A Lefschetz-type coincidence theorem by Saveliev
A Lefschetz-type coincidence theorem by Peter Saveliev
Fundamenta Mathematicae, 162 (1999) 1-2, 65-89, also a talk at the Joint Mathematics Meeting in January 1999. Reviews: MR 2000j:55005, ZM 934.55003.
Consider the Coincidence Problem: "If $X$ and $Y$ are topological spaces and $f,g:X\rightarrow X$ are maps, what can be said about the set $Coin(f,g)$ of $x \in X$ such that $f(x)=g(x)$?" (cf. the Fixed Point Problem). While the coincidence theory of maps between manifolds of the same dimension is well developed, very little is known about the case when the dimensions are different or one of the spaces is not a manifold. In this paper a Lefschetz-type coincidence theorem for two maps $f,g:X\rightarrow Y$ from an arbitrary topological space $X$ to a manifold $Y$ is given: $$I(f,g)=L(f,g),$$ i.e., the coincidence index is equal to the Lefschetz number. It follows that if $L(f,g)$ is not equal to zero then there is an $x \in X$ such that $f(x)=g(x)$ (i.e., $Coin(f,g)$ is nonempty). In particular, the theorem contains some well known coincidence results for these cases:
- $X,Y$ are $n$-manifolds and $f$ is boundary-preserving, and
- $Y$ is Euclidean and $f$ has acyclic fibers.
The latter includes the Eilenberg-Montgomery Fixed Point Theorem. We also provide several examples of how to use our results to detect coincidences under the circumstances that would not be covered by the classical theory. The simplest examples are:
- the map of the disk to the sphere by identification of the boundary to the point, and
- the projection of the torus to the circle.
The results have been generalized in the next paper.
Full text: A Lefschetz-type coincidence theorem (23 pages)