Can one define relative Hurewicz maps using the Dold-Thom theorem

Question

Let $A\to X$ be a (Hurewicz) cofibration of path-connected topological spaces. Then we have a long homotopy sequence $$ \dots\to \pi_i(A)\to \pi_i(X)\to \pi_i(X,A)\to \dots; $$ here we fix a base point inside $A$. It maps into the homotopy sequence for the pair $(SP(X),SP(A))$ for infinite symmetric powers, does it? Can one apply the Dold-Thom theorem here to obtain the relative Hurewicz maps $\pi_i(X,A)\to H_i(X,A)$ along with the morphism between the corresponding exact sequences?

My doubts are caused by the book "Algebraic Topology from a Homotopical Viewpoint" by Marcelo Aguilar, Samuel Gitler, and Carlos Prieto where homology is defined via the Dold-Thom theorem but a different definition of relative Hurewicz maps is used. Is "my" idea fine; is it discussed somewhere?

Answer 1

In the absolute case, $\pi_*(X) \rightarrow \pi_*(SP(X))$ can be identified with the Hurewicz map because one can explicitly compute $\mathbb{Z}=\pi_n(S^n) \rightarrow \pi_n(SP(S^n))=\mathbb{Z}$ and see it is an isomorphism (one way is to examine a cell structure). In the relative case, one can consider the commutative diagram

$\require{AMScd}$ \begin{CD} \pi_n(D^{n+1},S^n) @>{\partial}>> \pi_n(S^n)\\ @VVV @VVV\\ \pi_n(SP(D^{n+1}),SP(S^n)) @>{\partial }>> \pi_n(SP(S^n)) \end{CD}

By the previous analyis, the right hand map is an isomorphism, and both the top and bottom maps must be isomorphisms since they are sandwiched between contractible spaces, so we deduce that the left hand map is an isomorphism. Since relative homotopy groups are represent by $(D^{n+1},S^n)$, we deduce that the map $\pi_n(X,A) \rightarrow \pi_n(SP(X),SP(A))$ is $\pm$ the Hurewicz map (simply because it is an isomorphism on representing object). To eliminate the negative possibility, one can just examine the case $\pi_n(S^n) \rightarrow \pi_n(SP(S^n))$ a little more closely.