Abstract
This paper is concerned with the situation in which the topology of space (or space-time) changes to produce a new manifold that is cobordant with, but not necessarily of the same homotopy type as, the original manifold. The relevance to kink field theories is discussed. It is shown that whenever the concept of degree of mapping is applicable then the degree is conserved under the bordism relation. This has the consequence that certain (topological) fermions arising in general relativity are always conserved in number, even when changes in topology are permitted.
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