Ju n 20 06 EULER HOMOLOGY

We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented cobordism ring N∗(X) of a topological space X. This homology theory Eh∗ has coefficients Z/2 in every nonnegative dimension. There exists a natural transformation N∗(X) → Eh∗(X) that for X = pt assigns to each smooth manifold its Euler characteristic mod 2. The homology theory is constructed using cobordism of stratifolds, which are singular objects defined below. An isomorphism Eh∗(X) ∼= H∗(X;Z/2) ⊗Z/2 Z/2[t] of graded N∗modules is shown for any CW-complex X. For discrete groups G, we also define an equivariant version of the homology theory Eh∗, generalizing the equivariant Euler characteristic.