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Stratified and unstratified bordism of pseudomanifolds. (English) Zbl 1331.57026

Piecewise linear pseudomanifolds have natural stratifications. Also, there is a natural stratification of pseudomanifolds. The paper under review compares the two bordism theories that correspond to a pseudomanifold, first simply considered as a PL-pseudomanifold and second as a stratified pseudomanifold. The main result of the paper is that the two bordism theories coincide for a large class of stratified pseudomanifolds. That class is the class of intrinsic weak stratified pseudomanifolds (IWP). The key technical result is that there is an explicit construction of a bordism between a stratified pseudomanifold \(X\) and the pseudomanifold \(X^*\) which is the same pseudomanifold with the intrinsic stratification. That is the main tool for the proof that the forgetful map \({\Omega}^{\mathcal{C}}_n \to {\Omega}^{|\mathcal{C}|}_n\) is an isomorphism. Here \(\mathcal{C}\) denotes the class of IWP pseudomanifolds, \({\Omega}^{\mathcal{C}}_n\) denotes the bordism classes of stratified pseudomanifolds and \({\Omega}^{|\mathcal{C}|}_n\) denotes the bordism classes of pseudomanifolds. Similar constructions are applied for the study of the homology theories that are associated to these bordism theories. For a class of stratified pseudomanifold singularities \(\mathcal{E}\), the forgetful map \({\Omega}^{\mathcal{E}}(\cdot ) \to {\Omega}^{|\mathcal{E}|}(\cdot )\) is an isomorphism of homology theories, where the symbols have the corresponding meaning as before. The definition of the homology theories follows the construction of E. Akin [Trans. Am. Math. Soc. 205, 342–359 (1975; Zbl 0369.57006)].
The class IWP contains, among others, the class of all compact classical \(\partial\)-stratified pseudomanifolds, the Witt spaces [P. H. Siegel, Am. J. Math. 105, 1067–1105 (1983; Zbl 0547.57019)], the IP spaces [W. L. Pardon, Comment. Math. Helv. 65, No. 2, 198–233 (1990; Zbl 0707.57017)], \(\bar{s}\)-duality spaces [M. Goresky and W. Pardon, Topology 28, No. 3, 325–367 (1989; Zbl 0696.57011)], LSF spaces [M. Goresky and W. Pardon, Topology 28, No. 3, 325–367 (1989; Zbl 0696.57011)].
Also, the author states certain open problems that are consequences of the methods used in the paper. In general, he asks if there are constructions that generalize the results to the topological pseudomanifolds. Also, he asks if his constructions can be generalized to larger classes of pseudomanifolds. In the same spirit, he is asking if there is an example that the result fails in general classes of pseudomanifolds.

MSC:

57N80 Stratifications in topological manifolds
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55N33 Intersection homology and cohomology in algebraic topology
57Q20 Cobordism in PL-topology
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References:

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