I am trying to prove the following statement :

Let $\{ * \} \subset A \subset B \subset X$ be a chain of topological spaces (all subsets have the subspace topology). It is given that $A \to X$ and $B \to X$ are cofibrations. If $A$ and $B$ are homotopy equivalent rel $\{ * \}$, then the map $id_X : (X,A,*) \to (X,B,*)$ is a homotopy equivalence.

I seem to be able to prove it, but wish to know whether my following proof is correct or not. Could you please verify this ?

My attempt at proof : Let $r : B \to A $ be the homotopy inverse of $i : A \to B$. Let $ L : B \times I \to B $ be a homotopy between $id_B$ and $ i \circ r $. Since $B \to X$ is a cofibration, there is an extension to $X \times I$ of the map $ <id_X, L> : X \times \{ 0 \} \cup B \times I \to X$. Let us denote this extension by $H$. Thus we get a map $H_1 : (X,B,*) \to (X, A, *)$. Clearly the composition $ (X,B,*) \xrightarrow {H_1}(X,A,*) \xrightarrow{id} (X,B,*)$ is homotopic rel $\{ * \}$ as a map of pairs to to $(X,B,*) \xrightarrow{id} (X,B,*)$. Similarly we can find a left inverse to $ (X,A,*) \xrightarrow{i} (X,B,*)$. Hence proved.

The reason why I need this result is to understand a statement of p.153 of Algebraic Topology by Tammo tom Dieck. The statement says: The map $ \pi_n(A \times (0,1] \cup X, A \times (0,1), *) \to \pi_n(A \times (0,1] \cup X, A \times (0,1], *) $ is an isomorphism by homotopy invariance. (Here $A$ is a subspace of X and $(a,1) \sim a$). The book does not give any justification for this so I believe it must be obvious to see in some manner. Is there any other way of seeing why this statement is true without the above proof ? Thanks.


1 Answer 1


There is a result on homotopy equivalences of pairs of spaces which I state in my answer to this mathstackexchange question. You can extend this result by induction to a result on a map $f:X \to Y$ each with a chain of subspaces $X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n$, $Y_0 \subseteq Y_1 \subseteq \cdots \subseteq Y_n$ such that for all $i$ $f(X_i) \subseteq Y_i$ and all inclusions $X_i \to X_{i+1}, Y_i \to Y_{i+1}$ are cofibrations. What can you say if you know that for all $i$ $f_i: X_i \to Y_i$ is a homotopy equivalence? How relevant is this to your actual question?


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