## Obstruction theory and minimal number of coincidences for maps from a complex into a manifold.(English)Zbl 1045.55001

Let $$f,g: X\to Y$$ be maps and denote by $$MC[f, g]$$ the minimum number of coincidences, that is solutions to $$f'(x)= g'(x)$$, for all maps $$f'$$ homotopic to $$f$$ and $$g'$$ homotopic to $$g$$. In 1955, Schirmer proved that if $$X$$ and $$Y$$ are compact orientable manifolds of the same dimension $$n\geq3$$, then the Nielsen coincidence number $$N(f, g)$$, a natural generalization of the Nielsen fixed point number, calculates $$MC[f, g]$$. Subsequent work of many researchers has sought to extend Schirmer’s result to more general classes of spaces. In this paper, $$Y$$ is a compact but not necessarily orientable $$n$$-manifold and $$X$$ is an $$n$$-dimensional finite simplicial complex. The authors demonstrate that the coincidence problem can be restricted to a subcomplex of $$X$$ in which all maximal simplices are $$n$$-dimensional and there are no free $$(n- 1)$$-faces. In this setting, a homotopy invariant $$NO(f,g;X)$$ is defined that depends on the primary obstruction to deforming $$f$$ and $$g$$ to be coincidence free. If $$n\geq 3$$ and $$X$$ satisfies some mild geometric conditions, then $$NO(f,g;X)= MC[f,g]$$. The authors use their theory to investigate $$W\{X, Y\}$$, the maximum of $$MC[f, g] -N(f, g)$$ over all homotopy classes of pairs of maps from $$X$$ to $$Y$$. They prove that, if $$Y$$ is simply connected, then $$W\{X, Y\}$$ is finite (though not in general zero). They show by example that $$W\{X,Y\}$$ can be infinite. They also consider the interesting case where $$X$$ is a finite union of closed $$n$$-manifolds.

### MSC:

 55M20 Fixed points and coincidences in algebraic topology 54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: