An alternative approach to homology (Q2707509)

The author proposes an alternative definition of homology theory associated to a multiplicative cohomology theory \(h^*\): for any space \(X\), the geometric homology \(h_*(X)\) is generated by all triples \((M,x,f)\) -- where \(M\) is a smooth compact manifold with \(h^*\)-structure, \(x\in h^*(M)\) is a cohomology class and \(f: M\to X\) is a continuous map -- modulo the smallest relation which contains the bordism relation for such triples and an equivalence relation which compares triples where the manifolds and cohomology classes have different dimensions. A version of this construction is given by the same author in [Manuscr. Math. 96, No. 1, 67-80 (1998; Zbl 0897.55004)]. In the present paper he constructs a generalized homology functor on the category of all pairs of topological spaces with values in graded abelian groups, which satisfies the Eilenberg-Steenrod axioms except the dimension axiom. The author also shows that this functor has Poincaré duality and that the homology theories \(h_*\) and \(sph_*\) are naturally equivalent (where \(sph_*\) is the unreduced homology theory associated to \(h^*\)).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00041].