English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

A system of axioms for the Stiefel-Whitney classes of certain type of singular spaces is established. The main examples of these singular spaces are Euler manifolds mod 2 and homology manifolds mod 2. As a consequence, it is shown that on homology manifolds mod 2 the generalized Stiefel conjecture holds. The purpose of this paper is to set up a system of axioms which describe in a unique way the Stiefel-Whitney (S.W.) homology classes of some class of singular spaces, called in this paper allowable class. The main examples include Euler manifolds mod 2 and homology manifolds mod 2. This axiomatic characterization of S.W. homology classes then gives as corollaries affirmative answers to generalized Stiefel conjectures. The classical Stiefel conjecture says that on a smooth manifold Mn, the Poincare dual of the ith cohomology S.W. class is represented by the cycle mod 2 which is equal to the sum (mod 2) of all (n i)-simplices in the first barycentric subdivision of some triangulation of M [21]. In the generalized Stiefel conjecture we deal with Steenrod squares of Wu classes instead of ordinary cohomology S.W. classes. The question about a possible axiomatic description of S.W. classes for Euler manifolds or for more general spaces was posed by Blanton and Schweitzer [4]. Our corollary on homology manifolds mod 2 was also obtained by Taylor [19], but from a completely different viewpoint. The main tools in proving that our axioms determine a unique class are block bundle transversality and a description of cohomology classes as morphisms on bordism groups. These techniques then provide the "transversality classes" T(99) of an embedding 9p of our singular space X into the interior of a PL manifold M. T(99) lies in H*(M, aM; Z2) and determines characteristics (mod 2) of transversal intersections of X with singular manifolds in M. These ideas are based on the work of Latour [13]. In ?1 we give some basic facts about Euler manifolds mod 2 and their S.W. homology classes, which then motivates ??2 and 3 where we introduce allowable classes of spaces and introduce axioms for their S.W. classes. In ?4 we prove that the axioms determine unique classes and derive the above-mentioned consequences. Received by the editors July 10, 1980 and, in revised form, April 26, 1982. 1980 Mathematics Subject Classification. Primary 57P05, 55N40, 55R40; Secondary 55R60, 57R20.

Axioms for Stiefel-Whitney homology classes of some singular spaces