Homology of real projective space

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

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

Thus the key difference between even and odd dimensional projective spaces is that the top homology vanishes in even-dimensional projective spaces. This is related to the fact that even-dimensional projective space is non-orientable, while odd-dimensional projective space is orientable.

Odd-dimensional projective space with coefficients in an abelian group or module
$$H_p(\mathbb{P}^n(\R);M) = \left\lbrace \begin{array}{rl} M, & \qquad p=0,n\\ M/2M, & \qquad p \ \operatorname{odd}, 0 < p < n\\ T, & \qquad p \ \operatorname{even}, 0 < p < n \\ 0, & \qquad \operatorname{otherwise}\end{array}\right.$$

where $$T$$ is the 2-torsion submodule of $$M$$, i.e. the submodule comprising elements whose double is zero.

Even-dimensional projective space with coefficients in an abelian group or module $$M$$
$$H_p(\mathbb{P}^n(\R);M) = \left\lbrace \begin{array}{rl} M, & \qquad p=0\\ M/2M, &\qquad p \ \operatorname{odd}, 0 < p < n\\ T, & \qquad p \ \operatorname{even}, 0 < p \le n \\ 0, & \qquad \operatorname{otherwise}\end{array}\right.$$

where $$T$$ is the 2-torsion submodule of $$M$$, i.e. the submodule comprising elements whose double is zero.

Coefficients in a 2-divisible ring
If we consider the homology 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.

Homology groups with integer coefficients in tabular form
We illustrate how the homology groups work for small values of $$n$$. Note that for $$p > n$$, all homology groups $$H_p$$ are zero, so we omit those cells for visual ease.

Related invariants
These are all invariants that can be computed in terms of the homology groups.

Facts used

 * 1) uses::CW structure of real projective space

Explication of chain complex
The proof follows from fact (1). By fact (1), we note that the homology of real projective space $$\mathbb{P}^n(\R)$$ is the same as the homology of the following chain complex, obtained as its cellular chain complex:


 * For $$n$$ even:

$$0 \to 0 \to \dots \to 0 \to \mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z} \to \dots \stackrel{\cdot 2}{\to} \mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z}$$

where the largest nonzero chain group is the $$n^{th}$$ chain group.


 * For $$n$$ odd:

$$0 \to 0 \to \dots \to 0 \to \mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \to \dots \stackrel{\cdot 2}{\to} \mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z}$$

Note that the multiplication maps alternate between multiplication by two and multiplication by zero. In particular, in both cases, the map is multiplication by two if going down from an even to an odd index and multiplication by zero if going down from an odd to an even index. The key difference between the odd and even case is whether we start with a multiplication by two map or a multiplication by zero map.

Homology computation over an abelian group or module $$M$$
The chain complex remains the same, but each $$\mathbb{Z}$$ is replaced by $$M$$.

Denote by $$T$$ the 2-torsion submodule of $$M$$ and by $$M/2M$$ the quotient of $$M$$ by the submodule $$2M$$ comprising the doubles of elements.