Timeline for Is the top Stiefel-Whitney number of a topological manifold the Euler characteristic mod two?

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Sep 11, 2018 at 10:49	comment	added	Mark Grant		Topological manifolds have tangent microbundles, and Thom classes which live in $H^*(M\times M, M\times M-\Delta)$. See this note, for example: ams.org/journals/bull/1966-72-03/S0002-9904-1966-11537-9/… or Ch 14 of Switzer's book.
Sep 10, 2018 at 20:56	comment	added	mme		It sounds like even basic questions are too hard for me :) The Thom class comes from the tangent bundle, which at first blush sounds like we're out of luck. But topological manifolds are Poincare Duality spaces and so they have a Spivak normal fibration as a weak replacement. Maybe one can define Euler classes using that, but it is beyond my pay grade.
Sep 10, 2018 at 20:47	comment	added	Bombyx mori		@MikeMiller: I actually have something very basic to ask: If I recall correctly, Milnor-Stasheff defined the Euler class via the Thom class. Is this still doable for topological manifolds? If it is, what is the difficulty to extend the classical proof to this case?
Sep 10, 2018 at 20:43	comment	added	mme		It seems this is the same argument, with the clever reduction that $v_n = 0$ implies all vectors square to zero, and hence $H^k$ is a symplectic vector space. The proof I give above is basically the usual proof that a symplectic vector space is equivalent to a standard one. In any case very nice preprint.
Sep 10, 2018 at 20:38	history	edited	Bombyx mori	CC BY-SA 4.0	added 57 characters in body
Sep 10, 2018 at 20:33	history	answered	Bombyx mori	CC BY-SA 4.0