Reduced K-theory exact sequence of a cofibration

Since $q\circ i$ is the constant map to the collapsed basepoint, [F1] gives $i^{\ast }{q}^{\ast }=0$ on reduced groups. Hence the image of $q^{\ast }$ lies in the kernel of $i^{\ast }$.

Let $u\in {\tilde{K}}^0(X)$ satisfy $i^{\ast }u=0$, and let ${\rho }_u:X\to \mathbb{Z}$ be its locally constant virtual-rank function. The nonzero-rank locus $C={\rho }_u^{-1}(\mathbb{Z}\setminus \{0\})$ is clopen and disjoint from $A$; put $D=X\setminus C$. Both are compact, and $q$ restricts to a homeomorphism $C\cong q(C)$, with $q(C)$ clopen in $X\mathrm{/}A$. Thus the summand $u{\mathrm{\mid }}_C$ already descends across $q$, and it remains to descend the zero-rank class $u{\mathrm{\mid }}_D$.

Write $u{\mathrm{\mid }}_D=[P]-[Q]$. Refine the finite clopen rank decompositions of $P$ and $Q$; zero virtual rank says their ranks agree on each piece. Apply [F3]'s complement theorem on every piece and enlarge the finitely many trivial ambient bundles to one common dimension $n$. The piecewise complements glue to a bundle $Q^{\mathrm{\prime }}$ with $Q\oplus Q^{\mathrm{\prime }}\cong {\epsilon }^n$, while $E=P\oplus Q^{\mathrm{\prime }}$ has the constant rank $n$. Hence $u{\mathrm{\mid }}_D=[E]-[{\epsilon }^n]$. Since $i^{\ast }u=0$, [F3]'s stable-isomorphism criterion on $A$ allows a further common trivial summand so that $E{\mathrm{\mid }}_A\cong A\times {\mathbb{C}}^n$ by an actual supplied trivialization $\tau$; rename the enlarged rank as $n$.

Extend the bundle map $\tau :A\times {\mathbb{C}}^n\to E{\mathrm{\mid }}_A$ to a neighborhood of $A$ in $D$. Concretely, choose finitely many bundle charts over $D$, express the finitely many frame vectors of $\tau$ by bounded real and imaginary coordinate functions on the closed chart pieces, extend those functions by [F4], and combine the local extensions by the finite partition in [F5]. The resulting $n$ sections agree with $\tau$ on $A$. Their exterior product is nonzero on $A$, so continuity gives an open neighborhood $U$ of $A$ in $D$ on which they are a frame. Thus $E{\mathrm{\mid }}_U$ has a trivialization extending $\tau$. Here [F6] discharges the DC hypotheses from the single assumption AC.

Form a bundle $\bar{E}$ on $q(D)=D\mathrm{/}A$: away from the collapsed point use the charts of $E$ on $D\setminus A$, and over $q(U)$ use the trivialization in step 3.1, identifying every fiber above $A$ with the same copy of ${\mathbb{C}}^n$. On overlaps the transition matrices are the old continuous ones, expressed in this frame, so the quotient charts glue to a rank-$n$ bundle. Pulling back gives $q^{\ast }\bar{E}\cong E$ over $D$, with the chosen trivialization over $A$. Therefore $v_D=[\bar{E}]-[{\epsilon }^n]$ is reduced at the quotient basepoint and pulls back to $u{\mathrm{\mid }}_D$. On the disjoint clopen part $q(C)$ put $v_C=((q{\mathrm{\mid }}_C{)}^{-1}{)}^{\ast }(u{\mathrm{\mid }}_C)$. The two virtual bundles assemble over the finite clopen decomposition $X\mathrm{/}A=q(C)⨿q(D)$ to a reduced class $v$ with $q^{\ast }v=u$. This proves $\ker i^{\ast }\subseteq \mathrm{im}{q}^{\ast }$ and, with step 1.1, exactness.

Replace $i$ by its mapping-cylinder inclusion, which is a closed cofibration with the same homotopy cofiber. The cone base inside each successive reduced mapping cone is again a closed cofibration of compact Hausdorff well-pointed CGWH spaces. Applying steps 1.1–4.1 at every stage gives exactness at every term. The quotient identifications in [F2] identify the successive quotients with reduced suspensions; using its reflection maps gives exactly the recorded signs $-\mathrm{\Sigma }i$, $-\mathrm{\Sigma }q$, and thereafter their suspended alternation. Homotopy invariance in [F1] transports exactness across these identifications.

A finite CW subcomplex inclusion satisfies the stated compactness, Hausdorff, CGWH, well-pointed, and closed-cofibration hypotheses, so the result specializes to finite CW pairs.