Cobordism groups of simple branched coverings

Abstract

We consider branched coverings which are simple in the sense that any point of the target has at most one singular preimage. The cobordism classes of k-fold simple branched coverings between n-manifolds form an abelian group $${{\rm Cob}^1(n, k)}$$ . Moreover, $${{\rm Cob}^1(*, k) = \bigoplus_{n = 0}^{\infty}{\rm Cob}^1(n, k)}$$ is a module over $${\Omega^{SO}_{*}}$$ . We construct a universal k-fold simple branched covering, and use it to compute this module rationally. As a corollary, we determine the rank of the groups $${{\rm Cob}^1(n, k)}$$ . In the case n = 2 we compute the group $${{\rm Cob}^1(2, k)}$$ , give a complete set of invariants and construct generators.