# Cohomology of real projective space

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is cohomology group and the topological space/family is real projective space
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## Cohomology groups in piecewise form





### Odd-dimensional projective space with coefficients in an abelian group

For an abelian group , the cohomology is given by:



Here,  denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.

### Even-dimensional projective space with coefficients in an abelian group

For an abelian group , the cohomology is given by:



Here,  denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.

### Coefficients in a module over a 2-divisible ring

If we consider the cohomology with coefficients in a module  over a ring  where 2 is invertible, then we have:



In particular, these results are valid over the field of rational numbers or over any field of characteristic zero, or indeed any characteristic other than 2.

### Coefficients in characteristic two

Suppose  is an elementary abelian 2-group, i.e., a group in which the double of every element is zero. Then,  (so ) and , and we get:



This in particular applies to the case that  is the group , i.e., when we are taking coefficients in the field of two elements.

## Cohomology groups in tabular form

### Coefficients in integers

We illustrate how the cohomology groups work for small values of . Note that for , all cohomology groups  are zero, so we omit those cells for visual ease.

 Real projective space  Orientable?      
1 circle Yes  
2 real projective plane No  0 
3 RP^3 Yes  0  
4 RP^4 No  0  0 
5 RP^5 Yes  0  0  

### Coefficients in an abelian group

We let the abelian group be . Denote by  the 2-torsion of  and by  the submodule comprising doubles of elements.

 Real projective space  Orientable?      
1 circle Yes  
2 real projective plane No   
3 RP^3 Yes    
4 RP^4 No     
5 RP^5 Yes      

### Coefficients in a module over a 2-divisible ring

Suppose  has the structure of a module over a unital ring  where 2 is invertible. Then, in particular, we know that  and . Thus, both  and  are equal to . We get:

 Real projective space  Orientable?      
1 circle Yes  
2 real projective plane No  0 0
3 RP^3 Yes  0 0 
4 RP^4 No  0 0 0 0
5 RP^5 Yes  0 0 0 0 

### Coefficients in characteristic two

Suppose  is an elementary abelian 2-group, i.e., a group in which the double of every element is zero. Then,  (so ) and , and we get:

 Real projective space  Orientable?      
1 circle Yes  
2 real projective plane No   
3 RP^3 Yes    
4 RP^4 No     
5 RP^5 Yes      

## Cohomology ring structure

### Over the integers for even 

The cohomology ring  is the ring , where  is the unique non-identity element in .  in turn is the unique non-identity element in  for . The coefficients ring (i.e., the constant terms) is .

Note that that is almost the same as the ring , with the only difference being that for the constant terms, we are allowed to use the ring  rather than the quotient ring .

### Over the integers for odd 

The cohomology ring  is the ring  where  is the unique non-identity element in  and  is a generator of .  in turn is the unique non-identity element in  for . The coefficients ring (i.e., the constant terms) is .

Note that that is almost the same as the ring , with the only difference being that for the constant terms, we are allowed to use the ring  rather than the quotient ring .

## Reality checks

For any compact connected -dimensional manifold, the top cohomology group  is  if the space is orientable and is (?) (finite group?) otherwise.  odd: In this case, the space is obtained by taking the quotient of the orientable manifold  by the antipodal action, which is orientation-preserving (one way of seeing it is that is given by a scalar matrix of s in dimension , so has determinant 1). The quotient is thus also orientable. Indeed, for  odd, the top cohomology is .
 even: In this case, the space is obtained by taking the quotient of the orientable manifold  by the antipodal action, which is orientation-reversing (one way of seeing it is that is given by a scalar matrix of s in dimension , so has determinant -1). The quotient is thus non-orientable. Indeed, for  even, the top cohomology is .
?
For a compact connected orientable manifold of dimension , the Poincare duality theorem says that the homology group of dimension  is isomorphic to the cohomology group of dimension . Case  odd: As noted above, the manifold is orientable. The top and bottom homology and cohomology groups are isomorphic to . The even-dimensional cohomology groups and odd-dimensional homology groups are both isomorphic to . The odd-dimensional cohomology groups and even-dimensional homology groups are both zero groups. homology of real projective space

## Proof using homology groups

### Case of odd dimension

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### Case of even dimension

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## Proof using cochain complex constructed from CW structure

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