Relative and equivariant coincidence theory

Guo, Jianhan (1996) Relative and equivariant coincidence theory. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

In this thesis, we develop relative coincidence theory on the complement and equivariant coincidence theory. For two maps f and g from one pair of manifolds (X, A) to another (Y, B), a Nielsen number N(f, g; X- A) is introduced which serves as a homotopy invariant lower bound for the number of coincidence points of f and g on X - A. We provide a method for computing the Nielsen numbers N(f, g) and N(f, g; X- A) when gπ is onto and fπ(π₁(X)) ⊂ J(f). These results are also generalized to manifolds with boundary. -- To estimate the number of coincidence points for equivariant maps, some Nielsen type invariants are developed. These invariants are introduced for the general cases first, and then explored further for the special case, when the fixed point set of the action is nonempty. A method is provided to compute these numbers and give an estimate of the number of coincidence points of a pair of equivariant maps. Finally, minimality is discussed for both relative and equivariant cases and we prove in some cases that these numbers are attainable within the appropriate homotopy classes.

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