Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $X$ be an $n-1$ connected space. Why does a map $X\rightarrow K(\pi_n(X),n)$ that induces an isomorphism on $\pi_n$ exist and what is this map?

share|improve this question

3 Answers 3

If $X$ is a CW-complex, then you can build a $K(\pi_n(X),n)$ by attaching cells (of dimension $n+2$ and above) to $X$ to kill off the higher homotopy groups. The map you are thinking of is then an inclusion of $X$ into $K(\pi_n(X),n)$ as a sub-complex.

share|improve this answer

As stated, the statement is false. For example, let $X$ be the Hawaiian earring. Then $X$ is 0-connected. But $X$ can't have a map to an Eilenberg-Maclane space that's an isomorphism on $\pi_1$, because as a compact space it would have to have compact image, which would necessarily have finitely-generated $\pi_1$ in a CW complex like an Eilenberg-Maclane space, and $X$ has infinitely generated fundamental group. The statement is true for if $X$ is a CW-complex, as wckronholm explains.

share|improve this answer
    
It's a bad habit that a lot of us get into--thinking that all spaces are (almost) CW-complexes. –  jd.r May 17 '11 at 22:28
1  
Indeed - whenever I see a homotopy theoretic claim about "all spaces", I immediately think, "Can I break this with the topologist's sine curve or the Hawaiian earring?" –  MartianInvader May 18 '11 at 13:42

Let me assume $n>1$ and —so that your claim is true— that $X$ is a CW complex.

The Hurewicz theorem, in view of the hypothesis, gives us an isomorphism $$\phi:H_n(X,\mathbb Z)\to\pi_n(X).$$ On the other hand, in view of the hypothesis made on $X$, the universal coefficient theorem for cohomology gives us an isomorphism $$\alpha:H^n(X,\pi_n(X))\to \hom(H_n(X,\mathbb Z), \pi_n(X)) $$ Moreover, there is a canonical bijection $$\beta:[X,K(\pi_n(X), n)]\to H^n(X,\pi_n(X))$$ with $[X,K(\pi_n(X), n)]$ the set of homotopy classes of maps $X\to K(\pi_n(X),n)$.

The map you are looking for is any $f:X\to K(\pi_n(X),n)]$ whose homotopy class $[f]$ is such that $\alpha(\beta([f]))=\phi$.

(This works for any $K(\pi_n(X),n)$ that you pick)

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.