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edited Apr 17, 2013 at 2:50

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I liked very much the comment/answer of Jeff Strom. And this result seems to be, indeed, essentialy, a consequence of what he mentioned. At first glance, the result seems to be a consequence of the relative Hurewicz isomorphism (and, obviously, Whitehead theorem). But, since Hurewicz Isomorphism is a consequence of homotopy excision, we can prove the result using homotopy excision without passing to homology.

Assuming $\Sigma f $ is a weak equivalence between simply connected spaces, we get, by the homotopy excision (pag 81, May's Concise Course), that $\Sigma C_f\equiv C_{\Sigma f} $ is weakly equivalent to a point. Now, we complete by induction. We already know that $f$ is a $1$-equivalence. By induction, we assume that $ f $ is a $n$-equivalence. And, again, by homotopy excision, we know that this hypothesis implies that $(M_f,X)\to C_f $ is $(n+2)$-equivalence. So $C_f$ is $n$ connected. And, then, by Freudenthal theorem, $\Sigma :\pi_q( C_f)\to \pi _{q+1}(\Sigma C_f) $ is an isomorphism for $q< 2n+1 $. In particular, $C_f$ is $2n$-connected. Therefore, by the $(n+2)$-equivalence, we conclude that $f$ is a $(n+2)$-connected space. And this concludes our induction.

Concisely, this proof is about two lemmas: one is that commented by Jeff Strom (which can be proved using excision). The other lemma is a consequence of Freudenthal/excision: If $\sum X $ is $n$-connected and $X$ is simply connected, then $X$ is $n-1$ connected.

I liked very much the comment/answer of Jeff Strom. And this result seems to be, indeed, essentialy, a consequence of what he mentioned. At first glance, the result seems to be a consequence of the relative Hurewicz isomorphism (and, obviously, Whitehead theorem). But, since Hurewicz Isomorphism is a consequence of homotopy excision, we can prove the result using homotopy excision without passing to homology.

Assuming $\Sigma f $ is a weak equivalence between simply connected spaces, we get, by the homotopy excision (pag 81, May's Concise Course), that $\Sigma C_f\equiv C_{\Sigma f} $ is weakly equivalent to a point. Now, we complete by induction. We already know that $f$ is a $1$-equivalence. By induction, we assume that $ f $ is a $n$-equivalence. And, again, by homotopy excision, we know that this hypothesis implies that $(M_f,X)\to C_f $ is $(n+2)$-equivalence. So $C_f$ is $n$ connected. And, then, by Freudenthal theorem, $\Sigma :\pi_q( C_f)\to \pi _{q+1}(\Sigma C_f) $ is an isomorphism for $q< 2n+1 $. In particular, $C_f$ is $2n$-connected. Therefore, by the $(n+2)$-equivalence, we conclude that $f$ is a $(n+2)$-connected space. And this concludes our induction.

Concisely, this proof is about two lemmas: one is that commented by Jeff Strom (which can be proved using excision). The other lemma is a consequence of Freudenthal/excision: If $\sum X $ is $n$-connected and $X$ is simply connected, then $X$ is $n-1$ connected.

I liked very much the comment/answer of Jeff Strom. And this result seems to be, indeed, essentialy, a consequence of what he mentioned. At first glance, the result seems to be a consequence of the relative Hurewicz isomorphism (and, obviously, Whitehead theorem). But we can prove the result using homotopy excision without passing to homology.

Assuming $\Sigma f $ is a weak equivalence between simply connected spaces, we get, by the homotopy excision (pag 81, May's Concise Course), that $\Sigma C_f\equiv C_{\Sigma f} $ is weakly equivalent to a point. Now, we complete by induction. We already know that $f$ is a $1$-equivalence. By induction, we assume that $ f $ is a $n$-equivalence. And, again, by homotopy excision, we know that this hypothesis implies that $(M_f,X)\to C_f $ is $(n+2)$-equivalence. So $C_f$ is $n$ connected. And, then, by Freudenthal theorem, $\Sigma :\pi_q( C_f)\to \pi _{q+1}(\Sigma C_f) $ is an isomorphism for $q< 2n+1 $. In particular, $C_f$ is $2n$-connected. Therefore, by the $(n+2)$-equivalence, we conclude that $f$ is a $(n+2)$-connected space. And this concludes our induction.

Concisely, this proof is about two lemmas: one is that commented by Jeff Strom (which can be proved using excision). The other lemma is a consequence of Freudenthal/excision: If $\sum X $ is $n$-connected and $X$ is simply connected, then $X$ is $n-1$ connected.

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created Apr 17, 2013 at 2:45

875
5
22

I liked very much the comment/answer of Jeff Strom. And this result seems to be, indeed, essentialy, a consequence of what he mentioned. At first glance, the result seems to be a consequence of the relative Hurewicz isomorphism (and, obviously, Whitehead theorem). But, since Hurewicz Isomorphism is a consequence of homotopy excision, we can prove the result using homotopy excision without passing to homology.

Assuming $\Sigma f $ is a weak equivalence between simply connected spaces, we get, by the homotopy excision (pag 81, May's Concise Course), that $\Sigma C_f\equiv C_{\Sigma f} $ is weakly equivalent to a point. Now, we complete by induction. We already know that $f$ is a $1$-equivalence. By induction, we assume that $ f $ is a $n$-equivalence. And, again, by homotopy excision, we know that this hypothesis implies that $(M_f,X)\to C_f $ is $(n+2)$-equivalence. So $C_f$ is $n$ connected. And, then, by Freudenthal theorem, $\Sigma :\pi_q( C_f)\to \pi _{q+1}(\Sigma C_f) $ is an isomorphism for $q< 2n+1 $. In particular, $C_f$ is $2n$-connected. Therefore, by the $(n+2)$-equivalence, we conclude that $f$ is a $(n+2)$-connected space. And this concludes our induction.

Concisely, this proof is about two lemmas: one is that commented by Jeff Strom (which can be proved using excision). The other lemma is a consequence of Freudenthal/excision: If $\sum X $ is $n$-connected and $X$ is simply connected, then $X$ is $n-1$ connected.