Vietoris–Begle mapping theorem

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The Vietoris–Begle mapping theorem is a result in the mathematical field of algebraic topology. It is named for Leopold Vietoris and Edward G. Begle. The statement of the theorem, below, is as formulated by Stephen Smale.


Let and be compact metric spaces, and let be surjective and continuous. Suppose that the fibers of are acyclic, so that

for all and all ,

with denoting the th reduced homology group. Then, the induced homomorphism

is an isomorphism for and a surjection for .