Finding the cofibration of the map $S^{1} \rightarrow *. $

algebraic-topologycategory-theorycofibrationshomotopy-theorypushforward

I want to answer this question:

What is the homotopy cofibre of the unique map $S^{1} \rightarrow * $ ? describe the homotopy cofibre of $ X \rightarrow * $ in general.

My attempt:

I got a hint that I should find the cofibration of the map $S^{1} \rightarrow * .$ but I looked at the definition of "The Cofiber of a Map" in the book "Modern Classical Homotopy Theory" by Jeffery Strom, which is given below:

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But still I do not understand how to find the cofibration of my given map, could anyone help me in this please?

Also, I looked at the word "homotopy cofibre" definition in AT and "Modern Classical Homotopy Theory" by Jeffery Strom but I did not find it, could anyone tell me under which title can I find this word? or specifically at which page in either of the 2 books?

Also, I got a hint of solving this question by forming the weak homotopy pushout square and the strong homotopy pushout square but I do not know the relation of those to homotopy cofibration.

Also, I was given a hint of those diagrams:

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1-Actually, for the first diagram, which is a pushout diagram, I do not understand why $X \bigsqcup {*} = X$? does this because $X$ is a pointed space?

2-For the second diagram, I do not understand which side of the given pushout square represents our given map $S^{1} \rightarrow * ,$ is it the upper side or the left side? and why we should construct a diagram containing 2 "*"?

Could anyone help me answer this questions please? I want to arrange my thoughts to conclude the solution.

Best Answer

Question 1: The pushout can in fact be constructed as a quotient of the disjoint sum. Of course we have $X \sqcup * \ne X$, but we indentify $a = i(a) \in X$ with $p(a) = *$ for all $a \in A$, thus we obtain $X/A$.

Question 2: The homotopy cofiber is not obtained as the pushot of your diagram. Let us more generally consider a map $f : X \to Y$. In general it is no cofibration, but the inclusion $j : X \to M_f$ embedding $X$ as the top of the mapping cylinder is one and we have $r \circ j = f$, where $r : M_f \to Y$ is the canonical strong deformation retraction. The homotopy cofiber of $f$ is then defined as the pushout

$\require{AMScd}$ \begin{CD} X @>{j}>> M_f \\ @V{p}VV @V{p'}VV \\ * @>{f}>> C_f \end{CD}

Note that if $f$ is a cofibration, then one easily show that $C_f$ is homotopy equivalent to $X/A$.

In your case $f : S^1 \to *$ we get $M_f \approx D^2$ and $C_f \approx D^2/S^1 \approx S^2$.

Edited: As Jason DeVito pointed out in his comment, for any $f : X \to *$ we have $(M_f,X) \approx (CX,X)$, thus $C_f \approx CX/X \approx \Sigma X$.

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