1. ## Homotopy Functor

Show that the relative singular homology group is a covariant homotopy functor from the category
of pairs of topological spaces (X,A) to the category of graded abelian groups.

Thanks a ton!!

2. Originally Posted by sorrow
Show that the relative singular homology group is a covariant homotopy(homology ?) functor from the category
of pairs of topological spaces (X,A) to the category of graded abelian groups.

Thanks a ton!!
Let X be a topological space and A be a subspace of X.
For a pair (X, A), we can consider the following infinite sequence of homology groups and their homomorphisms.

$\cdots H_n(A) \longrightarrow H_n(X) \longrightarrow H_n(X,A)$ $\longrightarrow H_{n-1}(A) \longrightarrow H_{n-1}(X) \cdots \longrightarrow H_0(X, A) \longrightarrow 0$.

$H_n$ is a functor from the category of topological spaces to the category of (graded) abelian groups such that $H_n:Top \rightarrow Ab$ given by $X \mapsto H_n(X)$. To check it is a covariant functor, consider the topological inclusion map $i:A \rightarrow X$. It induces an abelian group homomorphism $H_n(i):H_n(A) \rightarrow H_n(X)$ (If it induces $H_n(X) \rightarrow H_n(A)$, then $H_n$ is a contravariant functor).

In a similar vein, consider a topological pair (X, A). $H_n$ assigns (X, A) to $H_n(X, A)$ and $fX,A) \rightarrow (Y,B)" alt="fX,A) \rightarrow (Y,B)" /> to $H_n(f):H_n(X,A) \rightarrow H_n(Y,B)$.You also need to check $H_n$ satisfies the functoriality (link).

3. thank you for your help