Difference between revisions of "Homotopy type of connected sum depends on choice of gluing map"

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(Created page with "==Statement== It is possible to find an example of compact connected orientable manifolds <math>M_1</math> and <math>M_2</math> such that the homotopy type of the [[connecte...")
 
(Proof)
 
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==Proof==
 
==Proof==
  
To construct an example, we need to find a case where both <math>M_1</math> and <math>M_2</math> are orientable but neither of them has an orientation-reversing self-homeomorphism. One simple choice, by Fact (1), is to set both <math>M_1</math> and <math>M_2</math> as isomorphic to the [[complex projective plane]] <math>\mathbb{P}^2(\mathbb{C})</math>.  
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To construct an example, we need to find a case where both <math>M_1</math> and <math>M_2</math> are orientable but neither of them has an orientation-reversing self-homeomorphism. One simple choice, by Fact (1), is to set both <math>M_1</math> and <math>M_2</math> as homeomorphic to the [[complex projective plane]] <math>\mathbb{P}^2(\mathbb{C})</math> which has real dimension 4.  
  
 
There are two possible connected sums:
 
There are two possible connected sums:
  
* [[Connected sum of two complex projective planes with same orientation]]: This has cohomology ring isomorphic to <math>\mathbb{Z}[x,y]/(x^2 - y^2,x^3,y^3)</math>, where <math>x,y</math> are additive generators of the free abelian group <math>H^2</math> and <math>x^2 = y^2</math> is the additive generator for <math>H^4</math>.
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* [[Connected sum of two complex projective planes with same orientation]]: This has cohomology ring isomorphic to <math>\mathbb{Z}[x,y]/(x^2 - y^2,xy,x^3,y^3)</math>, where <math>x,y</math> are additive generators of the free abelian group <math>H^2</math> and <math>x^2 = y^2</math> is the additive generator for <math>H^4</math>.
* [[Connected sum of two complex projective planes with opposite orientation]]: This has cohomology ring isomorphic to <math>\mathbb{Z}[x,y]/(x^2 + y^2,x^3,y^3)</math>, where <math>x,y</math> are additive generators of the free abelian group <math>H^2</math> and <math>x^2 = -y^2</math> is the additive generator for <math>H^4</math>.
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* [[Connected sum of two complex projective planes with opposite orientation]]: This has cohomology ring isomorphic to <math>\mathbb{Z}[x,y]/(x^2 + y^2,xy,x^3,y^3)</math>, where <math>x,y</math> are additive generators of the free abelian group <math>H^2</math> and <math>x^2 = -y^2</math> is the additive generator for <math>H^4</math>.

Latest revision as of 18:18, 15 December 2016

Statement

It is possible to find an example of compact connected orientable manifolds M_1 and M_2 such that the homotopy type of the connected sum M_1 \# M_2 is not well defined, i.e., we can get connected sums of different homotopy types depending on the choice of the gluing map.

Facts used

  1. Complex projective space has orientation-reversing self-homeomorphism iff it has odd complex dimension

Proof

To construct an example, we need to find a case where both M_1 and M_2 are orientable but neither of them has an orientation-reversing self-homeomorphism. One simple choice, by Fact (1), is to set both M_1 and M_2 as homeomorphic to the complex projective plane \mathbb{P}^2(\mathbb{C}) which has real dimension 4.

There are two possible connected sums: