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The algebraic topology book I am reading recently covered the following theorem named after Whitehead and corresponding direct consequence.

THEOREM. If X is a CW complex of dimension less than n and $e: Y \rightarrow Z$ is a n-equivalence then we have a induced bijection $e_{*} : [X,Y] \rightarrow [X,Z]$.

FOLLOW UP THEOREM. If $e$ above is a n-equivalence of CW complexes of dimension less than $n$ than we have that $e$ is a homotopy equivalence.

I understand the theorems but the author makes the following statement immediately after the theorems.

"If X is a finite CW complex, in the sense that it has finitely many cells, and if dim X > 1 and X is not contractible, then it is known that X has infinitely many non-zero homotopy groups."

I was wondering if anyone could shed some light on this.

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    $\begingroup$ This is not a question about the Whitehead theorem. $\endgroup$ Sep 7, 2014 at 15:41
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    $\begingroup$ This is false. The torus has exactly one non-trivial homotopy group $\pi_1(T^2)\cong \mathbb{Z}^2$, $\pi_k(T^2)=0,\forall k\geq 2$. $\endgroup$
    – Dan Rust
    Sep 7, 2014 at 15:59
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    $\begingroup$ It seems that the result is true for simply connected such spaces, and follows from a result of Serre in 1953 (according to this MO question by John Baez). $\endgroup$ Sep 7, 2014 at 16:07
  • $\begingroup$ Thanks for the Serre reference. I'll take a look. $\endgroup$
    – user135520
    Sep 7, 2014 at 16:19

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Not a full answer, but too long for a comment. There is a theorem of Serre, in his 1953 article Cohomologie modulo 2 des complexes d'Eilenberg-MacLane (it's theorem 10 in there). The theorem in question is stated as follows:

Theorem [Serre]: Let $X$ be a path-connected, simply connected space such that:

  • $H_i(X; \mathbb{Z})$ is a finitely generated abelian group for $i > 0$;
  • $H_i(X; \mathbb{Z}/2\mathbb{Z}) = 0$ for big enough $i$;
  • $H_i(X; \mathbb{Z}/2\mathbb{Z}) \neq 0$ for at least one $i \neq 0$.

Then there is an infinite number of integers $i$ such that $\pi_i(X)$ contains a subgroup isomorphic to $\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z}$.

The first two conditions are easily seen to be satisfied for a finite CW complexe. I'm not sure if the third condition is also always satisfied, though. It's clear that it is for closed manifolds, but I don't know about general finite CW complexes...

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