Let $X$ and $Y$ be topological spaces and $A$ and $B$ be subspaces of $X$ and $Y$ respectively. We write $f:(X,A)\to (Y, B)$ as a shorthand for writing $f:X\to Y$ and $f(A)\subseteq B$.

Now Proposition 2.19 in Hatcher's Algebraic Topology reads:

If two maps $f(X, A)\to (Y, B)$ are homotopic through maps of pairs $(X, A)\to (Y, B)$, then $f_*=g_*:H_n(X, A)\to H_n(Y, B)$.

I do not understand what is meant by ... homotopic through maps of pairs (X, A)\to (Y, B) ...

Can somebody please write out the formal definition?


1 Answer 1


It is a map $f_t: X \times I \to Y$ such that $f_0 = f$, $f_1 = g$, and $f_t (A \times I )\subset B$. Conceptually "the image of $A$ is always inside $B$."


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