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.

[edit] Theorem

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

for all and all ,

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

is an isomorphism for and a surjection for r = n.

[edit] References

• "Leopold Vietoris (1891–2002)", Notices of the American Mathematical Society, vol. 49, no. 10 (November 2002) by Heinrich Reitberger