In May, Concise Algebraic Topology, p. 108-109, for a cofibration $A \rightarrow X$ a "topological boundary map" is defined as the composite:
$X/A \xrightarrow{\psi^{-1}} Ci \xrightarrow{\pi} \Sigma A$
where the homotopy equivalence $\psi$ collapses $CA$ inside the mapping cone $Ci$. Then he says:
In this context, $\tilde{E}$ is a generalized reduced homology theory.
Questions:
Does the Corollary simply say: the induced map fits into the long exact sequence?
How does one show the Corollary? Because $\tilde{E}$ is a generalized homology theory, I don't know what a diagram chase would look like; I can't rely on concrete cycles as in, e.g., singular homology.
The topological boundary map and the suspension isomorphism make use of identifications of $Ci/X$ with $CA/A$ with $\Sigma A$. I thought it was implied each of these preserve the cone/suspension coordinate but if this is the case then the corollary as stated is only correct to a sign.
To illustrate this take $(X,A)=(D^n, S^{n-1})$ and assume the each of the identifications preserves the cone/suspension coordinate. We have a map $\xi_n: (D^n,S^{n-1}) \to (CS^{n-1},S^{n-1})$ given by $\xi_n(tx_1, \cdots, tx_n)=(x_1, \cdots, x_n) \wedge (1-t)$. By naturality of the boundary homomorphism in homology for a pair we get a commutative diagram $\require{AMScd}$ \begin{CD} \tilde{E}_q(D^n/S^{n-1}) @>{\partial_{D^n/S^{n-1}}}>> \tilde{E}_{q-1}(S^{n-1}) \\ @VVV @VVV\\ \tilde{E}_q(CS^{n-1}/S^{n-1}) @>{\partial_{CS^{n-1}/S^{n-1}}}>> \tilde{E}_{q-1}(S^{n-1}) \end{CD} The vertical map on the right is the identity since $\xi_n$ restricts to the identity on $S^{n-1}$. The map along the bottom is $\Sigma^{-1}$ under the suspension/cone coordinate preserving identification of $CS^{n-1}/S^{n-1}$ with $\Sigma S^{n-1}$. The vertical map on the left side is induced by the map $D^n/S^{n-1} \to CS^{n-1}/S^{n-1}$ induced by $\xi_n$. Following May we will continue to write $\xi_n$ for the induced map $D^n/S^{n-1} \to CS^{n-1}/S^{n-1}$. It follows that $\Sigma^{-1} \circ (\xi_n)_*=\partial_{D^n/S^{n-1}}$ is the boundary homomorphism.
This is very similar to what we want but it turns out $-\xi_n$ is the topological boundary map for $D^n/S^{n-1}$. This is more or less immediate using results from the previous chapter: There we find a lemma
and the assertion that $\iota_n \circ \nu_n = -\xi_n$. Because $\psi$ is a homotopy equivalence we can reverse it. Following the diagram clockwise from the bottom left to the top right is the topological boundary map. Then by homotopy commutativity the topological boundary map is (homotopic to) $\iota_n \circ \nu_n = -\xi_n$.
Thus if each of the identifications between $Ci/X$ with $CA/A$ with $\Sigma A$ preserve the suspension coordinate we need to change the statement of the corollary. The correct statement in this case should reverse the suspension coordinate after the topological boundary map. Explicitly the boundary homomorphism should be the composite $$\tilde{E}_q(X/A) \xrightarrow{\partial_*} \tilde{E}_q(\Sigma A) \xrightarrow{\rho_*} \tilde{E}_q(\Sigma A) \xrightarrow{\Sigma^{-1}} \tilde{E}_{q-1}(A)$$ where $\rho(x \wedge t) = x \wedge (1-t)$.
Here's a proof: We will use naturality of the boundary homomorphism together with maps $(X,A) \to (Mi, A) \to (CA, A)$ that are constant on $A$ to construct a diagram.
$\require{AMScd}$ \begin{CD} \tilde{E}_q(X/A) @>{\partial}>> \tilde{E}_{q-1}(A) \\ @VV{(\psi^{-1})_*}V @VV=V\\ \tilde{E}_q(Ci) @>{\partial}>> \tilde{E}_{q-1}(A) \\ @VV{(\rho \circ \pi)_*}V @VV=V\\ \tilde{E}_q(\Sigma A) @>{\partial = \Sigma^{-1}}>> \tilde{E}_{q-1}(A) \end{CD}
Note: Although $Mi = X \cup A \times I / ((a,0) \sim a)$ the copy of $A$ in the pair $(Mi, A)$ is the one at the top coming from points $(a,1) \in A \times I$.
The vertical maps on the left are induced by the induced maps $X/A \to Mi/A = Ci$ and $Ci \to CA/A = \Sigma A$. We will show the map $X/A \to Mi/A$ is (homotopic to) $\psi^{-1}$ while the map $Ci \to \Sigma A$ is $-\pi = \rho \circ \pi$.
I should define what these maps are. The easy one is $(Mi, A) \to (CA, A)$. All points of $X$ are sent to the cone point while pairs $(a,t)$ are sent to $(a, 1-t)$. The map on quotients this induces is $-\pi = \rho \circ \pi$. The harder one to define is the map $(X,A) \to (Mi, A)$. To define this we need to use results from Chapter 5 Section 5 on Cofiber Homotopy Equivalences.
The projection map $Mi \to X$ is a homotopy equivalence which is also a map under $A$. Its a proposition from that section that homotopy equivalences that are maps under A are cofiber homotopy equivalences. Thus there is a cofiber homotopy inverse $(X,A) \to (Mi, A)$. Because these maps and homotopies are all under $A$ the induced maps on quotients $X/A \to Mi/A = Ci$ and $Ci \to X/A$ are also homotopy inverses. Moreover the induced map $Ci \to X/A$, being induced by the projection map, coincides with $\psi$. It follows that $(X,A) \to (Mi, A)$ induces $\psi^{-1}:X/A \to Ci$. This concludes the proof.