with intersection chains rather
than ordinary chains would lead to an extension of the symmetric signature to
pseudomanifolds with only even codimensional strata and, more generally, to
Witt spaces; see e.g. [60], [16], [6]. In the context of their Witt package
program [1], Albin, Leichtnam, Mazzeo and Piazza applied this symmetric
signature in defining a $C^{*}$-algebraic Witt symmetric signature in
$K_{*}(C^{*}_{r}\pi)$ which agrees rationally with the index class of the
signature operator $\eth_{\operatorname{sign}}$. In [10], Laures, McClure and
the author adopt the approach outlined in [6] to construct the symmetric
signature of Witt and integral intersection Poincaré spaces: The morphism
$\tau:\operatorname{MWITT}\to\mathbb{L}(\mathbb{Q})$ of ring spectra yields an
$\mathbb{L}(\mathbb{Q})$-homology fundamental class for $n$-dimensional closed
Witt spaces $X$ by setting

$[X]_{\mathbb{L}}:=\tau[\operatorname{id}_{X}]\in\mathbb{L}(\mathbb{Q})_{n}(X),$

$[\operatorname{id}_{X}]\in\Omega^{\operatorname{Witt}}_{n}(X)$. This
fundamental class yields the symmetric signature

$\sigma^{*}(X)=A[X]_{\mathbb{L}}\in L^{n}(\mathbb{Q}[\pi_{1}X])$

under the assembly map

$A:\mathbb{L}(\mathbb{Q})_{n}(X)\longrightarrow L^{n}(\mathbb{Q}[\pi_{1}X]).$

A detailed account of extending Mishchenko’s approach to intersection chains
has been provided by Friedman and McClure in [24]. By [10, Thm. 10.12], the
above symmetric signature $\sigma^{*}(X)$ agrees with the construction of
Friedman-McClure. According to [24, Prop. 5.20], the homomorphism
$L^{n}(\mathbb{Q}[\pi_{1}X])\to L^{n}(\mathbb{Q})$ for $n=4k$ maps the
symmetric signature $\sigma^{*}(X)$ to the Witt class $w(X)$ of the
intersection form on $IH^{\bar{m}}_{2k}(X;\mathbb{Q})$. Under the localization
homomorphism $L^{n}(\mathbb{Q})\to
L^{n}(\mathbb{Q})\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}],$ $w(X)$ maps to the
ordinary signature, $w(X)_{(\operatorname{odd})}=\sigma(X)\cdot x^{k}$,
$n=4k$, as in the manifold case of Remark 3.2. The image of $[X]_{\mathbb{L}}$
under the localization homomorphism
$\mathbb{L}(\mathbb{Q})_{n}(X)\to\mathbb{L}(\mathbb{Q})_{n}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$
will again be denoted by $[X]_{\mathbb{L}}$.

######  Proposition 5.1.

In positive degrees, the composition

$\Omega^{\operatorname{Witt}}_{*}(\operatorname{pt})\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}\mathbb{L}(\mathbb{Q})_{*}(\operatorname{pt})\stackrel{{\scriptstyle
A}}{{=}}L^{*}(\mathbb{Q})$

agrees with the map $w:\Omega^{\operatorname{Witt}}_{*}(\operatorname{pt})\to
L^{*}(\mathbb{Q})$ which sends the bordism class of a $4k$-dimensional Witt
space $X$ to the Witt class $w(X)$ of its intersection form on
$IH^{\bar{m}}_{2k}(X;\mathbb{Q})$ and is zero in degrees not divisible by $4$.

###### Proof.

The group $\Omega^{\operatorname{Witt}}_{i}(\operatorname{pt})$ is zero in
degrees that are not divisible by $4$. Thus $A\circ\tau$ agrees trivially with
$w$ in such degrees. Let
$[X]\in\Omega^{\operatorname{Witt}}_{4k}(\operatorname{pt}),$ $k>0,$ be any
element. Let $c:X\to\operatorname{pt}$ denote the constant map and consider
the diagram

(8)
$\textstyle{\Omega^{\operatorname{Witt}}_{4k}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{c_{*}}$$\textstyle{\mathbb{L}(\mathbb{Q})_{4k}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\scriptstyle{c_{*}}$$\textstyle{L^{4k}(\mathbb{Q}[\pi_{1}X])\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{*}}$$\textstyle{\Omega^{\operatorname{Witt}}_{4k}(\operatorname{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\mathbb{L}(\mathbb{Q})_{4k}(\operatorname{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\textstyle{L^{4k}(\mathbb{Q}).}$

Since both $\tau$ and the assembly map are natural, the diagram commutes. The
right hand vertical map sends the symmetric signature $\sigma^{*}(X)\in
L^{4k}(\mathbb{Q}[\pi_{1}X])$ to $w(X)$. Therefore,

$\displaystyle A\tau[X]$ $\displaystyle=A\tau
c_{*}[\operatorname{id}_{X}]=c_{*}A\tau[\operatorname{id}_{X}]$
$\displaystyle=c_{*}A[X]_{\mathbb{L}}=c_{*}\sigma^{*}(X)=w(X).$

∎

The morphism $\tau:\operatorname{MWITT}\to\mathbb{L}(\mathbb{Q})$ is not an
equivalence. One reason is that $\operatorname{MWITT}$ is connective whereas
$\mathbb{L}(\mathbb{Q})$ is periodic. Let $t_{\geq m}E\to E$ be the
$(m-1)$-connective cover of a spectrum $E$. A morphism $\phi:E\to F$ of
spectra lifts, uniquely up to homotopy, to a morphism $t_{\geq m}\phi:t_{\geq
m}E\to t_{\geq m}F$. The lift $t_{\geq 0}\tau$ to the connective cover
$t_{\geq 0}\mathbb{L}(\mathbb{Q})$ is still no equivalence, as
$\pi_{0}\operatorname{MWITT}=\Omega^{\operatorname{Witt}}_{0}(\operatorname{pt})=\mathbb{Z}$,
while $\pi_{0}\mathbb{L}(\mathbb{Q})=L^{0}(\mathbb{Q})$ contains an infinitely
generated amount of $2$-primary torsion. Degree zero is, however, the only
offending nonnegative degree:

######  Theorem 5.2.

The lift $t_{\geq 1}\tau:t_{\geq 1}\operatorname{MWITT}\to t_{\geq
1}\mathbb{L}(\mathbb{Q})$ is a weak equivalence.

###### Proof.

By Proposition 5.1, the diagram

$\textstyle{\pi_{i}(t_{\geq
1}\operatorname{MWITT})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{\geq
1}\tau}$$\scriptstyle{\cong}$$\textstyle{\pi_{i}(t_{\geq
1}\mathbb{L}(\mathbb{Q}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\pi_{i}(\operatorname{MWITT})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{w}$$\textstyle{\pi_{i}(\mathbb{L}(\mathbb{Q}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\textstyle{L^{i}(\mathbb{Q})}$

commutes for $i\geq 1$. Siegel’s Witt bordism calculation [55, Prop 1.1, p.
1098] asserts that $w$ is an isomorphism. ∎

The central construction of the present paper is the following lift of the
Siegel-Sullivan orientation introduced in [55] to the ring-spectrum level.

######  Definition 5.3.

The _ring-spectrum level Siegel-Sullivan orientation_

$\Delta:\operatorname{MWITT}\longrightarrow{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$

is the morphism of ring spectra given by the composition

$\operatorname{MWITT}\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}\mathbb{L}(\mathbb{Q})[\mbox{$\frac{1}{2}$}]=\mathbb{L}()[\mbox{$\frac{1}{2}$}]\stackrel{{\scriptstyle\kappa^{-1}}}{{\simeq}}{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$

of the Witt-orientation introduced by Laures, McClure and the author with a
ring equivalence $\kappa^{-1}$ inverse to the ring equivalence $\kappa$ of
Proposition 2.1.

Diagram (7) then embeds into the homotopy commutative diagram

$\textstyle{\operatorname{MSPL}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\scriptstyle{\sigma^{*}}$$\scriptstyle{\phi_{W}}$$\textstyle{\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{MWITT}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\scriptstyle{\tau}$$\textstyle{\mathbb{L}(\mathbb{Q})[\mbox{$\frac{1}{2}$}]}$$\textstyle{{\operatorname{KO}}[\mbox{$\frac{1}{2}$}].\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$

Thus the ring-spectrum level Siegel-Sullivan orientation
$\Delta:\operatorname{MWITT}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$
restricts under $\phi_{W}:\operatorname{MSPL}\to\operatorname{MWITT}$ to the
Sullivan orientation

$\operatorname{MSPL}\stackrel{{\scriptstyle\Delta}}{{\longrightarrow}}{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$

of Definition 3.5. In order to describe the induced map
$\Delta_{*}:\operatorname{MWITT}_{*}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]_{*}$
on coefficients, we shall employ the symmetric signature of Witt spaces. We
observe that (2) extends from the manifold case to Witt spaces:

######  Proposition 5.4.

The ring-spectrum level Siegel-Sullivan orientation $\Delta$ induces on
homotopy groups the homomorphism

$\Delta_{*}:\Omega^{\operatorname{Witt}}_{4k}=\operatorname{MWITT}_{4k}\longrightarrow{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]_{4k}=\mathbb{Z}[\mbox{$\frac{1}{2}$}]\langle
a^{k}\rangle$

given by

(9) $\Delta_{*}[X^{4k}]=\sigma(X)\cdot a^{k},$

where $\sigma(X)$ is the signature of the intersection form on the
intersection homology groups $IH_{2k}(X;\mathbb{Q})$ of $X$.

###### Proof.

Let $[X]\in\Omega^{\operatorname{Witt}}_{4k}$ be any element. The constant map
$c:X\to\operatorname{pt}$ induces a commutative diagram

$\textstyle{\Omega^{\operatorname{Witt}}_{n}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{c_{*}}$$\textstyle{\mathbb{L}(\mathbb{Q})_{n}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A[\frac{1}{2}]}$$\scriptstyle{c_{*}}$$\textstyle{L^{n}(\mathbb{Q}[\pi_{1}X])\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{*}}$$\textstyle{\Omega^{\operatorname{Witt}}_{n}(\operatorname{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\mathbb{L}(\mathbb{Q})_{n}(\operatorname{pt})\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A[\frac{1}{2}]}$$\textstyle{L^{n}(\mathbb{Q})\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].}$

This is quite similar to Diagram (8), except that here we map into L-theory
away from $2$. The claim is established by the calculation

$\displaystyle\kappa_{*}\Delta_{*}[X]$
$\displaystyle=\tau_{*}[X]=\tau_{*}c_{*}[\operatorname{id}_{X}]=c_{*}\tau_{*}[\operatorname{id}_{X}]=c_{*}[X]_{\mathbb{L}}$
$\displaystyle=c_{*}(A[\mbox{$\frac{1}{2}$}])[X]_{\mathbb{L}}=c_{*}\sigma^{*}(X)_{(\operatorname{odd})}=w(X)_{(\operatorname{odd})}=\sigma(X)\cdot
x^{k},$

so that

$\Delta_{*}[X]=\sigma(X)\cdot\kappa^{-1}_{*}(x^{k})=\sigma(X)\cdot a^{k}.$

∎

The ring-spectrum level Siegel-Sullivan orientation allows in particular for
the following description of the Siegel-Sullivan orientation class of a Witt
space:

######  Definition 5.5.

The _Siegel-Sullivan orientation class_ of a compact $n$-dimensional Witt
space $(X,\partial X)$ is given by the image

$\Delta(X):=\Delta_{*}[\operatorname{id}_{X}]\in{\operatorname{KO}}_{n}(X,\partial
X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$

of the Witt bordism class of the identity on $X$ under the homomorphism
$\Delta_{*}:\Omega^{\operatorname{Witt}}_{n}(X,\partial
X)\to({\operatorname{KO}}[\mbox{$\frac{1}{2}$}])_{n}(X,\partial X)$ induced by
the ring-spectrum level Siegel-Sullivan orientation $\Delta$.

We will see in Proposition 5.7 below that this terminology is justified, i.e.
that $\Delta(X)=\mu_{X}$. By our construction,

(10)
$\kappa_{*}\Delta(X)=\kappa_{*}\Delta_{*}[\operatorname{id}_{X}]=\tau[\operatorname{id}_{X}]=[X]_{\mathbb{L}}\in(\mathbb{L}\mathbb{Q}[\mbox{$\frac{1}{2}$}])_{n}(X,\partial
X).$

Since
$\Delta:\operatorname{MWITT}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$
restricts to the Sullivan orientation
$\Delta:\operatorname{MSPL}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$, the
Siegel-Sullivan orientation $\Delta(X)$ agrees with the Sullivan orientation
$\Delta_{\operatorname{SPL}}(X)$ when $X$ is a PL-manifold. As a final point
of business in setting up the spectrum level Siegel-Sullivan orientation, we
shall verify that the induced transformation of homology theories agrees with
the classical construction as given by Siegel and outlined in Section 4. The
argument is based on a result ([9, Prop. 2, p. 597]) of Cappell, Shaneson and
the author which shows that at odd primes, Witt bordism classes are
representable by smooth oriented bordism classes. Let us review this result
briefly.

For an integer $j$, let $\bar{j}$ denote its residue class in
$\mathbb{Z}/_{4}$. Using $4$-fold periodicity, we may view
${\operatorname{KO}}\mbox{$\frac{1}{2}$}$-homology as
$\mathbb{Z}/_{4}$-graded. On the groups
$C_{\bar{j}}(X,Y):=\bigoplus_{k\in\mathbb{Z}}\Omega^{\operatorname{SO}}_{j+4k}(X,Y)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$,
define an equivalence relation by

$[M^{j+4k}\times
N^{4i}\stackrel{{\scriptstyle\operatorname{proj}}}{{\longrightarrow}}M\stackrel{{\scriptstyle
f}}{{\longrightarrow}}X]\sim\sigma(N)\cdot[M^{j+4k}\stackrel{{\scriptstyle
f}}{{\longrightarrow}}X],$

where $\sigma(N)$ is the signature of the manifold $N$. (See also [32, p.
193].) The periodicity relations (6) imply that Sullivan’s orientation
$\Delta_{{\operatorname{SO}}*}$ induces a well-defined homomorphism

(11)
$\Delta_{{\operatorname{SO}}*}:Q_{\bar{j}}(X,Y)\longrightarrow({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{\bar{j}}(X,Y)$

on the quotient $Q_{\bar{j}}(X,Y):=C_{\bar{j}}(X,Y)/\sim$. This is a natural
transformation of functors, and Sullivan proves that it is an isomorphism for
compact PL pairs $(X,Y)$ ([56], [39, 4.15, p. 85]). This shows in particular
that $(X,Y)\mapsto Q_{\bar{j}}(X,Y)$ is a ($\mathbb{Z}/_{4}$-graded) homology
theory on compact PL pairs. Let $Z$ denote the ring
$Z=\mathbb{Z}[\mbox{$\frac{1}{2}$}]$. The Laurent polynomial ring
$Z[t,t^{-1}]$ is a $\mathbb{Z}$-graded ring with $\deg(t)=4$. There is a
canonical subring inclusion $Z[t]\subset Z[t,t^{-1}],~{}t\mapsto t.$ Via this
inclusion, $Z[t,t^{-1}]$ becomes a $Z[t]$-module and Panov observes in [44]
that this module is flat. As the connective spectrum
${\operatorname{ko}}\mbox{$\frac{1}{2}$}$ is a ring spectrum,
$({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{*}(X)$ is in particular a right
$({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{*}(\operatorname{pt})=Z[a]$-module,
where $a\in({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{4}(\operatorname{pt})=Z$
is the generator which complexifies to the square of the complex Bott element,
as in Section 2. For periodic ${\operatorname{KO}}$, we have
$({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(\operatorname{pt})=Z[a,a^{-1}],$
and the canonical map
$({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{*}(\operatorname{pt})\to({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(\operatorname{pt})$
is given by the inclusion $Z[a]\subset Z[a,a^{-1}],~{}a\mapsto a.$ Using the
isomorphism $Z[t]\cong Z[a],~{}t\mapsto a,$
$({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{*}(X)$ and
$({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(X)$ become right
$Z[t]$-modules. We may therefore form the tensor product of $Z[t]$-modules

$({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{*}(X)\otimes_{Z[t]}Z[t,t^{-1}].$

Since the functor $-\otimes_{Z[t]}Z[t,t^{-1}]$ is exact by Panov’s
observation, the functor

$(X,Y)\mapsto({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{*}(X,Y)\otimes_{Z[t]}Z[t,t^{-1}]$

is exact and thus a homology theory (on CW pairs $(X,Y)$), which is
$\mathbb{Z}$-graded by $\deg(x\otimes_{Z[t]}rt^{k})=n+4k,$
$x\in({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{n}(X,Y),$ $r\in Z$. Now
consider ${\operatorname{KO}}_{*}(-)$ as a $\mathbb{Z}$-graded (not
$\mathbb{Z}/_{4}$-graded) theory. The connective cover
$({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{*}(-)\to({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(-)$
is $Z[t]$-linear and hence induces a natural transformation

$\Phi:({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{*}(-)\otimes_{Z[t]}Z[t,t^{-1}]\longrightarrow({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(-)$

of $\mathbb{Z}$-graded homology theories. On a point, $\Phi$ is the identity
map. Thus $\Phi$ is a natural equivalence of homology theories. This shows how
to reconstruct periodic ${\operatorname{KO}}\mbox{$\frac{1}{2}$}$-homology
from connective ${\operatorname{ko}}\mbox{$\frac{1}{2}$}$-homology and the
action of $Z[t]$ on it. Similarly, we can turn Witt bordism, which is a
connective theory, into a periodic version: Denote Witt bordism away from $2$
by

$W_{*}(X):=\Omega^{\operatorname{Witt}}_{*}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].$

This is a $\mathbb{Z}$-graded homology theory with coefficients

$W_{*}(\operatorname{pt})=Z[c],~{}c:=[\mathbb{C}P^{2}]\otimes 1\in
W_{4}(\operatorname{pt}).$

(After inverting $2$, only the signature survives as an invariant; the $2$\-
and $4$-torsion is killed.) The $\mathbb{Z}$-graded abelian group $W_{*}(X)$
is a right module over the ring $W_{*}(\operatorname{pt})$ as usual. The ring
isomorphism $Z[t]\to W_{*}(\operatorname{pt})$ induced by $t\mapsto c\otimes
1\in W_{4}(\operatorname{pt})$ makes $W_{*}(X)$ into a right $Z[t]$-module. We
may thus form the tensor product

$\overline{W}_{*}(X):=W_{*}(X)\otimes_{Z[t]}Z[t,t^{-1}],$

which is $\mathbb{Z}$-graded by $\deg(x\otimes_{Z[t]}rt^{k})=n+4k,$ $x\in
W_{n}(X),$ $r\in Z.$ Panov’s observation shows that $\overline{W}_{*}(-)$ is a
homology theory. It is naturally a right $Z[t,t^{-1}]$-module and right
multiplication with $t$ is an isomorphism with inverse given by right
multiplication with $t^{-1}$. This shows that $\overline{W}_{*}(-)$ is
$4$-periodic so that we may call it _periodic Witt-bordism at odd primes_. The
inclusion $Z[t]\subset Z[t,t^{-1}]$ induces a natural map

$i_{*}:W_{*}(X)=W_{*}(X)\otimes_{Z[t]}Z[t]\longrightarrow\overline{W}_{*}(X).$

Again, the periodicity relations (6) imply that

(12) $\mu^{\operatorname{Witt}}([f:V^{j-4k}\to
X]\cdot[M^{4k}])=(\mu^{\operatorname{Witt}}[f])\cdot\sigma(M)a^{k}\in({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{j}(X),$

$[f]\in\Omega^{\operatorname{Witt}}_{j-4k}(X),$
$[M]\in\Omega^{\operatorname{SO}}_{4k}(\operatorname{pt})$, which shows that
$\mu^{\operatorname{Witt}}$ is a homomorphism of $Z[t]$-modules, as is
$\Delta_{{\operatorname{SO}}*}$ in the manifold case. Tensoring over $Z[t]$
with $Z[t,t^{-1}]$, we get a natural isomorphism

$\mu^{\operatorname{Witt}}\otimes_{Z[t]}\operatorname{id}:\overline{W}_{*}(X)=W_{*}(X)\otimes_{Z[t]}Z[t,t^{-1}]\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{*}(X)\otimes_{Z[t]}Z[t,t^{-1}]$

of $\mathbb{Z}$-graded homology theories by Siegel’s Theorem 4.2. For any
$j\in\mathbb{Z}$, a well-defined map

$\omega:Q_{\bar{j}}(X)\otimes
Z\longrightarrow(W_{*}(X)\otimes_{Z[t]}Z[t,t^{-1}])_{j}$

is given by setting

$\omega([g:M^{j-4k}\to X]\otimes_{\mathbb{Z}}r):=[g]\otimes_{Z[t]}rt^{k}\in
W_{j-4k}(X)\otimes_{Z[t]}Z\langle t^{k}\rangle,~{}k\in\mathbb{Z},~{}r\in Z,$

where one views the closed oriented smooth manifold $M$ as a Witt space via
its canonical PL structure. On compact PL spaces $X$, the diagram

$\textstyle{Q_{\bar{j}}(X)\otimes
Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\Delta_{{\operatorname{SO}}*}}$$\scriptstyle{\omega}$$\textstyle{({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{\bar{j}}(X)}$$\textstyle{\overline{W}_{j}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\mu^{\operatorname{Witt}}\otimes_{Z[t]}\operatorname{id}}$$\textstyle{(({\operatorname{ko}}\mbox{$\frac{1}{2}$})_{*}(X)\otimes_{Z[t]}Z[t,t^{-1}])_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\Phi}$

commutes for every $j\in\mathbb{Z}$ by Proposition 4.3. In particular,
$\omega$ is an isomorphism, from which representability of Witt bordism
classes by smooth manifolds, away from $2$, can be deduced. The following
consequence will be used in the proof of Proposition 5.7:

######  Proposition 5.6.

1\. Given a $Z[t]$-linear map
$\alpha_{*}:W_{*}(X)\to({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(X),$
there exists a unique extension of $\alpha_{*}$ to a homomorphism

$\overline{\alpha}_{*}:\overline{W}_{*}(X)\longrightarrow({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(X)$

of $Z[t,t^{-1}]$-modules. If $\alpha_{*}$ is a natural transformation of
homology theories, then so is $\overline{\alpha}_{*}$.

2\. Let
$\alpha_{*},\beta_{*}:W_{*}(X)\to({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(X)$
be $Z[t]$-linear natural transformations of homology theories on compact PL
$X$. If $\alpha_{*}([g:M\to X]\otimes 1)=\beta_{*}([g]\otimes 1)$ for every
$g$ on smooth manifolds $M$, then $\alpha_{*}=\beta_{*}$ on $W_{*}(X)$, and
$\overline{\alpha}_{*}=\overline{\beta}_{*}$ on $\overline{W}_{*}(X)$ for
their periodic versions.

###### Proof.

We prove statement 1: Since $\alpha_{*}$ is $Z[t]$-linear, it induces a map

$\overline{\alpha}_{*}:(W_{*}(X)\otimes_{Z[t]}Z[t,t^{-1}])_{j}\longrightarrow({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{\bar{j}}(X),~{}j\in\mathbb{Z},$

by setting, for $p\in Z[t,t^{-1}],$

$\overline{\alpha}_{*}([f]\otimes_{Z[t]}p):=(\alpha_{*}[f])\cdot p,$

where on the right hand side, we interpret $p$ as an element of $Z[a,a^{-1}]$
by substituting $t\mapsto a$. Then $\overline{\alpha}_{*}$ is
$Z[t,t^{-1}]$-linear, and the diagram

(13)
$\textstyle{W_{*}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{*}}$$\scriptstyle{i_{*}}$$\textstyle{\overline{W}_{*}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\alpha}_{*}}$$\textstyle{({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(X)}$

commutes.

We turn to the proof of uniqueness. Suppose that
$\beta_{*}:\overline{W}_{*}(X)\to({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(X)$
is any $Z[t,t^{-1}]$-linear extension of $\alpha_{*}$, i.e. $\beta_{*}\circ
i_{*}=\alpha_{*}$. Then

$\displaystyle\beta_{*}([f]\otimes_{Z[t]}p)$
$\displaystyle=\beta_{*}([f]\otimes_{Z[t]}(1\cdot
p))=\beta_{*}(([f]\otimes_{Z[t]}1)\cdot p)$
$\displaystyle=(\beta_{*}([f]\otimes_{Z[t]}1))\cdot p=(\beta_{*}i_{*}[f])\cdot
p$ $\displaystyle=\alpha_{*}[f]\cdot
p=\overline{\alpha}_{*}([f]\otimes_{Z[t]}p).$

Hence $\overline{\alpha}_{*}$ is unique. If $\alpha_{*}$ is natural in $X$ and
commutes with suspension isomorphisms, then $\overline{\alpha}_{*}$ inherits
these properties.

We prove statement 2: Since $\alpha_{*}$ and $\beta_{*}$ are $Z[t]$-linear,
they induce uniquely $Z[t,t^{-1}]$-linear transformations

$\overline{\alpha}_{*},\overline{\beta}_{*}:(W_{*}(X)\otimes_{Z[t]}Z[t,t^{-1}])_{j}\longrightarrow({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{\bar{j}}(X),~{}j\in\mathbb{Z},$

as explained in statement 1. Given an element

$[f:V^{j-4k}\to
X]\otimes_{Z[t]}rt^{k}\in(W_{*}(X)\otimes_{Z[t]}Z[t,t^{-1}])_{j}=\overline{W}_{j}(X)$

$k\in\mathbb{Z},$ $r\in Z$, there exists a (unique) element $q\in
Q_{\bar{j}}(X)\otimes Z$ with $\omega(q)=[f]\otimes_{Z[t]}rt^{k}$, as $\omega$
is an isomorphism. Such an element is represented in the quotient
$Q_{\bar{j}}(X)\otimes Z$ by an element of the form

$q=\sum_{i=1}^{m}[g_{i}:M_{i}^{j-4k_{i}}\to X]\otimes
r_{i},~{}[g_{i}]\in\Omega^{\operatorname{SO}}_{j-4k_{i}}(X),~{}r_{i}\in
Z,~{}k_{i}\in\mathbb{Z}.$

By the definition of $\omega,$
$\omega([g_{i}]\otimes_{\mathbb{Z}}r_{i})=[g_{i}]\otimes_{Z[t]}r_{i}t^{k_{i}},$
so that

$[f]\otimes_{Z[t]}rt^{k}=\sum_{i=1}^{m}[g_{i}]\otimes_{Z[t]}r_{i}t^{k_{i}}$

and consequently,

$\displaystyle\overline{\alpha}_{*}([f]\otimes_{Z[t]}rt^{k})$
$\displaystyle=\sum_{i=1}^{m}\overline{\alpha}_{*}([g_{i}]\otimes_{Z[t]}r_{i}t^{k_{i}})=\sum_{i=1}^{m}(\alpha_{*}[g_{i}])\cdot
r_{i}a^{k_{i}}$ $\displaystyle=\sum_{i=1}^{m}(\beta_{*}[g_{i}])\cdot
r_{i}a^{k_{i}}=\sum_{i=1}^{m}\overline{\beta}_{*}([g_{i}]\otimes_{Z[t]}r_{i}t^{k_{i}})$
$\displaystyle=\overline{\beta}_{*}([f]\otimes_{Z[t]}rt^{k}).$

This proves that the periodic versions agree on $\overline{W}_{*}(X)$,
$\overline{\alpha}_{*}=\overline{\beta}_{*}$. Using the commutativity of (13)
we deduce $\alpha_{*}=\overline{\alpha}_{*}\circ
i_{*}=\overline{\beta}_{*}\circ i_{*}=\beta_{*}.$ ∎

######  Proposition 5.7.

The natural transformation of homology theories induced by
$\Delta:\operatorname{MWITT}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$
agrees with the transformation $\mu^{\operatorname{Witt}},$

$\Delta_{*}=\mu^{\operatorname{Witt}}:\Omega^{\operatorname{Witt}}_{*}(Y,B)\longrightarrow{\operatorname{KO}}_{*}(Y,B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$

on compact PL pairs $(Y,B)$. In particular, $\Delta(X)=\mu_{X}$ for a compact
Witt space $(X,\partial X)$.

###### Proof.

The ring-spectrum level Siegel-Sullivan orientation
$\Delta:\operatorname{MWITT}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$
restricts to the orientation
$\Delta:\operatorname{MSPL}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$ of
Definition 3.5, and then further to
$\Delta|:\operatorname{MSO}\to\operatorname{MSPL}\stackrel{{\scriptstyle\Delta}}{{\longrightarrow}}{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$.
By Proposition 3.3, and since $\Delta_{\operatorname{SPL}}$ extends
$\Delta_{\operatorname{SO}}$, the induced map

$\Delta|_{*}:\Omega^{\operatorname{SO}}_{*}(Y,B)\longrightarrow{\operatorname{KO}}_{*}(Y,B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$

agrees with

$\Delta_{{\operatorname{SO}}*}:\Omega^{\operatorname{SO}}_{*}(Y,B)\longrightarrow{\operatorname{KO}}_{*}(Y,B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$

on every compact PL pair $(Y,B)$. Since
$\Delta_{*}:\Omega^{\operatorname{Witt}}_{*}(Y,B)\to{\operatorname{KO}}_{*}(Y,B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$
is odd prime local, it factors uniquely through the odd-primary localization
$W_{*}(-)$ of $\Omega^{\operatorname{Witt}}_{*}(-)$. The resulting
homomorphism

(14)
$\Delta_{*}:W_{*}(Y,B)\longrightarrow({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(Y,B)$

is a natural transformation of homology theories. On a point, it is given by

$\Delta_{*}(c^{k})=\Delta_{{\operatorname{SO}}*}(c^{k})=\Delta_{{\operatorname{SO}}*}[\mathbb{C}P^{2}]^{k}=\sigma(\mathbb{C}P^{2})^{k}\cdot
a^{k}=a^{k},$

using the signature relation (1). Since
$\Delta:\operatorname{MWITT}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$ is a
morphism of ring spectra, this shows that (14) is $Z[t]$-linear. Siegel’s
transformation

$\mu^{\operatorname{Witt}}:W_{*}(Y,B)\longrightarrow({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{*}(Y,B)$

is a natural transformation of homology theories as well, and it is
$Z[t]$-linear by the periodicity relation (12). Thus, in order to prove
equality of the two transformations on $W_{*}(Y,B)$, it remains by Proposition
5.6 to show that they agree on compact smooth manifolds mapping into $(Y,B)$.
Let $g:(M,\partial M)\to(Y,B)$ be a smooth manifold in $(Y,B)$. By Proposition
4.3, $\mu_{M}=\Delta_{\operatorname{SO}}(M)$ and hence

$\Delta_{*}([g]\otimes
1)=\Delta_{{\operatorname{SO}}*}[g]=g_{*}\Delta_{\operatorname{SO}}(M)=g_{*}\mu_{M}=\mu^{\operatorname{Witt}}([g]\otimes
1),$

as was to be shown. ∎

##  6\. Multiplicativity of Orientation Classes

The classical Sullivan orientation $\Delta_{\operatorname{SO}}(M\times N)$ of
a product $M\times N$ of closed smooth manifolds is known to satisfy the
multiplicativity property

(15) $\Delta_{\operatorname{SO}}(M\times
N)=\Delta_{\operatorname{SO}}(M)\times\Delta_{\operatorname{SO}}(N)$

with respect to the external product on
${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$-homology. One way to see this is
to argue L-theoretically using Ranicki’s multiplicative morphism
$\sigma^{*}:\operatorname{MSPL}\to\mathbb{L}(\mathbb{Z})$: A ring morphism
$\phi:E\to F$ between ring spectra preserves external products, i.e. the
diagram

(16) $\textstyle{E_{*}(X)\otimes
E_{*}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi\otimes\phi}$$\scriptstyle{\times}$$\textstyle{F_{*}(X)\otimes
F_{*}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\times}$$\textstyle{E_{*}(X\times
Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{F_{*}(X\times
Y)}$

commutes. Applying this to $\sigma^{*}$, one obtains the multiplicativity of
the $\mathbb{L}$-homology orientation of PL manifolds,

$\displaystyle[M]_{\mathbb{L}}\times[N]_{\mathbb{L}}$
$\displaystyle=\sigma^{*}[\operatorname{id}_{M}]\times\sigma^{*}[\operatorname{id}_{N}]=\sigma^{*}([\operatorname{id}_{M}]\times[\operatorname{id}_{N}])$
$\displaystyle=\sigma^{*}[\operatorname{id}_{M\times N}]=[M\times
N]_{\mathbb{L}}.$

(Via the ring morphism $\operatorname{MSTOP}\to\mathbb{L}(\mathbb{Z})$, this
works just as well for topological manifolds.) Using the ring equivalence
$\kappa^{-1}:\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]\simeq{\operatorname{KO}}[\mbox{$\frac{1}{2}$}],$
it follows that $\Delta(M\times N)=\Delta(M)\times\Delta(N)$ for
$\Delta:\operatorname{MSPL}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$ as in
Definition 3.5. Equation (15) is then a consequence of Proposition 3.3.
Alternatively, and more directly, one may also deduce (15) from the
multiplicativity of Sullivan’s universal elements

$\Delta_{4n}\in{\operatorname{KO}}^{4n}(\operatorname{MSO}_{4n})\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].$

Indeed, he shows ([56, p. 202, g)]) that the canonical homomorphism, induced
by the classifying map,

$\widetilde{{\operatorname{KO}}\mbox{$\frac{1}{2}$}}^{4(q+r)}(\operatorname{MSO}_{4(q+r)})\longrightarrow\widetilde{{\operatorname{KO}}\mbox{$\frac{1}{2}$}}^{4(q+r)}(\operatorname{MSO}_{4q}\wedge\operatorname{MSO}_{4r})$

sends $\Delta_{4(q+r)}\mapsto\Delta_{4q}\times\Delta_{4r}.$ This, together
with naturality, implies the relation

$\Delta_{\operatorname{SO}}(\xi\times\eta)=\Delta_{\operatorname{SO}}(\xi)\times\Delta_{\operatorname{SO}}(\eta)\in\widetilde{{\operatorname{KO}}\mbox{$\frac{1}{2}$}}^{*}(\operatorname{Th}(\xi\times\eta))=\widetilde{{\operatorname{KO}}\mbox{$\frac{1}{2}$}}^{*}(\operatorname{Th}\xi\wedge\operatorname{Th}\eta)$

for the Sullivan Thom class
$\Delta_{\operatorname{SO}}(\xi)\in\widetilde{{\operatorname{KO}}\mbox{$\frac{1}{2}$}}(\operatorname{Th}\xi)$
of an oriented vector bundle $\xi$ over a finite complex; see also Madsen-
Milgram [39, p. 116]. Finally, one applies this to the stable normal bundles
$\xi=\nu_{M},$ $\eta=\nu_{N}$ and uses Alexander duality.

######  Remark 6.1.

As Rosenberg and Weinberger point out in [52, p. 51, Lemma 6], it is _not_
true that the class $[D_{M}]$ of the signature operator $D_{M}$ in Kasparov
K-homology is multiplicative under external Kasparov product. If one of the
dimensions of these classes is even, then the class is multiplicative, but if
both dimensions are odd, then $[D_{M\times N}]=2[D_{M}]\times[D_{N}]$. See
also [34, Theorem 8.5].

A first immediate application of our approach, then, is a proof of full
cartesian multiplicativity of the Siegel-Sullivan orientation class
$\mu_{X}=\Delta(X)\in{\operatorname{KO}}_{n}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$
for Witt spaces $X$, generalizing the manifold multiplicativity. (This was not
established in [55].)

######  Theorem 6.2.

Let $X$ and $Y$ be closed Witt spaces. Then the Siegel-Sullivan orientation of
the Witt space $X\times Y$ is given by

$\Delta(X\times Y)=\Delta(X)\times\Delta(Y),$

using the external product
$({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{m}(X)\otimes({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{n}(Y)\to({\operatorname{KO}}\mbox{$\frac{1}{2}$})_{m+n}(X\times
Y)$.

###### Proof.

Applying diagram (16) to the ring morphism
$\Delta:\operatorname{MWITT}\to{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$, we
obtain for the orientation classes

$\displaystyle\Delta(X\times Y)$
$\displaystyle=\Delta_{*}[\operatorname{id}_{X\times
Y}]=\Delta_{*}([\operatorname{id}_{X}]\times[\operatorname{id}_{Y}])$
$\displaystyle=\Delta_{*}[\operatorname{id}_{X}]\times\Delta_{*}[\operatorname{id}_{Y}]=\Delta(X)\times\Delta(Y).$

∎

##  7\. Bundle Transfer of the Siegel-Sullivan Orientation

The equivalence of $\mathbb{E}_{\infty}$-ring spectra
$\kappa:{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\simeq\mathbb{L}()[\mbox{$\frac{1}{2}$}]$
constructed in Proposition 2.1, together with $\mathbb{L}$-theoretic results
of [8], allows for a treatment of bundle transfers of Siegel-Sullivan classes.
We begin with recollections on homological block bundle transfer homomorphisms
and use $\kappa$ to relate the transfers on ${\operatorname{KO}}$\- and
$\mathbb{L}$-homology.

Let $F$ be a closed oriented PL manifold of dimension $d$ and let $K$ be a
finite ball complex with associated polyhedron $B=|K|$. Let $b$ denote the
dimension of $B$. Let $\xi$ be an oriented PL $F$-block bundle over $K$ with
total space $X=E(\xi)$, $\dim X=b+d$. The theory of $F$-block bundles has been
developed by Casson in [19]. PL-locally trivial PL fiber bundles $X\to B$ with
pointwise fiber $F$ are a special case.

Let $E$ be a ring spectrum equipped with a ring map $\operatorname{MSPL}\to
E$. Then the block bundle $\xi$ has an associated block transfer homomorphism

(17) $\xi^{!}:E_{n}(B)\longrightarrow E_{n+d}(X).$

In [8], we described $\xi^{!}$ as a composition

$E_{n}(B)\stackrel{{\scriptstyle\sigma}}{{\cong}}\widetilde{E}_{n+s}(S^{s}B^{+})\stackrel{{\scriptstyle
T(\xi)_{*}}}{{\longrightarrow}}\widetilde{E}_{n+s}(\operatorname{Th}(\nu))\stackrel{{\scriptstyle\Phi}}{{\longrightarrow}}E_{n+s-(s-d)}(X).$

Here, $\sigma$ is the suspension isomorphism and
$T(\xi):S^{s}B^{+}\to\operatorname{Th}(\nu)$ the Umkehr map (i.e. Pontrjagin-
Thom collapse) associated to a $\xi$-block preserving PL embedding
$X\hookrightarrow B\times^{s}$ for large $s$. Such an embedding can be shown
to have a regular neighborhood that is the total space of an $(s-d)$-disc
block bundle $\nu$ over $X$, see e.g. [8, Section 2]. This normal disc block
bundle $\nu$ represents the stable vertical normal bundle of $\xi$ and can be
taken to be a PL microbundle (or PL $(^{s-d},0)$-bundle) since
$\operatorname{BSPL}\simeq\operatorname{B}\widetilde{\operatorname{SPL}}$ for
the stable classifying spaces. (Such a stable vertical normal bundle exists
even for mock bundles with manifold blocks, see [15, IV.2, p. 83].) The image
of the $\operatorname{MSPL}$-cohomology Thom class of $\nu$ under the ring map
$\operatorname{MSPL}\to E$ yields an $E$-cohomology Thom class of $\nu$. Cap
product with this class defines the Thom homomorphism $\Phi$.

We shall consider the block transfer $\xi^{!}$ in the cases where the ring
spectrum $E$ is ${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$,
$\mathbb{L}(\mathbb{Q})$ or
$\mathbb{L}(\mathbb{Q})[\mbox{$\frac{1}{2}$}]=\mathbb{L}(\mathbb{Z})[\mbox{$\frac{1}{2}$}]$,
and the ring maps $\operatorname{MSPL}\to E$ are the orientations considered
earlier. The compatibility of these transfers will be established in Lemma 7.3
and is essentially a consequence of the multiplicativity of the map $\kappa$.
We need to be more precise about the involved Thom homomorphisms $\Phi$. Our
arguments involve the three Thom classes discussed in Section 3: The class
$u_{\operatorname{SPL}}(\alpha)$ in $\operatorname{MSPL}$-cohomology,
$u_{\mathbb{L}}(\alpha)$ in $\mathbb{L}$-cohomology, and the class
$\Delta(\alpha)$ in ${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$-cohomology.
Let $E$ be a ring spectrum and let $m=s-d$ denote the rank of the
aforementioned representative of the stable vertical normal disc block bundle
$\nu$. The reduced $E$-cohomology group of the Thom space can be expressed as
a relative group,

$\widetilde{E}^{m}(\operatorname{Th}(\nu))\cong E^{m}(N,\partial N),$

where $N$ is the total space of $\nu$ and $\partial N$ the total space of the
sphere bundle of $\nu$. Let

$\rho_{*}:E_{*}(N)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}E_{*}(X)$

be the inverse of the isomorphism induced on $E$-homology by the inclusion
$X\hookrightarrow N$ of the zero section. If $\nu$ is $E$-orientable, then
using the cap product

$\cap:E^{m}(N,\partial N)\otimes E_{n}(N,\partial N)\longrightarrow
E_{n-m}(N)\stackrel{{\scriptstyle\rho_{*}}}{{\cong}}E_{n-m}(X)$

with a Thom class ($E$-orientation) $u\in E^{m}(N,\partial N)$ for $\nu$, we
obtain the _Thom homomorphism_

$\Phi:=\rho_{*}(u\cap-):\widetilde{E}_{n}(\operatorname{Th}(\nu))\cong
E_{n}(N,\partial N)\longrightarrow E_{n-m}(X).$

Since $\Delta(\nu)$ is a
${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$-cohomology orientation of $\nu$
with $\Delta(\nu)=\Delta_{*}u_{\operatorname{SPL}}(\nu)$ (Definition 3.6), we
get for the ring spectrum $E={\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$ the
Thom homomorphism

$\Phi=\rho_{*}(\Delta(\nu)\cap-):\widetilde{{\operatorname{KO}}}_{n}(\operatorname{Th}(\nu))\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\longrightarrow{\operatorname{KO}}_{n-m}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].$

Similarly, since $u_{\mathbb{L}}(\nu)$ is an
$\mathbb{L}[\mbox{$\frac{1}{2}$}]$-cohomology orientation of $\nu$ with
$u_{\mathbb{L}}(\nu)=\sigma^{*}u_{\operatorname{SPL}}(\nu),$ we receive for
the ring spectrum $E=\mathbb{L}[\mbox{$\frac{1}{2}$}]$ the Thom homomorphism

$\Phi=\rho_{*}(u_{\mathbb{L}}(\nu)\cap-):\widetilde{\mathbb{L}}_{n}(\operatorname{Th}(\nu))\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\longrightarrow\mathbb{L}_{n-m}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].$

######  Lemma 7.1.

The Thom homomorphisms $\Phi$ on $\mathbb{L}[\mbox{$\frac{1}{2}$}]$-homology
and ${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$-homology agree under the
natural isomorphism $\kappa$, that is, the diagram

$\textstyle{\widetilde{{\operatorname{KO}}}_{n}(\operatorname{Th}(\nu))\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{{\operatorname{KO}}_{n-m}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\widetilde{\mathbb{L}}_{n}(\operatorname{Th}(\nu))\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{L}_{n-m}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]}$

commutes.

###### Proof.

As $\kappa_{*}$ is a natural transformation of homology theories, it commutes
with the isomorphism $\rho_{*}$. Since $\kappa$ is a morphism of ring spectra,
it respects cap products, i.e. the diagram

$\textstyle{({\operatorname{KO}}[\mbox{$\frac{1}{2}$}])^{m}(Y,A)\otimes({\operatorname{KO}}[\mbox{$\frac{1}{2}$}])_{n}(Y,A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cap}$$\scriptstyle{\kappa_{*}\otimes\kappa_{*}}$$\textstyle{({\operatorname{KO}}[\mbox{$\frac{1}{2}$}])_{n-m}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{*}}$$\textstyle{(\mathbb{L}[\mbox{$\frac{1}{2}$}])^{m}(Y,A)\otimes(\mathbb{L}[\mbox{$\frac{1}{2}$}])_{n}(Y,A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cap}$$\textstyle{(\mathbb{L}[\mbox{$\frac{1}{2}$}])_{n-m}(Y)}$

commutes. By Lemma 3.7, $\kappa_{*}(\Delta(\nu))=u_{\mathbb{L}}(\nu).$
Therefore,

$\displaystyle\kappa_{*}\Phi$
$\displaystyle=\kappa_{*}\rho_{*}(\Delta(\nu)\cap-)=\rho_{*}\kappa_{*}(\Delta(\nu)\cap-)$
$\displaystyle=\rho_{*}(\kappa_{*}(\Delta(\nu))\cap\kappa_{*}(-))=\rho_{*}(u_{\mathbb{L}}(\nu)\cap\kappa_{*}(-))=\Phi\kappa_{*}.$

∎

######  Remark 7.2.

Since localization is multiplicative on the spectrum level, it takes the Thom
homomorphism $\Phi$ defined by capping with the Thom class
$u_{\mathbb{L}}(\nu)\in(\widetilde{\mathbb{L}\mathbb{Q}})^{m}(\operatorname{Th}(\nu))$
to the Thom homomorphism $\Phi$ defined by capping with the localized class
$u_{\mathbb{L}}(\nu)\in(\widetilde{\mathbb{L}\mathbb{Q}}[\mbox{$\frac{1}{2}$}])^{m}(\operatorname{Th}(\nu))$,
that is, the diagram

$\textstyle{(\widetilde{\mathbb{L}\mathbb{Q}})_{n}(\operatorname{Th}(\nu))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{(\mathbb{L}\mathbb{Q})_{n-m}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widetilde{\mathbb{L}\mathbb{Q}}[\mbox{$\frac{1}{2}$}])_{n}(\operatorname{Th}(\nu))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{(\mathbb{L}\mathbb{Q}[\mbox{$\frac{1}{2}$}])_{n-m}(X)}$

commutes. Thus $\mathbb{L}$-theoretic transfers also commute with localization
away from $2$.

Using the Thom homomorphisms $\Phi$ appearing in Lemma 7.1, there are in
particular transfers

$\xi^{!}:\mathbb{L}_{n}(B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\longrightarrow\mathbb{L}_{n+d}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]$

and

$\xi^{!}:{\operatorname{KO}}_{n}(B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\longrightarrow{\operatorname{KO}}_{n+d}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].$

######  Lemma 7.3.

The diagram of transfers

$\textstyle{{\operatorname{KO}}_{*+d}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\mathbb{L}_{*+d}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]}$$\textstyle{{\operatorname{KO}}_{*}(B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi^{!}}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\mathbb{L}_{*}(B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi^{!}}$

commutes.

###### Proof.

By construction of the block transfer $\xi^{!}$, the diagram factors as

$\textstyle{{\operatorname{KO}}_{*+d}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\mathbb{L}_{*+d}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]}$$\textstyle{\widetilde{{\operatorname{KO}}}_{*+s}(\operatorname{Th}(\nu))\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\widetilde{\mathbb{L}}_{*+s}(\operatorname{Th}(\nu))\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\widetilde{{\operatorname{KO}}}_{*+s}(S^{s}B^{+})\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T(\xi)_{*}}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\widetilde{\mathbb{L}}_{*+s}(S^{s}B^{+})\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T(\xi)_{*}}$$\textstyle{{\operatorname{KO}}_{*}(B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\mathbb{L}_{*}(B)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$

The bottom and middle squares commute, as $\kappa_{*}$ is a natural
transformation of homology theories, while the top square involving the Thom
homomorphisms commutes by Lemma 7.1. ∎

The material on block bundle transfer homomorphisms developed above enables us
to establish our main result on bundle transfer of Siegel-Sullivan
orientations:

######  Theorem 7.4.

If $\xi$ is an oriented PL $F$-block bundle with closed oriented PL manifold
fiber $F$ over a closed Witt base $B$, then the Siegel-Sullivan orientations
of base and total space $X$ are related under block bundle transfer by

$\xi^{!}\Delta(B)=\Delta(X).$

###### Proof.

Remark 7.2 implies that transfer commutes with localization away from $2$: the
diagram

$\textstyle{(\mathbb{L}\mathbb{Q})_{*+d}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{L}\mathbb{Q}[\mbox{$\frac{1}{2}$}])_{*+d}(X)}$$\textstyle{(\mathbb{L}\mathbb{Q})_{*}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi^{!}}$$\textstyle{(\mathbb{L}\mathbb{Q}[\mbox{$\frac{1}{2}$}])_{*}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi^{!}}$

commutes. In [8, Theorem 7.1] we showed that the left hand transfer sends
$[B]_{\mathbb{L}}$ to $[X]_{\mathbb{L}}$. Thus

$\xi^{!}[B]_{\mathbb{L}}=[X]_{\mathbb{L}}$

for the right hand transfer as well. By (10),

$\kappa_{*}\Delta(X)=[X]_{\mathbb{L}}\in(\mathbb{L}\mathbb{Q}[\mbox{$\frac{1}{2}$}])_{b+d}(X),~{}\kappa_{*}\Delta(B)=[B]_{\mathbb{L}}\in(\mathbb{L}\mathbb{Q}[\mbox{$\frac{1}{2}$}])_{b}(B).$

Using Lemma 7.3,

$\kappa_{*}\xi^{!}\Delta(B)=\xi^{!}\kappa_{*}\Delta(B)=\xi^{!}[B]_{\mathbb{L}}=[X]_{\mathbb{L}}=\kappa_{*}\Delta(X).$

It follows that $\xi^{!}\Delta(B)=\Delta(X)$, as $\kappa_{*}$ is an
isomorphism. ∎

##  8\. Gysin Restriction of the Siegel-Sullivan Orientation

Our method based on the equivalence of $\mathbb{E}_{\infty}$-ring spectra
$\kappa:{\operatorname{KO}}[\mbox{$\frac{1}{2}$}]\simeq\mathbb{L}()[\mbox{$\frac{1}{2}$}]$
constructed in Proposition 2.1, together with results of [7], allows for a
treatment of Gysin restrictions of Siegel-Sullivan classes under normally
nonsingular inclusions of singular spaces in a fashion parallel to our
analysis of bundle transfers in the previous section. An inclusion
$g:Y\hookrightarrow X$ of stratified spaces is _normally nonsingular_ if $Y$
has an open tubular neighborhood that can be equipped in a stratum preserving
manner with the structure of a vector bundle $\nu$ over $Y$ such that $Y$ is
identified with the zero section. For example, the transverse intersection of
a smooth submanifold with a Whitney stratified set $X$ in an ambient smooth
manifold is normally nonsingular in $X$ ([27, p. 47, Thm. 1.11]).

Let $g:Y^{n-c}\hookrightarrow X^{n}$ be a codimension $c$ normally nonsingular
inclusion of closed Witt spaces with normal bundle $\nu$ of rank $c$. Let $E$
be a ring spectrum such that $\nu$ has an $E$-orientation $u$. We describe the
Gysin restriction on $E$-homology associated to $g$. The canonical map
$j:X^{+}\to\operatorname{Th}(\nu)$ induces a homomorphism

$j_{*}:E_{*}(X)\longrightarrow\widetilde{E}_{*}(\operatorname{Th}(\nu)).$

As in the previous section, cap product with the $E$-orientation $u$ yields
the Thom homomorphism

$\Phi=\rho_{*}(u\cap-):\widetilde{E}_{*}(\operatorname{Th}(\nu))\longrightarrow
E_{*-c}(Y).$

Composition defines the _Gysin restriction_

$g^{!}=\Phi\circ j_{*}:E_{*}(X)\longrightarrow E_{*-c}(Y).$

Now suppose that $\nu$ is $H\mathbb{Z}$-oriented, compatibly with the
orientations of $X$ and $Y$. Applying the above general description of $g^{!}$
to $E={\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$ with $u=\Delta(\nu)$, we
obtain the Gysin homomorphism

$g^{!}:{\operatorname{KO}}_{*}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\longrightarrow{\operatorname{KO}}_{*-c}(Y)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}],$

and applying it to $E=\mathbb{L}[\mbox{$\frac{1}{2}$}]$ with
$u=u_{\mathbb{L}}(\nu)$, we obtain the Gysin homomorphism

$g^{!}:\mathbb{L}_{*}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\longrightarrow\mathbb{L}_{*-c}(Y)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].$

######  Lemma 8.1.

The diagram of Gysin restrictions

$\textstyle{{\operatorname{KO}}_{*-c}(Y)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\mathbb{L}_{*-c}(Y)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]}$$\textstyle{{\operatorname{KO}}_{*}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{!}}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\mathbb{L}_{*}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{!}}$

commutes.

###### Proof.

By construction of the restrictions $g^{!}$, the diagram factors as

$\textstyle{{\operatorname{KO}}_{*-c}(Y)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\mathbb{L}_{*-c}(Y)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]}$$\textstyle{\widetilde{{\operatorname{KO}}}_{*}(\operatorname{Th}(\nu))\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\widetilde{\mathbb{L}}_{*}(\operatorname{Th}(\nu))\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{{\operatorname{KO}}_{*}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{*}}$$\scriptstyle{\kappa_{*}}$$\scriptstyle{\cong}$$\textstyle{\mathbb{L}_{*}(X)\otimes\mathbb{Z}[\mbox{$\frac{1}{2}$}].\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{*}}$

The bottom square commutes, as $\kappa_{*}$ is a natural transformation of
homology theories, while the top square involving the Thom homomorphisms
commutes by Lemma 7.1. ∎

The Siegel-Sullivan orientation behaves under normally nonsingular Gysin
restrictions as follows.

######  Theorem 8.2.

Let $g:Y^{n-c}\hookrightarrow X^{n}$ be an oriented normally nonsingular
inclusion of closed Witt spaces. The
${\operatorname{KO}}[\mbox{$\frac{1}{2}$}]$-homology Gysin map $g^{!}$ of $g$
sends the Siegel-Sullivan orientation of $X$ to the Siegel-Sullivan
orientation of $Y$:

$g^{!}\Delta(X)=\Delta(Y).$

###### Proof.

The proof is analogous to the one of Theorem 7.4. Remark 7.2 implies that
Gysin restriction commutes with localization away from $2$: the diagram

$\textstyle{(\mathbb{L}\mathbb{Q})_{*-c}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathbb{L}\mathbb{Q}[\mbox{$\frac{1}{2}$}])_{*-c}(Y)}$$\textstyle{(\mathbb{L}\mathbb{Q})_{*}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{!}}$$\textstyle{(\mathbb{L}\mathbb{Q}[\mbox{$\frac{1}{2}$}])_{*}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{!}}$

commutes. In [7, Theorem 3.17] we showed that the left hand restriction sends
$[X]_{\mathbb{L}}$ to $[Y]_{\mathbb{L}}$. Thus

$g^{!}[X]_{\mathbb{L}}=[Y]_{\mathbb{L}}$

for the right hand restriction as well. By (10),

$\kappa_{*}\Delta(X)=[X]_{\mathbb{L}}\in(\mathbb{L}\mathbb{Q}[\mbox{$\frac{1}{2}$}])_{n}(X),~{}\kappa_{*}\Delta(Y)=[Y]_{\mathbb{L}}\in(\mathbb{L}\mathbb{Q}[\mbox{$\frac{1}{2}$}])_{n-c}(Y).$

Using Lemma 8.1,

$\kappa_{*}g^{!}\Delta(X)=g^{!}\kappa_{*}\Delta(X)=g^{!}[X]_{\mathbb{L}}=[Y]_{\mathbb{L}}=\kappa_{*}\Delta(Y).$

It follows that $g^{!}\Delta(X)=\Delta(Y)$, as $\kappa_{*}$ is an isomorphism.
∎

In tandem, Theorems 8.2 and 7.4 show that the transfer associated to a
normally nonsingular map $Y\to B$ (Goresky-MacPherson [29, 5.4.3], Fulton-
MacPherson [26]) sends $\Delta(B)$ to $\Delta(Y)$.

## References

  * [1] P. Albin, E. Leichtnam, R. Mazzeo, P. Piazza, The Signature Package on Witt Spaces, Annales Scientifiques de l’École normale supérieure 45 (2012), 241 – 310.
  * [2] P. Albin, M. Banagl, E. Leichtnam, R. Mazzeo, P. Piazza, Refined Intersection Homology on non-Witt Spaces, J. Topol. Anal. 7 (2015), no. 1, 105 – 133.
  * [3] M. Banagl, Extending Intersection Homology Type Invariants to non-Witt Spaces, Memoirs Amer. Math. Soc. 160 (2002), no. 760, 1 – 83.
  * [4] M. Banagl, The L-Class of non-Witt Spaces, Annals of Math. 163 (2006), 1 – 24.
  * [5] M. Banagl, Topological Invariants of Stratified Spaces, Springer Monographs in Math., Springer Verlag, 2007.
  * [6] M. Banagl, The Signature of Singular Spaces and its Refinements to Generalized Homology Theories, in: Topology of Stratified Spaces, eds. G. Friedman, E. Hunsicker, A. Libgober, L. Maxim, Mathematical Sciences Research Institute Publications, vol. 58, Cambridge University Press, 2011, 223 – 249.
  * [7] M. Banagl, Gysin Restriction of Topological and Hodge-theoretic Characteristic Classes for Singular Spaces, New York J. of Math. 26 (2020), 1273 – 1337.
  * [8] M. Banagl, Bundle Transfer of the Goresky-MacPherson L-Class for Singular Spaces, preprint (2022), arXiv:2210.05432.
  * [9] M. Banagl, S. Cappell and J. Shaneson, Computing Twisted Signatures and L-Classes of Stratified Spaces, Math. Ann. 326 no. 3 (2003), 589 – 623.
  * [10] M. Banagl, G. Laures, J. McClure, The L-Homology Fundamental Class for IP-Spaces and the Stratified Novikov Conjecture, Selecta Math. 25:7 (2019), pp. 1 - 104.
  * [11] M. Banagl, D. Wrazidlo, The Uniqueness Theorem for Gysin Coherent Characteristic Classes of Singular Spaces, preprint (2022), arXiv:2210.13009.
  * [12] J. C. Becker, D. H. Gottlieb, The Transfer Map and Fiber Bundles, Topology 14 (1975), 1 – 12.
  * [13] A. Borel et al., Intersection Cohomology, Progress in Math., vol. 50, Birkhäuser, 1984.
  * [14] J.-P. Brasselet, J. Schürmann, S. Yokura, Hirzebruch Classes and Motivic Chern Classes for Singular Spaces, J. Topol. Analysis 2 (2010), 1 – 55.
  * [15] S. Buoncristiano, C. P. Rourke, B. J. Sanderson, A Geometric Approach to Homology Theory, London Math. Soc. Lecture Note Series, vol. 18, Cambridge Univ. Press, 1976.
  * [16] S. E. Cappell, J. L. Shaneson, Stratifiable Maps and Topological Invariants, J. of the Amer. Math. Soc. 4 (1991), 521 – 551.
  * [17] S. Cappell, J. Shaneson, S. Weinberger, Classes topologiques caractéristiques pour les actions de groupes sur les espaces singuliers, C. R. Acad. Sci. Paris 313 (1991), 293 – 295.
  * [18] S. E. Cappell and S. Weinberger, _Classification de certaines espaces stratifiés_ , C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 399–401.
  * [19] A. J. Casson, Generalisations and Applications of Block Bundles, Trinity College, Cambridge fellowship dissertation, 1967. In: The Hauptvermutung Book (ed. A. Ranicki), 33 – 68.
  * [20] S. J. Curran, Intersection Homology and Free Group Actions on Witt Spaces, Michigan Math. J. 39 (1992), 111 – 127.
  * [21] J. M. Davies, The Structure Sheaf of the Moduli of Oriented $p$-divisible Groups, preprint (2020), arXiv:2007.00482v1.
  * [22] J. M. Davies, Constructing and Calculating Adams Operations on Topological Modular Forms, preprint (2021), arXiv:2104.13407v1
  * [23] G. Friedman, Singular Intersection Homology, new mathematical monographs, vol. 33, Cambridge University Press, 2020.
  * [24] G. Friedman, J. McClure, The Symmetric Signature of a Witt Space, Journal of Topology and Analysis 5 (2013), 121 – 159.
  * [25] W. Fulton, Intersection Theory, 2nd ed., Springer Verlag, 1998.
  * [26] W. Fulton, R. MacPherson, Categorical Framework for the Study of Singular Spaces, Memoirs Amer. Math. Soc. 31 (1981), no. 243.
  * [27] M. Goresky, R. MacPherson, Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 14, Springer-Verlag, 1988.
  * [28] M. Goresky, R. MacPherson, Intersection Homology Theory, Topology 19 (1980), 135 – 162.
  * [29] M. Goresky, R. MacPherson, Intersection Homology II, Invent. Math. 71 (1983), 77 – 129.
  * [30] R. M. Goresky, P. H. Siegel, Linking Pairings on Singular Spaces, Comm. Math. Helv. 58 (1983), 96 – 110.
  * [31] F. Kirwan, J. Woolf, An Introduction to Intersection Homology Theory, 2nd edition, Chapman & Hall/CRC, 2006.
  * [32] M. Kreck, W. Lück, The Novikov Conjecture, Oberwolfach Seminars, Birkhäuser Verlag, 2005.
  * [33] M. Land, T. Nikolaus, On the Relation between K- and L-Theory of $C^{*}$-Algebras, Math. Annalen 371 (2018), 517 – 563.
  * [34] M. Land, T. Nikolaus, M. Schlichting, L-Theory of $C^{*}$-Algebras, preprint, arXiv:2208.10556v1.
  * [35] G. Laures, J. McClure, Multiplicative Properties of Quinn Spectra, Forum Math. 26 (2014), no.4, 1117 – 1185.
  * [36] L. G. Maxim, Intersection Homology & Perverse Sheaves, Graduate Texts in Math. 281, Springer-Verlag, 2019.
  * [37] W. Lück, A. Ranicki, Surgery Obstructions of Fibre Bundles, J. of Pure and Applied Algebra 81 (1992), 139 – 189.
  * [38] J. Lurie, Lecture 25 on Algebraic L-Theory and Surgery: The Hirzebruch Signature Formula (2011), https://www.math.ias.edu/l̃urie/287x.html
  * [39] I. Madsen, R. J. Milgram, The Classifying Spaces for Surgery and Cobordism of Manifolds, Annals of Math. Studies 92, Princeton University Press, 1979.
  * [40] C. McCrory, Cone Complexes and PL Transversality, Transactions of the American Math. Soc. 207 (1975), 269 – 291.
  * [41] J. Peter May, $E_{\infty}$ Ring Spaces and $E_{\infty}$ Ring Spectra, Lecture Notes in Math. 577 (1977), Springer-Verlag.
  * [42] A. S. Mishchenko, Homotopy Invariants of non-simply connected Manifolds. III. Higher Signatures, Izv. Akad. Nauk SSSR, ser. mat. 35 (1971), 1316–1355.
  * [43] J. W. Morgan, D. P. Sullivan, The Transversality Characteristic Class and Linking Cycles in Surgery Theory, Annals of Math. 99 (1974), no. 3, 463 – 544.
  * [44] N. V. Panov, Universal Coefficient Formulas for Stable K-Theory, Mat. Zametki 11 (1972), 53 – 60 = Math. Notes 11 (1972), 36 – 40.
  * [45] W. L. Pardon, Intersection homology, Poincaré spaces and the characteristic variety theorem, Comment. Math. Helv. 65 (1990), 198 – 233.
  * [46] O. Randal-Williams, The Family Signature Theorem, Proceedings of the Royal Society of Edinburgh (Ranicki memorial issue), to appear.
  * [47] A. A. Ranicki, Algebraic L-Theory and Topological Manifolds, Cambridge Tracts in Math., vol. 102, Cambridge Univ. Press, 1992.
  * [48] A. A. Ranicki, The Algebraic Theory of Surgery,
https://www.maths.ed.ac.uk/~v1ranick/papers/ats.pdf (1978).

  * [49] A. Ranicki, The Algebraic Theory of Surgery II. Applications to Topology, Proc. London Math. Soc. 40 (1980), 193 – 283.
  * [50] A. A. Ranicki, The Total Surgery Obstruction, Proc. 1978 Aarhus Topology Conference, Lect. Notes in Math. 763, Springer, 1979, 275 – 316.
  * [51] J. Rosenberg, Analytic Novikov for Topologists, in Novikov Conjectures, Index Theorems and Rigidity, vol. 1, eds. S. C. Ferry, A. Ranicki, J. Rosenberg, London Math. Soc. Lecture note series 226, Cambridge University Press, 1995, 338 – 372.
  * [52] J. Rosenberg, S. Weinberger, The Signature Operator at 2, Topology 45 (2006), 47 – 63.
  * [53] Y. B. Rudyak, On Thom Spectra, Orientability, and Cobordism, 2nd printing, Springer Monographs in Math., Springer Verlag, 2008.
  * [54] J. Schürmann, Characteristic Classes of Mixed Hodge Modules, in: Topology of Stratified Spaces (Eds. Friedman, Hunsicker, Libgober, Maxim), MSRI publications vol. 58, Cambridge Univ. Press, 2011, 419 – 470.
  * [55] P. H. Siegel, Witt Spaces: A Geometric Cycle Theory for $KO$-Homology at Odd Primes, Amer. J. of Math. 105 (1983), 1067 – 1105.
  * [56] D. Sullivan, Geometric Topology: Localization, Periodicity, and Galois Symmetry, MIT notes, 1970.
  * [57] D. Sullivan, Singularities in Spaces, Proc. of Liverpool Singularities Symposium II, Lecture Notes in Math. 209, 196 – 206, Springer-Verlag, 1971.
  * [58] R. M. Switzer, Algebraic Topology — Homotopy and Homology, Grundlehren der math. Wissenschaften, vol. 212, Springer-Verlag, 1975.
  * [59] L. Taylor, B. Williams, Surgery Spaces: Formulae and Structure, Proceedings 1978 Waterloo Algebraic Topology Conference, Lecture Notes in Math. 741, Springer, 170 – 195 (1979).
  * [60] S. Weinberger, The Topological Classification of Stratified Spaces, Chicago Lectures in Math., The University of Chicago Press, 1994.