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The symmetric signature of a Witt space. (English) Zbl 1276.55010

The symmetric signature of a closed oriented \(n\)-dimensional manifold \(M\) is an element \(\sigma^\ast (M)\) in the symmetric \(L\)-group \(L^n (\mathbb{Z}\pi_1 (M))\). It was originally introduced by A. S. Miščenko [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1316–1355 (1971; Zbl 0248.57014)]. In fact, any \(n\)-dimensional Poincaré space \(X\) has a symmetric signature \(\sigma^\ast (X)\in L^n (\mathbb{Z} \pi_1 (X))\), which is homotopy invariant; see also [A. A. Ranicki, Algebraic L-theory and topological manifolds. Cambridge Tracts in Mathematics. 102. Cambridge: Cambridge University Press. (1992; Zbl 0767.57002)]. Naturally, it is desirable to define the symmetric signature even on singular spaces. A Witt space is a stratified pseudomanifold \(X\) such that the rational middle-dimensional, middle-perversity intersection homology of all links of odd-codimensional strata vanishes. Since this condition implies the self-duality of the intersection chain sheaf, a Witt space possesses at least a signature \(\sigma (X)\in \mathbb{Z}\), which makes such spaces a natural choice of singular spaces for which to construct a symmetric signature. The symmetric signature of a Witt space appeared first in the work of [S. Cappell, J. Shaneson and S. Weinberger, C. R. Acad. Sci., Paris, Sér. I 313, No. 5, 293–295 (1991; Zbl 0742.57023)].
For integral Poincaré duality pseudomanifolds \(X\), [T. Eppelmann, Signature Homology and Symmetric L-Theory. Heidelberg: Univ. Heidelberg, Naturwissenschaftlich-Mathematische Gesamtfakultät (Dissertation). (2007; Zbl 1137.57305)] sketches a far-reaching method on the level of spectra, which, as is observed in [M. Banagl, Mathematical Sciences Research Institute Publications 58, 223–249 (2011; Zbl 1236.55001)], yields an \(\mathbb{L}^\bullet\)-homology orientation class \([X]_{\mathbb{L}}\in H_n (X; \mathbb{L}^\bullet)\), where \(\mathbb{L}^\bullet\) is the symmetric L-spectrum. Consequently, by composing with the assemby map \(A:H_n (X; \mathbb{L}^\bullet) \to L^n (\mathbb{Z} \pi_1 (X))\), one obtains a symmetric signature \(\sigma^\ast (X)=A([X]_{\mathbb{L}})\). In fact, according to Section 12.3 of [S. Weinberger, The topological classification of stratified spaces, Chicago, IL: University of Chicago Press. (1994; Zbl 0826.57001)], one can associate to any self-dual sheaf complex a characteristic class in L-homology.
The paper under review gives a more direct description of the symmetric signature of a Witt space by adapting Miščenko’s techniques to intersection homology. This involves verifying allowability conditions for the diagonal map and cross product on the intersection chain complex. This approach also yields stratified homotopy invariance of the symmetric signature.

MSC:

55N33 Intersection homology and cohomology in algebraic topology
57R67 Surgery obstructions, Wall groups
57N80 Stratifications in topological manifolds
19J25 Surgery obstructions (\(K\)-theoretic aspects)
19G24 \(L\)-theory of group rings
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