math.AT — Algebraic Topology
Axiomatic degree theory on topological manifolds
In this note we study extension of standard Euclidean degree in nonlinear analysis to the setting of general oriented topological manifolds. The axiomatization of Euclidean degree is adapted to this setting. Existence and uniqueness are proven. The existence does not rely on smooth approximations as in the Euclidean case -- this is irrelevant for manifolds which do not admit a smooth structure and could be ambiguous for manifolds admitting many smooth structures. Rather, we rely on a cohomological approach using only topological information. When compared to the Euclidean case, we think this approach makes the proofs simpler and more transparent. The standard notion of a degree of maps between closed oriented manifolds studied in algebraic topology is recovered as a special case. A generalization of the notion of winding number is developed and related to the notion of degree. The familiar composition formula of the Euclidean degree is also obtained in this context.