Cohomology of real projective space

Odd-dimensional projective space with coefficients in integers
$$H^p(\mathbb{P}^n(\R); \mathbb{Z}) = \left\lbrace \begin{array}{rl} \Z, & \qquad p=0,n\\ \Z/2\Z, &\qquad p \ \operatorname{even}, 0 < p < n\\ 0, & \qquad \operatorname{otherwise}\end{array}\right.$$

Even-dimensional projective space with coefficients in integers
$$H^p(\mathbb{P}^n(\R); \mathbb{Z}) = \left\lbrace \begin{array}{rl} \Z, & \qquad p=0\\ \Z/2\Z, & \qquad p \ \operatorname{even}, 0 < p \le n\\ 0, & \qquad \operatorname{otherwise}\end{array}\right.$$

Odd-dimensional projective space with coefficients in an abelian group
For an abelian group $$M$$, the cohomology is given by:

$$H^p(\mathbb{P}^n(\R); M) = \left\lbrace \begin{array}{rl} M, & \qquad p=0,n\\ M/2M, &\qquad p \ \operatorname{even}, 0 < p < n\\ T, & \qquad p \ \operatorname{odd}, 0 < p < n \\ 0, & \qquad \operatorname{otherwise}\end{array}\right.$$

Here, $$T$$ 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 $$M$$, the cohomology is given by:

$$H^p(\mathbb{P}^n(\R); M) = \left\lbrace \begin{array}{rl} M, & \qquad p=0\\ M/2M, & \qquad p \ \operatorname{even}, 0 < p \le n\\ T, & \qquad p \ \operatorname{odd}, 0 < p < n \\ 0, & \qquad \operatorname{otherwise}\end{array}\right.$$

Here, $$T$$ 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 $$M$$ over a ring $$R$$ where 2 is invertible, then we have:

$$H^p(\mathbb{P}^n(\R);M) := \left\lbrace\begin{array}{rl} M, & p = 0 \\ M, & p = n, n \ \operatorname{odd}\\ 0, & p = n, n \ \operatorname{even}\\ 0, & p \ne 0,n \\\end{array}\right.$$

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 $$M$$ is an elementary abelian 2-group, i.e., a group in which the double of every element is zero. Then, $$2M = 0$$ (so $$M/2M \cong M$$) and $$T = M$$, and we get:

$$H^p(\mathbb{P}^n(\R);M) = \left\lbrace\begin{array}{rl}M, & \qquad 0 \le p \le n \\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.$$

This in particular applies to the case that $$M$$ is the group $$\mathbb{Z}/2\mathbb{Z}$$, i.e., when we are taking coefficients in the field of two elements.

Coefficients in integers
We illustrate how the cohomology groups work for small values of $$n$$. Note that for $$p > n$$, all cohomology groups $$H^p$$ are zero, so we omit those cells for visual ease.

Coefficients in an abelian group
We let the abelian group be $$M$$. Denote by $$T$$ the 2-torsion of $$M$$ and by $$2M$$ the submodule comprising doubles of elements.

Coefficients in a module over a 2-divisible ring
Suppose $$M$$ has the structure of a module over a unital ring $$R$$ where 2 is invertible. Then, in particular, we know that $$2M = M$$ and $$T = 0$$. Thus, both $$M/2M$$ and $$T$$ are equal to $$0$$. We get:

Coefficients in characteristic two
Suppose $$M$$ is an elementary abelian 2-group, i.e., a group in which the double of every element is zero. Then, $$2M = 0$$ (so $$M/2M \cong M$$) and $$T = M$$, and we get:

Over the integers for even $$n$$
The cohomology ring $$H^*(\mathbb{R}\mathbb{P}^n;\mathbb{Z})$$ is the ring $$\mathbb{Z}[x]/\langle 2x, x^{(n/2)+1} \rangle$$, where $$x$$ is the unique non-identity element in $$H^2(\mathbb{R}\mathbb{P}^n;\mathbb{Z})$$. $$x^r$$ in turn is the unique non-identity element in $$H^{2r}(\mathbb{R}\mathbb{P}^n;\mathbb{Z})$$ for $$1 \le r \le n/2$$. The coefficients ring (i.e., the constant terms) is $$H^0$$.

Note that that is almost the same as the ring $$\mathbb{Z}/2\mathbb{Z}[x]/\langle x^{(n/2)+1} \rangle$$, with the only difference being that for the constant terms, we are allowed to use the ring $$\mathbb{Z}$$ rather than the quotient ring $$\mathbb{Z}/2\mathbb{Z}$$.

Over the integers for odd $$n$$
The cohomology ring $$H^*(\mathbb{R}\mathbb{P}^n;\mathbb{Z})$$ is the ring $$\mathbb{Z}[x,y]/\langle 2x, x^{(n+1)/2}, xy, y^2\rangle$$ where $$x$$ is the unique non-identity element in $$H^2(\mathbb{R}\mathbb{P}^n;\mathbb{Z})$$ and $$y$$ is a generator of $$H^n(\mathbb{R}\mathbb{P}^n;\mathbb{Z})$$. $$x^r$$ in turn is the unique non-identity element in $$H^{2r}(\mathbb{R}\mathbb{P}^n;\mathbb{Z})$$ for $$1 \le r \le (n-1)/2$$. The coefficients ring (i.e., the constant terms) is $$H^0$$.

Note that that is almost the same as the ring $$\mathbb{Z}/2\mathbb{Z}[x,y]/\langle x^{(n/2)+1},y^2,xy \rangle$$, with the only difference being that for the constant terms, we are allowed to use the ring $$\mathbb{Z}$$ rather than the quotient ring $$\mathbb{Z}/2\mathbb{Z}$$.

Facts used

 * 1) uses::Homology of real projective space
 * 2) uses::Dual universal coefficients theorem
 * 3) uses::CW structure of real projective space