Homotopy and homology groups of finite CW complex

If $X$ is a finite CW complex, then $H_n$ and $\pi_n$ are finitely generated for $n\geq 2$.

I think there could be two possible interpretation of finite CW complex. First one is $X$ is $n$-dimensional CW complex and second is $X$ has finitely many cells. Obviously first case is more general.

But if $X$ is just $n$-dimensional CW complex then I think infinitely many disjoint $S^2$ with usual CW structure shows $H_2$ is not finitely generated. For $\pi_n$ case, maybe not.

Now suppose $X$ has finitely many cells. For homology case, considering cellular homology, since subgroup of finitely generated free abelian group is finitely generated and quotient of finitely generated group is also finitely generated, $H_n$ is finitely generated for $n\geq 2$. For homotopy group case, I doubt that.

• If $X$ is $n$-dimensional CW complex (first case) then are $\pi_n$ finitely generated for $n\geq 2$?
• Does my argument on homology for second case (pure algebra argument) works?
• If $X$ has finitely many cells, are $\pi_n$ finitely generated for $n\geq 2$?

Although I think finite CW complex means $X$ has finitely many cells, I wonder the case for $n$-dimensional CW complex.

Solution 1:

First of all, "finite CW complex" always refers to a CW complex with only finitely many cells. A CW complex in which the dimensions of the cells are bounded is instead called "finite-dimensional".

You are correct that a finite CW complex has finitely generated homology by just looking at the cellular complex. (Or even more simply, you can prove it by induction on the number of cells in the complex using the long exact sequences in homology.) However, it is not true that the homotopy groups of a finite CW complex have to be finite. For instance, consider $X=S^1\vee S^2$. Its universal cover $Y$ has the homotopy type of an infinite wedge of copies of $S^2$ (the universal cover of $S^1$ is $\mathbb{R}$ and then you attach a copy of $S^2$ at each integer in $\mathbb{R}$). For $n\geq 2$, we have $\pi_n(X)\cong \pi_n(Y)$. By Hurewicz, $\pi_2(Y)\cong H_2(Y)\cong \mathbb{Z}^{\oplus\mathbb{N}}$ is not finitely generated.

If $X$ is a finite simply connected CW complex then $\pi_n(X)$ is finitely generated for all $n$. This can be proven using the Serre spectral sequence and a Whitehead tower for $X$. Specifically, you prove by induction using the Serre spectral sequence that each space in the Whitehead tower for $X$ has finitely generated homology in each degree, and then it follows by Hurewicz that each homotopy group of $X$ is finitely generated. (Slightly more generally, if $X$ is a finite CW complex such that $\pi_1(X)$ is finite, then $\pi_n(X)$ is still finitely generated for all $n$, since you can just pass to the universal cover which is still a finite CW complex since $\pi_1(X)$ is finite.)