# Are homotopic maps over a cofibration homotopic about the cofibration?

Allow $X$ be a Hausdorff room and also $A$ a shut subspace. Intend the incorporation $A \hookrightarrow X$ is a cofibration. Allow $f, g: X \to Y$ be maps that settle on $A$ and also which are homotopic. Are they homotopic about $A$?

My inspiration for asking this inquiry originates from the adhering to outcome:

Let $i: A \to X, j: A \to Y$ be cofibrations. Intend $f: X \to Y$ is a map that makes the all-natural triangular commutative. Intend $f$ is a homotopy equivalence. After that $f$ is a cofiber homotopy equivalence.

On the various other hand, I'm having problem adjusting the evidence in Peter May is publication of this to the inquiry I asked. However, the typical instances of sets of maps which are homotopic yet not relative to which some part on which they concur (claim, the identification map of a comb room and also its falling down to an accordingly picked factor), do not appear to entail NDR sets.

I assume the equipment of blockage concept manage the grandfather clause where the rooms are skeleta of a CW - facility. Below is the arrangement (I'm primarily simply replicating from pp. 6 - 7 of Mosher & Tangora below):

Let $Y$ be merely - attached for simpleness. First, allow $B$ be a facility and also $A$ be a subcomplex. Allow $f:A\cup B^n\rightarrow Y$. After that we get an *blockage cochain * $c(f)\in C^{n+1}(B,A;\pi_n(Y))$ (i.e. a function on loved one $(n+1)$ - cells with values in $\pi_n(Y)$). In a similar way, allow $K$ be an intricate ; after that for any kind of 2 maps $f,g:K\rightarrow Y$ that settle on $K^{n-1}$, we in a similar way get a *distinction cochain * $d(f,g)\in C^n(K;\pi_n(Y))$.

Below are both outcomes.

- Theory: There is a map $g:A\cup B^{n+1}\rightarrow Y$ concurring with $f$ on $A\cup B^{n-1}$ iff $[c(f)]=0 \in H^{n+1}(B,A;\pi_n(Y))$.
- Theory: The constraints of $f$ and also $g$ to $X^n$ are homotopic rel $K^{n-1}$ iff $d(f,g)=0 \in C^n(K;\pi_n(Y))$. They are homotopic rel $K^{n-2}$ iff $[d(f,g)]=0 \in H^n(K;\pi_n(Y))$.

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