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Removing coincidences of maps between manifolds of different dimensions by Saveliev

Removing coincidences of maps between manifolds of different dimensions by Peter Saveliev

Topological Methods in Nonlinear Analysis, 22 (2003) 1, 105-114, also a talk at the conference Topological Methods in Nonlinear Analysis, Będlewo, Poland, 18-23 June, 2001.

This is further development of the study in my previous paper Lefschetz coincidence theory for maps between spaces of different dimensions by Saveliev applied to manifolds. Consider the Coincidence problem: "If $X$ and $Y$ are a manifolds of dimension $n+m$ and $n$ respectively and $f,g:X\rightarrow X$ are maps, what can be said about the set $Coin(f,g)$ of coincidences, i.e., the set of $x \in X$ such that $f(x)=g(x)$?" While the coincidence theory for codimension $m=0$ is well developed, very little is known about the case $m>0$. We have defined the Lefschetz homomorphism and the coincidence homomorphism, where the Lefschetz homomorphism is a certain graded homomorphism on the homology module of $Y$ of degree $(-n)$, and prove that they coincide. It follows that if the Lefschetz homomorphism is not identically 0 then $Coin(f,g)$ is nonempty.

Here we consider the converse of this theorem for the cohomology index. If the index is identically 0, does it mean that there are no coincidences? For $m=0$, the answer is yes under very mild restrictions. Under certain conditions, the answer is yes for $m>0$ as well. The coincidence index is the only obstruction to the removability for maps with fibers either acyclic or homeomorphic to spheres of certain dimensions. We also address the normalization property of the index and coincidence-producing maps. The results partially complement the work on codimenison 1 by Fuller, Jezierski, Dimovski and Geoghegan.

Full text: Removing coincidences of maps between manifolds of different dimensions (7 pages)