Mod Two Homology and Cohomology pp 127-200 | Cite as

# Products

## Abstract

So far, the reader may not have been impressed by the essential differences between homology and cohomology: the latter is dual to the former via the Kronecker pairing, so they are even isomorphic for spaces of finite homology type. However, cohomology is a definitely more powerful invariant than homology, thanks to its *cup product*, making \(H^*(-)\) a graded \({{\mathbb {Z}}}_2\)-algebra. Thus, the homotopy types of two spaces with isomorphic homology may sometimes be distinguished by the algebra-structure of their cohomology. Simple examples are provided by \({\mathbb R}P^{2}\) versus \(S^1\vee S^2\), or by the \(2\)-torus versus the Klein bottle.