Cobordism Exact Sequences for Differential and Combinatorial Manifolds

TL;DR: In this article, a combinatorial analogue to Thom's notion of t-regularity is defined and some basic lemmas on submanifolds representing double coverings are shown to be valid for topological manifolds if only some corresponding version of Lemma 7 could be established in that case.

Abstract: Several exact sequences have been established recently which relate cobordism groups of various types [1], [3], [8]. These have all been established in the differential case. The main object of this paper is to establish the same results in the combinatorial case. We first define a combinatorial analogue to Thom's notion of t-regularity, and prove some basic lemmas on submanifolds representing double coverings (Part I). After this, we find that the same proofs are valid as in the differential case, so we treat the two cases together. This gives us an opportunity to collect together all the geometrical arguments used in the proofs of these theorems, which have hitherto been somewhat scattered in the literature. We establish the three exact sequences in Theorem 1; the proofs are so simple that one could axiomatise a category of manifolds in which they work. The proof would be valid for topological manifolds if only some corresponding version of Lemma 7 could be established in that case. We then give a proof, in the same order of ideas, that one of our maps is a derivation. The behaviour of the others with respect to multiplication is rather more complicated, and we hope to return to it in a subsequent paper. We shall assume known the definitions of combinatorial and of differential manifolds (which may have boundary). All manifolds occurring in this paper will be compact. A compact manifold without boundary is called closed. To avoid logical difficulties concerning sets, we may regard all manifolds as imbedded in a finite dimensional subspace of a given Hilbert space; however, it will be more convenient to state our constructions for abstract manifolds. We shall usually denote manifolds by the symbols: V, W,M,N.