I want to calculate all the homology groups of $\mathbb{R}P^\infty = \bigcup\limits_{i=o}^\infty \mathbb{R}P^n$. The tools I have are Mayer-Vietoris sequence, cofibre long exact sequences (though I don't know a cofibration involving this space), and cellular homology.
First, I'm wondering how to interpret this union. Does it involve disjoint copies of the real projective plane in each dimension? Is it like a wedge sum at the basepoint? In particular, is $\mathbb{R}P^\infty$ path connected?
Second, I'm trying to calculate homology using the cellular chain complex. However, since $\mathbb{R}P^n$ has one cell in every dimension, wouldn't $\mathbb{R}P^\infty$ have infinitely many cells in every dimension? Is it still true that every other map $C_i(\mathbb{R}P^\infty)\rightarrow C_{i-1}(\mathbb{R}P^\infty)$ is the zero map or multiplication by two (like in cellular homology calculations of $\mathbb{R}P^n$ for finite $n$)? Why or why not?
Any help is appreciated!
Best Answer
The union is not a wedge, but a sequence of spaces $\mathbb{RP}^1\subset \mathbb{RP}^2\subset \dots$ where each space is a CW subcomplex of the next. In particular, $\mathbb{RP}^n$ is the $n$-skeleton of $\mathbb{RP}^\infty$.
It seems like you already understand the cellular chain complexes of each $\mathbb{RP}^n$, and the cellular chain complex of $\mathbb{RP}^\infty$ can be inferred from them: for each $n$ the inclusion $\mathbb{RP}^n \hookrightarrow \mathbb{RP}^\infty$ induces isomorphisms $C_k(\mathbb{RP}^n) \cong C_k(\mathbb{RP}^\infty)$ for each $k \leq n$, and the boundary operators $\partial_k\colon C_k(\mathbb{RP}^\infty)\to C_{k-1}(\mathbb{RP}^\infty)$ for $k\leq n$ agree with those of $\mathbb{RP}^n$ by naturality.
In general if $X$ is a CW complex and $X^{(n)}$ is its $n$-skeleton, then the inclusion $X^{(n)}\hookrightarrow X$ induces isomorphisms $H_k(X^{(n)}) \cong H_k(X)$ for all $k<n$ (exercise: prove this). In particular, if you want to know $H_k(\mathbb{RP}^\infty)$ you could choose an $n>k$ and compute $H_k(\mathbb{RP}^n)$.