Exercises:Saul - Algebraic Topology - 5/Exercise 5.6

Exercises

Exercise 5.6

Let $(X,\mathcal{ J })$ be a topological space and let $A\in\mathcal{P}(X)$ be a retract of $X$ (with the continuous map of the retraction being $r:X\rightarrow A$). Lastly take i $i:A\rightarrow X$ to be the inclusion map, $i:a\mapsto a$.

Show that: $H_*^s(X)\cong H_*^s(A)\oplus H_*^s(X,A)$

Possible solution

I have proved:

• If $f:X\rightarrow Y$ is a homotopy equivalence then $f_*:H_n(X)\rightarrow H_n(Y)$ is a group isomorphism for each $n\in\mathbb{N}_{\ge 0}$

Then as a corollary to the above

• If $A\in\mathcal{P}(X)$ is a retract of $X$ (and $r:X\rightarrow A$ is the continuous map of the retraction) then $i_*:H_*(A)\rightarrow H_*(X)$ is a monomorphism (injection) onto (as in surjective) a direct summand of $H_*(X)$
  • If $A$ is a deformation retraction of $X$ (Caveat:presumably strong?) then $i_*$ is an isomorphism.
• $i:A\rightarrow X$ is the inclusion mapping.

To prove this corollary I show:

• $H_*(X)\cong G\oplus H$ where:
  • $G:\eq\text{Im}(i_*)$ and $H:\eq\text{Ker}(r_*)$

The second part of the statement (the deformation retraction part) is an immediate result of the first theorem, the second bit is proved without reference to it. So I should be good!