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

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be compact metric spaces, and let ${\displaystyle f:X\to Y}$ be surjective and continuous. Suppose that the fibers of ${\displaystyle f}$ are acyclic, so that

${\displaystyle {\tilde {H}}_{r}(f^{-1}(y))=0,}$ for all ${\displaystyle 0\leq r\leq n-1}$ and all ${\displaystyle y\in Y}$,

with ${\displaystyle {\tilde {H}}_{r}}$ denoting the ${\displaystyle r}$th reduced homology group. Then, the induced homomorphism

${\displaystyle f_{*}:{\tilde {H}}_{r}(X)\to {\tilde {H}}_{r}(Y)}$

is an isomorphism for ${\displaystyle r\leq n-1}$ and a surjection for ${\displaystyle r=n}$.