# 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

Get more specific information about real projective space | Get more computations of cohomology group

## Contents

- 1 Cohomology groups in piecewise form
- 1.1 Odd-dimensional projective space with coefficients in integers
- 1.2 Even-dimensional projective space with coefficients in integers
- 1.3 Odd-dimensional projective space with coefficients in an abelian group
- 1.4 Even-dimensional projective space with coefficients in an abelian group
- 1.5 Coefficients in a module over a 2-divisible ring
- 1.6 Coefficients in characteristic two

- 2 Cohomology groups in tabular form
- 3 Cohomology ring structure
- 4 Reality checks
- 5 Facts used
- 6 Proof using homology groups
- 7 Proof using cochain complex constructed from CW structure

## Cohomology groups in piecewise form

### Odd-dimensional projective space with coefficients in integers

### Even-dimensional projective space with coefficients in integers

### 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

General assertion | Verification in this case | See also ... |
---|---|---|

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 |

## Facts used

- Homology of real projective space
- Dual universal coefficients theorem
- CW structure of real projective space

## Proof using homology groups

### Case of odd dimension

*Fill this in later*

### Case of even dimension

*Fill this in later*

## Proof using cochain complex constructed from CW structure

*Fill this in later*