Cobordism groups of simple branched coverings

2017/07/08 by Nagy, Csaba
#57M12 #57R45 #57R90 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1707.02450

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 Cob¹(n,k). Moreover, Cob¹(*,k) = \bigoplus_n=0^∞ Cob¹(n,k) is a module over Ω^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 Cob¹(n,k). In the case n = 2 we compute the group Cob¹(2,k), give a complete set of invariants and construct generators.

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