Vietoris–Begle mapping theorem

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.

Theorem

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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 Vietoris homology group. Then, the induced homomorphism

 

is an isomorphism for   and a surjection for  .

Note that as stated the theorem doesn't hold for homology theories like singular homology. For example, Vietoris homology groups of the closed topologist's sine curve and of a segment are isomorphic (since the first projects onto the second with acyclic fibers). But the singular homology differs, since the segment is path connected and the topologist's sine curve is not.

References

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