Vipul at 19:07, 2 April 2011

2011-04-02T19:07:51Z

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	==Statement==		==Statement==

	The [[connected sum of manifolds]] operation is not cancellative in any sense (up to homotopy, up to homeomorphism, up to diffeomorphism, etc.) Specifically, there is a natural number <math>n</math> such that we can find <math>n<math>-dimensional compact connected [[manifold]]s <math>A,B,C</math> such that <math>A \# B</math> and <math>A \# C</math> are homeomorphic (in fact, diffeomorphic if we put a differential structure) but <math>B</math> and <math>C</math> are not homeomorphic or even homotopy-equivalent.		The [[connected sum of manifolds]] operation is not cancellative in any sense (up to homotopy, up to homeomorphism, up to diffeomorphism, etc.) Specifically, there is a natural number <math>n</math> such that we can find <math>n</math>-dimensional compact connected [[manifold]]s <math>A,B,C</math> such that <math>A \# B</math> and <math>A \# C</math> are homeomorphic (in fact, diffeomorphic if we put a differential structure) but <math>B</math> and <math>C</math> are not homeomorphic or even homotopy-equivalent.

	==Proof==		==Proof==

Vipul: Created page with "==Statement== The connected sum of manifolds operation is not cancellative in any sense (up to homotopy, up to homeomorphism, up to diffeomorphism, etc.) Specifically, there..."

2011-04-02T19:07:07Z

Created page with "==Statement== The connected sum of manifolds operation is not cancellative in any sense (up to homotopy, up to homeomorphism, up to diffeomorphism, etc.) Specifically, there..."

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==Statement==

The [[connected sum of manifolds]] operation is not cancellative in any sense (up to homotopy, up to homeomorphism, up to diffeomorphism, etc.) Specifically, there is a natural number <math>n</math> such that we can find <math>n<math>-dimensional compact connected [[manifold]]s <math>A,B,C</math> such that <math>A \# B</math> and <math>A \# C</math> are homeomorphic (in fact, diffeomorphic if we put a differential structure) but <math>B</math> and <math>C</math> are not homeomorphic or even homotopy-equivalent.

==Proof==

===The case of <math>n = 2</math>===

{{further|[[Dyck's theorem]]}}

Here, we set <math>A</math> as the [[real projective plane]] <math>\mathbb{P}^2(\R)</math>, <math>B</math> as the [[Klein bottle]], and <math>C</math> as the [[2-torus]]. Both <math>A \# B</math> and <math>A \# C</math> are homeomorphic to what's called [[Dyck's surface]] (by a result called [[Dyck's theorem]]). However, the [[Klein bottle]] and the [[2-torus]] and not homeomorphic -- the former is non-orientable (and hence its second homology group vanishes) and the latter is orientable (and hence its second homology group is <math>\mathbb{Z}</math>).

Connected sum is not cancellative - Revision history

Vipul at 19:07, 2 April 2011

Vipul: Created page with "==Statement== The connected sum of manifolds operation is not cancellative in any sense (up to homotopy, up to homeomorphism, up to diffeomorphism, etc.) Specifically, there..."