Proprietés caractéristiques du cap-produit à coefficients locaux. (Characteristic properties of cap-product with local coefficients) (Q1067647)

This paper is devoted to characterizing the cap product with local coefficients, \(H_ i(X;M)\times H^ k(X;N)\to H_{i-k}(X;M\otimes N),\) in terms of its behavior for \(i=k=0\), for maps \(X\to Y\) and for short exact sequences of coefficients M'\(\to M\to M''\), N'\(\to N\to N''\). In fact, naturality under maps \(X\to Y\) is only used to reduce the general situation, via the Kan-Thurston theorem, to the case \(X=K(\pi,1)\). As the authors note, this reduction to group (co)-homology provides a formal analogy between the characterizing properties of the cap product and those of the cup product for the cohomology of finite groups [see \textit{E. Weiss}, Cohomology of groups (1969; Zbl 0192.342)]. It is interesting to note that the approach described here does not give a characterization of the cap product with ordinary coefficients. The authors mention this and remark that furthermore, in opposition to the situation of the cup product for finite group cohomology, properties analogous to those enunciated do not seem to characterize the cup product for spaces.