Is there anyway to define an Euler Characteristic for infinite CW complexes, like $\mathbb{R}P^{\infty}$?

Question

I ask this because of the following thing I have been thinking of:

We know $\mathbb{R}P^{\infty}$ can be thought of as an infinite CW complex with one cell in each dimension.

If you try to use the same formula as the normal Euler Characteristic, you see you have $1-1+1-1+1-....$

This really doesn't converge to a number, however there is that way in which you can view the sum of this series as being $1/2$.

What I am wondering is if there is any way to interpret the answer of $1/2$ in terms of the topological properties of the space $\mathbb{R}P^{\infty}$.

Thanks!

Answer 1

I'm not sure if this directly answer your question, but:

I don't believe you can really define an Euler characteristic. But one thing is actually interesting: the Poincaré series of the space. In general, for a graded vector space $V$, you define it as $L(t) = \sum \dim(L_n) t^n$. This is a formal power series. For a space $X$, you can use its cohomology with coefficients in $A$: $$L_X(t) = \sum \dim(H^n(X; A)) t^n$$

If $X$ is a finite CW-complex, then this is related to the Euler characteristic by $\chi(X) = L_X(-1)$.

The infinite dimensional real projective space $\mathbb{RP}^\infty$ is an Eilenberg–MacLane of type $K(\mathbb{Z/2Z}, 1)$. In his article Cohomologie modulo $2$ des complexes d'Eilenberg–MacLane (French), Serre described how the behavior of the Poincaré series (mod 2) of an Eilenberg–MacLane space can give you information about the space in question.

You should read it (I'm not about to sum up the whole article in this answer box), but there are very interesting results. In particular this theorem (Théorème 10) is proven through the use of the Poincaré series of Eilenberg-MacLane spaces, and more specifically their behavior at $1$:

Theorem: Let $X$ be a simply connected space, such that:

1. $H_i(X; \mathbb{Z})$ is a finitely generated abelian group for all $i > 0$;
2. $H_i(X; \mathbb{Z/2Z})$ is zero for big enough $i$;
3. $H_i(X; \mathbb{Z/2Z})$ is nonzero for at least one $i$.

Then there is an infinite number of integers $i$ such that $\pi_i(X)$ contains a subgroup isomorphic to either $\mathbb{Z}$ or $\mathbb{Z/Z}$.