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The signature of partially defined local coefficient systems. (English) Zbl 1165.32013

In [M. Banagl, S. E. Cappell and J. L. Shaneson, Math. Ann. 326, No. 3, 589–623 (2003; Zbl 1034.32021)] and [M. Banagl, Proc. Lond. Math. Soc., III. Ser. 92, No. 2, 428–470 (2006; Zbl 1089.57018)], formulas are provided for both the twisted signature \(\sigma(X; \mathbb S)\) and the twisted \(L\)-class \(L(X; \mathbb S)\) of a stratified space. Namely- we have
\[ \sigma(X; \mathbb S)=\varepsilon_{*}((\widetilde {\text{ch}}([\mathbb S_{K}])\cap L) \] and
\[ L(X; \mathbb S)= \widetilde {\text{ch}}([\mathbb S_{K}])\cap L). \]
In the former case, the result is applied for \(X\) a stratified normal Witt space and in the latter case for \(X\) a non-Witt spaces but with an extra hypothesis that it possess Lagrangian structures.
The present work contains two parts. The first part contains examples of spaces equipped with a local coefficient system defined over the top stratum, which does not extend to the entire space such that the characteristic class formula for the signature fails. These examples are \(4\)-dimensional closed oriented orbifolds \(X\) with finite singular set, endowed with an Hermitian local coefficient system on the top stratum, such that the link of every singularity is a lens space \(L_p(1,1)\) where \(p\geq 3\) is an integer. The result is:
Theorem 3.1 Let \(p\geq 3\) be an integer. There exists a \(4\)-dimensional, closed, orientable orbifold \(X\) with finite singular set \(\Sigma\), together with a Hermitian local coefficient system \(\mathbb S\) on \(X - \Sigma\), such that 6mm
(1)
The link of every singularity in \(\Sigma\) has finite fundamental group \(Z_p\).
(2)
The twisted signature \(\sigma(X; \mathbb S)\) does not satisfy an Atiyah-type multiplicative characteristic class formula:

\[ \sigma(X; \mathbb S)\neq \langle\widetilde {\text{ch}}[\mathbb S]_{K}, L(X)\rangle. \]
The second part considers codimension two embedding of manifolds, where the target is regarded as a stratified space with the bottom stratum the image of the embedding and the top stratum the complement. When the embedded manifold is a sphere, then various formulae that compute the twisted signature are provided, even in the case where the local system does not extend from the top stratum to the entire space. Formulae are presented for 6 different situations of embedding of \(S^n \hookrightarrow M^{n+2}\). To exemplify it is shown: Let \(f:S^2 \hookrightarrow S^4\) be a smooth \(2\)-knot and let \(\mathbb{S}\) be a Hermitian local coefficient system on \(S^4-\Sigma\) with structure group \(U(k,l)\). Then
\[ \sigma( S_f^4; \mathbb S)=-4\langle c_2(\mathbb S),[S^4]\rangle, \]
and \(\sigma (S_f^4;\mathbb S)\equiv 0\pmod 8\).

MSC:

32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
55N33 Intersection homology and cohomology in algebraic topology
57N25 Shapes (aspects of topological manifolds)
32Q40 Embedding theorems for complex manifolds
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References:

[1] DOI: 10.1017/S0305004100051872 · Zbl 0314.58016
[2] DOI: 10.1007/978-0-8176-4765-0
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