Exercises:Saul - Algebraic Topology - 5

Exercises

Exercise 5.6

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

Show that: [ilmath]H_*^s(X)\cong H_*^s(A)\oplus H_*^s(X,A)[/ilmath]

Possible solution

I have proved:

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

Then as a corollary to the above

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

To prove this corollary I show:

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

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!