Timeline for Cap product à la Poincaré

6 events

when toggle format	what		by	license	comment
Dec 18, 2014 at 13:32	comment	added	Danny Ruberman		I think you may be missing Liviu's point: the cap product is an algebraically defined operation; on the chain level it is the adjoint to the cup product. Its notation perhaps suggests intersections, but it is not defined in terms of intersecting anything. To interpret cap and cup products in terms of intersections requires being in a space with some kind of transversality properties, typically a manifold. To get integer-valued intersection numbers, as opposed to mod 2 intersections, requires some orientations.
Dec 17, 2014 at 20:43	comment	added	ARG		So if intersection is only counted mod 2, then there would be no problem whatsoever (except compactness)?
Dec 17, 2014 at 16:19	vote	accept	ARG
Dec 17, 2014 at 16:19	comment	added	Liviu Nicolaescu		The cap product $\cap: H^k\times H_m\to H_{m-k}$is defined for any (reasonably) behaved space. Only its intersection theoretic interpretation requires the space be oriented manifold.
Dec 17, 2014 at 15:08	history	edited	David E Speyer	CC BY-SA 3.0	edited body
Dec 17, 2014 at 13:43	history	answered	David E Speyer	CC BY-SA 3.0