Relationship between cohomology and higher-homotopy

Question

Let $M$ be a connected, compact, and orientable 3-manifold ($H^3(M)\cong\mathbb{Z}$), and let $G$ be a simple Lie group satisfying $\pi_1(G)=\pi_2(G)=0$. Let

$\pi_M(G)$ denote the set of homotopy classes of maps from $M\to G$
$\pi_3(G)$ denote the set of homotopy classes of maps from $\mathbb{S}^3\to G$

Suppose I have a map $f:M\to \mathbb{S}^3$ which induces an isomorphism on $H^3$, and for which $f_*:\pi_3(G)\to \pi_M(G)$ is surjective. $$f_*[g]:=[g\circ f]$$

Is this enough information to infer that $f_*$ is an isomorphism?

Have you tried to see if you can apply the Hurewicz Theorem? I’m not saying that it works, but it’s worth taking a look at. :) — Berrick Caleb Fillmore, Apr 19 '15 at 02:57
Is $M$ supposed to be a 3-manifold? Otherwise, I don't understand your parenthetical ($H^3(M) \cong \mathbb{Z}$). — Jason DeVito - on hiatus, Apr 19 '15 at 03:19
$g \in \pi_M(G)$ is a map $M \to G$. How do you intend to compose this with $f: M \to S^3$? The domain is wrong. You need to reverse an arrow somewhere here. — , Apr 19 '15 at 03:39

score 2 · Accepted Answer · edited Apr 13 '17 at 12:58

If $G$ is a simple Lie group, then $\pi_3(G) = \Bbb Z$ (see here). So the only way this homomorphism could fail to be an isomorphism is if $\pi_M(G)$ were finite. Now $G$ is homotopy equivalent to a space $G'$ with a single 0-cell, a single 3-cell, 4-cells whose boundary maps are the constant maps to the basepoint, and cells of higher dimension; this follows from theorem 4C.1 in Hatcher.

From this, cellular approximation shows that every map $M \to G'$ is homotopic to one lying entirely in the copy of $S^3$; Hopf's theorem says that these are classified by their degree; and therefore even after wedging on the extra copies of $S^4$ they are still not homotopic. (This is because the inclusion $S^3 \hookrightarrow G'$ induces an isomorphism on $H^3$.)

So there are infinitely many homotopy classes of maps $M \to G$ as desired. Your homomorphism is an isomorphism.

Relationship between cohomology and higher-homotopy

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