View source for Vietoris–Begle mapping theorem

{{short description|On the homology of continuous maps between compact metric spaces}}
The '''Vietoris–Begle mapping theorem''' is a result in the [[Mathematics|mathematical]] field of [[Algebraic topology|algebraic topology]]. It is named for [[Biography:Leopold Vietoris|Leopold Vietoris]] and Edward G. Begle. The statement of the theorem, below, is as formulated by [[Biography:Stephen Smale|Stephen Smale]].

==Theorem==
Let <math>X</math> and <math>Y</math> be [[Compact space|compact]] metric spaces, and let <math>f:X\to Y</math> be [[Surjective function|surjective]] and [[Continuous function|continuous]]. Suppose that the [[Image (mathematics)|fibers]] of <math>f</math> are acyclic, so that
:<math>\tilde H_r(f^{-1}(y)) = 0,</math> for all <math>0\leq r\leq n-1</math> and all <math>y\in Y</math>,
with <math>\tilde H_r</math> denoting the <math>r</math>th [[Reduced homology|reduced Vietoris homology]] [[Group (mathematics)|group]]. Then, the induced [[Homomorphism|homomorphism]]
:<math>f_*:\tilde H_r(X)\to\tilde H_r(Y)</math>
is an [[Isomorphism|isomorphism]] for <math>r\leq n-1</math> and a surjection for <math>r=n</math>.

Note that as stated the theorem doesn't hold for homology theories like [[Singular homology|singular homology]]. For example, Vietoris homology groups of the closed [[Topologist's sine curve|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==
*[http://www.ams.org/notices/200210/fea-vietoris.pdf "Leopold Vietoris (1891–2002)"], ''Notices of the American Mathematical Society'', vol. 49, no. 10 (November 2002) by Heinrich Reitberger

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[[Category:Theorems in algebraic topology]]



{{Sourceattribution|Vietoris–Begle mapping theorem}}