Difference between revisions of "Connected sum of two complex projective planes with same orientation"

From Topospaces
Jump to: navigation, search
(Definition)
(Algebraic topology)
 
Line 26: Line 26:
 
The cohomology ring is as follows:
 
The cohomology ring is as follows:
  
<math>H^*(M) = \mathbb{Z}[x,y]/(x^2 - y^2, x^3,y^3)</math>
+
<math>H^*(M) = \mathbb{Z}[x,y]/(x^2 - y^2, xy, x^3,y^3)</math>
  
 
===Homotopy groups===
 
===Homotopy groups===
  
 
The [[fundamental group]] is the trivial group.
 
The [[fundamental group]] is the trivial group.

Latest revision as of 16:40, 13 December 2016

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces


Definition

This topological space is defined as the connected sum of two copies of the complex projective plane \mathbb{P}^2(\mathbb{C}), where they are glued with the same orientation.

Related facts

Algebraic topology

Homology groups

The homology groups are as follows:

H_p(M) = \left\lbrace \begin{array}{rl}\mathbb{Z}, & \qquad p = 0,4 \\ \mathbb{Z} \oplus \mathbb{Z}, & \qquad p = 2 \\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.

Cohomology groups

The cohomology groups are as follows:

H^p(M) = \left\lbrace \begin{array}{rl}\mathbb{Z}, & \qquad p = 0,4 \\ \mathbb{Z} \oplus \mathbb{Z}, & \qquad p = 2 \\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.

The cohomology ring is as follows:

H^*(M) = \mathbb{Z}[x,y]/(x^2 - y^2, xy, x^3,y^3)

Homotopy groups

The fundamental group is the trivial group.