Lefschetz coincidence theory for maps between spaces of different dimensions by Saveliev

Lefschetz coincidence theory for maps between spaces of different dimensions by Peter Saveliev

Topology and Its Applications, 116 (2001) 1, 137-152, also a talk at the Joint Mathematics Meeting in January 2000.

This is further development of the study in my previous paper A Lefschetz-type coincidence theorem by Saveliev.

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 if the dimensions are different or one of the spaces is not a manifold.

For a given pair of maps $f,g:X\rightarrow Y$ from an arbitrary topological space (in particular, an $m$-manifold) to an $n$-manifold, we generalize the coincidence index and the Lefschetz coincidence number. The former is generalized by the coincidence homomorphism defined by R. Brooks and R. Brown. For the latter, we consider the Lefschetz homomorphism as a certain graded homomorphism of degree $(-n)$ that depends only on the homomorphisms generated by $f$ and $g$ on the homology groups. We prove that they coincide (the "graded Lefschetz coincidence theorem"). It follows that if the Lefschetz homomorphism is not identically 0 then there is an $x \in X$ such that $f(x)=g(x)$ (a coincidence).

The theorem contains previously known results concerning the following.

• (i) Coincidences for $n$-manifolds $X,Y$ with $f$ boundary-preserving.
• (ii) Fixed points of a multivalued map $G:Y\rightarrow Y$ with acyclic values; here $X=Graph(G)$, $f,g$ are the projections (in particular, consider the Eilenberg-Montgomery fixed point theorem).
• (iii) Fixed points of a parametrized map $F:T×Y\rightarrow Y$; here $g=F$, $f$ is the projection, and $Fix(F)=\{ (t,x):F(t,x)=x\} $.

I have addressed (i) and (ii) in the previous paper; here we concentrate on (iii) and significantly generalize some results of Geoghegan, Nicas and Oprea.

Full text: Lefschetz coincidence theory for maps between spaces of different dimensions (14 pages)