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Community Bot
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As pointed out by David White in when mapping cone is contractiblewhen mapping cone is contractible there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be the two loops in $X=S^1\vee S^1$ and glue in two $2$-cells along the words $a^5b^{−3}$ and $b^3(ab)^{−2}$.

My question is: Is this possible if the suspension, $\Sigma X$, of $X$ is homotopy equivalent to the unit interval $I=[0,1]$ rel ends?

More precisely: if $f:\Sigma X \to I$ is a homotopy equivalence with inverse $g$ and homotopies $h_1:fg\simeq id_I$ and $h_2:gf\simeq id_{\Sigma X}$ such that $g(i) = [X\times i]$ for $i=0,1$, $h_1$ is a homotopy rel {$0,1$} and $h_2$ is a homotopy rel {$[X\times 0]$, $[X\times 1]$} then does $X$ have to be contractible?

As pointed out by David White in when mapping cone is contractible there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be the two loops in $X=S^1\vee S^1$ and glue in two $2$-cells along the words $a^5b^{−3}$ and $b^3(ab)^{−2}$.

My question is: Is this possible if the suspension, $\Sigma X$, of $X$ is homotopy equivalent to the unit interval $I=[0,1]$ rel ends?

More precisely: if $f:\Sigma X \to I$ is a homotopy equivalence with inverse $g$ and homotopies $h_1:fg\simeq id_I$ and $h_2:gf\simeq id_{\Sigma X}$ such that $g(i) = [X\times i]$ for $i=0,1$, $h_1$ is a homotopy rel {$0,1$} and $h_2$ is a homotopy rel {$[X\times 0]$, $[X\times 1]$} then does $X$ have to be contractible?

As pointed out by David White in when mapping cone is contractible there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be the two loops in $X=S^1\vee S^1$ and glue in two $2$-cells along the words $a^5b^{−3}$ and $b^3(ab)^{−2}$.

My question is: Is this possible if the suspension, $\Sigma X$, of $X$ is homotopy equivalent to the unit interval $I=[0,1]$ rel ends?

More precisely: if $f:\Sigma X \to I$ is a homotopy equivalence with inverse $g$ and homotopies $h_1:fg\simeq id_I$ and $h_2:gf\simeq id_{\Sigma X}$ such that $g(i) = [X\times i]$ for $i=0,1$, $h_1$ is a homotopy rel {$0,1$} and $h_2$ is a homotopy rel {$[X\times 0]$, $[X\times 1]$} then does $X$ have to be contractible?

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Ricardo Andrade
Ricardo Andrade
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As pointed out by David White in when mapping cone is contractible there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be the two loops in $X=S1\vee S1$$X=S^1\vee S^1$ and glue in two $2$-cells along the words $a^5b^{−3}$ and $b^3(ab)^{−2}$.

My question is: Is this possible if the suspension, $\Sigma X$, of $X$ is homotopy equivalent to the unit interval $I=[0,1]$ rel ends?

More precisely: if $f:\Sigma X \to I$ is a homotopy equivalence with inverse $g$ and homotopies $h_1:fg\simeq id_I$ and $h_2:gf\simeq id_{\Sigma X}$ such that $g(i) = [X\times i]$ for $i=0,1$, $h_1$ is a homotopy rel {$0,1$} and $h_2$ is a homotopy rel {$[X\times 0]$, $[X\times 1]$} then does $X$ have to be contractible?

As pointed out by David White in when mapping cone is contractible there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be the two loops in $X=S1\vee S1$ and glue in two $2$-cells along the words $a^5b^{−3}$ and $b^3(ab)^{−2}$.

My question is: Is this possible if the suspension, $\Sigma X$, of $X$ is homotopy equivalent to the unit interval $I=[0,1]$ rel ends?

More precisely: if $f:\Sigma X \to I$ is a homotopy equivalence with inverse $g$ and homotopies $h_1:fg\simeq id_I$ and $h_2:gf\simeq id_{\Sigma X}$ such that $g(i) = [X\times i]$ for $i=0,1$, $h_1$ is a homotopy rel {$0,1$} and $h_2$ is a homotopy rel {$[X\times 0]$, $[X\times 1]$} then does $X$ have to be contractible?

As pointed out by David White in when mapping cone is contractible there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be the two loops in $X=S^1\vee S^1$ and glue in two $2$-cells along the words $a^5b^{−3}$ and $b^3(ab)^{−2}$.

My question is: Is this possible if the suspension, $\Sigma X$, of $X$ is homotopy equivalent to the unit interval $I=[0,1]$ rel ends?

More precisely: if $f:\Sigma X \to I$ is a homotopy equivalence with inverse $g$ and homotopies $h_1:fg\simeq id_I$ and $h_2:gf\simeq id_{\Sigma X}$ such that $g(i) = [X\times i]$ for $i=0,1$, $h_1$ is a homotopy rel {$0,1$} and $h_2$ is a homotopy rel {$[X\times 0]$, $[X\times 1]$} then does $X$ have to be contractible?

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Spiros Adams-Florou
Spiros Adams-Florou
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Does there exist a space X whose suspension is homotopy equivalent to [0,1] rel ends but where X is not contractible?

As pointed out by David White in when mapping cone is contractible there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be the two loops in $X=S1\vee S1$ and glue in two $2$-cells along the words $a^5b^{−3}$ and $b^3(ab)^{−2}$.

My question is: Is this possible if the suspension, $\Sigma X$, of $X$ is homotopy equivalent to the unit interval $I=[0,1]$ rel ends?

More precisely: if $f:\Sigma X \to I$ is a homotopy equivalence with inverse $g$ and homotopies $h_1:fg\simeq id_I$ and $h_2:gf\simeq id_{\Sigma X}$ such that $g(i) = [X\times i]$ for $i=0,1$, $h_1$ is a homotopy rel {$0,1$} and $h_2$ is a homotopy rel {$[X\times 0]$, $[X\times 1]$} then does $X$ have to be contractible?