We consider an inclusion $j: A \hookrightarrow X$. Let $A_j = A \times_X PX$ be the homotopy fiber (which is the fiber of the fibration associated to $j$). The space $PX$ there is the Moore path space $PX = \{ (\gamma,r) \in X^{[0,\infty)} \times (0,\infty): \gamma(t) = \gamma (r) ~ \forall t \geq r \}$ and $A \times_X PX = \{ (a,(\gamma,r)) \in A \times PX: \gamma (0) = a \}$ is the space of all paths starting in $A$.

The homotopy groups of $A_j$ are related to the relative homotopy groups of the pair $(X,A)$: $$ \pi_n (X,A) \cong \pi_{n-1} (A_j) $$ (see Theorem 5.1.8 in Piccinini, Renzo A., Lectures on homotopy theory, North-Holland Mathematics Studies. 171. Amsterdam etc.: North-Holland. xii, 293 p. (1992). ZBL0742.55001., for example).

My questions are:

  1. Is there also an isomorphism $H_n(X,A) \cong H_{n-1} (A_j)$ on homology?
  2. If not, can one specify some conditions such that this statement holds?

Maybe one sentence about the background of my question: I want to find a Sullivan minimal model for the pair $(X,A)$ such that the rational Hurewicz morphism $\text{hur}_{(X,A)}: \pi_* (X,A) \otimes \mathbb{Q} \to H_* (X,A; \mathbb{Q})$ can be decoded via this model (compare to Chapter 13(c) of Felix, Yves; Halperin, Stephen; Thomas, Jean-Claude, Rational homotopy theory, Graduate Texts in Mathematics. 205. New York, NY: Springer. xxxii, 535 p. (2001). ZBL0961.55002.).

Thank you very much in advance.

  • 3
    $\begingroup$ For 1., this will be true only in a range depending on the connectivity of the pair $(X,A)$. An example to think about is $A=*$ a point in $X$, then $A_j=\Omega X$, the space of loops based at $*$. There is a map $H_{n-1}(\Omega X)\to H_n(X,*)$ called homology suspension, which is an isomorphisms in a range depending on the connectivity of $X$. $\endgroup$ – Mark Grant Oct 9 at 9:36

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