# Cofiber Sequences in Reduced Homology Theory

While going through axiomatic treatments of homology theories I got a bit stuck on this problem. Consider given a reduced homology theory, i.e. functors $(\tilde{E}_q:Top_* \to Ab)_{q \in \mathbb{N}}$, from the category of (nice) based spaces to the category of Abelian groups such that:

If $f: X \to Y$ is a weak homotopy then $\tilde{E}_q (f)$ is an isomorphism $\forall q \in \mathbb{N}$.

If $i: A \to X$ is a cofibration then, $\tilde{E}_q(A) \to \tilde{E}_q(X) \to \tilde{E}_q(X/A)$ is an exact sequence.

For each integer $q$, there is a natural isomorphism $\tilde{E}_q(X) \cong \tilde{E}_{q+1}( \Sigma X)$.

If $X$ is a wedge of a set of (nice) based spaces, then the inclusions $X_i \to X$ induce an isomorphism $\sum_i \tilde{E}_q(X_i) \to \tilde{E}_q(X)$.

Then we want to construct a homology theory for pairs of spaces, by setting $E_*(X):=\tilde{E}_*(X_+), \ \ E_*(X,A):=\tilde{E}_*(C(i_+))$, where $X_+:=X \coprod *$ is $X$ with an adjoined disjoint point, and $C(i_+)$ is the homotopy cofiber of $\ i_+:A_+ \to X_+$. We define the connecting morphism $\partial_q: E_q(X,A) \to E_{q-1} (A)$ to be $\Sigma^{-1} \circ \partial_*: \tilde{E}_q (X_+ / A_+) \to \tilde{E}_q(\Sigma (A_+) \to \tilde{E}_{q-1}(A_+)$, where $\partial$ is the composition of a homotopy inverse of $\psi: C(i_+) \to X_+ / A_+$ with $\pi:C(i_+) \to \Sigma(A_+)$, which collapses $X_+$.

The question is: how to prove that, with these definitions, $\ldots \to \tilde{E}_q (A_+) \to \tilde{E}_q (X_+) \to E(X,A) \to \tilde{E}_{q-1} (A_+)\to \ldots$ is exact? For one composition, namely $\tilde{E}_q (A_+) \to \tilde{E}_q (X_+) \to E_q(X,A)$ it follows directly from the axioms for reduced homology and from $C(i_+)\simeq X_+ / A_+$.

Turning to the other compositions, I can show that both are zero (thanks to cofiber sequences), but how can I prove the rest?

The reference is http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf (pages 59 and 110). Thanks in advance.

• Perhaps $X_+ \to C(i_+) \to \Sigma A_+$ (this collapses $X_+$) is a cofiber sequence, and then apply the 2nd listed axiom? – ykm Jul 31 '14 at 12:17
• Sorry for having not specified it, but I have already managed to solve it. Thanks anyway! :) – Edoardo Lanari Aug 2 '14 at 12:19