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Topological Bordism of Singular Spaces and an Application to Stratified L-Classes. (arXiv:2206.06329v3 [math.AT] UPDATED)

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A generalized-homology bordism-theory is constructed, such that for certain
manifold homotopy stratified sets (MHSS; Quinn-spaces) homeomorphism-invariant
geometric fundamental-classes exist. The construction combines three ideas:
Firstly, instead of restricting geometric cycles by conditions on links only, a
more flexible framework is built directly via geometric properties, secondly,
controlled topology methods are used to give an accessible link-based criterion
to detect suitable cycles and thirdly, a geometric argument is used to show,
that these classes of cycles are suitable to study the transition to intrinsic
stratifications. As an application, we give a construction of topologically
(homeomorphism) invariant (homological) L-classes on MHSS Witt-spaces
satisfying conditions on Whitehead-groups of links and the dimensional spacing
of meeting strata. These L-classes agree, whenever those spaces are
additionally pl-pseudomanifolds, with the Goresky-MacPherson L-classes.

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