A so-called special generic map is by definition a map of smooth manifolds all of whose singularities are definite fold points. It is in general an open problem posed by Saeki in 1993 to determine the set of integers p for which a given homotopy sphere admits a special generic map into R p .By means of the technique of Stein factorization we introduce and study certain special generic maps of homotopy spheres into Euclidean spaces called standard. Modifying a construction due to Weiss, we show that standard special generic maps give naturally rise to a filtration of the group of homotopy spheres by subgroups that is strongly related to the Gromoll filtration. Finally, we apply our result to concrete homotopy spheres, which particularly answers Saeki's problem for the Milnor 7-sphere.

No abstract

By a theorem of Banagl-Chriestenson, intersection spaces of depth one pseudomanifolds exhibit generalized Poincaré duality of Betti numbers, provided that certain characteristic classes of the link bundles vanish. In this paper, we show that the middle-perversity intersection space of a depth one Witt space can be completed to a rational Poincaré duality space by means of a single cell attachment, provided that a certain rational Hurewicz homomorphism associated to the link bundles is surjective. Our approach continues previous work of Klimczak covering the case of isolated singularities with simply connected links. For every singular stratum, we show that our condition on the rational Hurewicz homomorphism implies that the Banagl-Chriestenson characteristic classes of the link bundle vanish. Moreover, using Sullivan minimal models, we show that the converse implication holds at least in the case that twice the dimension of the singular stratum is bounded by the dimension of the link. As an application, we compare the signature of our rational Poincaré duality space to the Goresky-MacPherson intersection homology signature of the given Witt space. We discuss our results for a class of Witt spaces having circles as their singular strata.

No abstract

No abstract