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
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
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Case of even dimension
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Proof using cochain complex constructed from CW structure
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