Rank of the fundamental group of a component of a function space

Details Rank of the fundamental group of a component of a function space Rank of the fundamental group of a component of a function space

2005-02-15

arxiv.org/abs/math/0502311v1

Mathematics

Algebraic Topology

Scientific paper

We compute the rank of the fundamental group of an arbitrary connected component of the space map(X, Y) for X and Y nilpotent CW complexes with X finite. For the general component corresponding to a homotopy class f : X --> Y, we give a formula directly computable from the Sullivan model for f. For the component of the constant map, our formula expresses the rank in terms of classical invariants of X and Y. Among other applications and calculations, we obtain the following: Let G be a compact simple Lie group with maximal torus T^n. Then the fundamental group of map(S^2, G/T^n; f) is a finite group if and only if f: S^2 --> G/T^n is essential.

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