Reduced cone suspension and cofiber sequence

For a well-pointed CGWH based space $(X,x_0)$ set $CX=(X\times I)\mathrm{/}(X\times \{1\}\cup \{x_0\}\times I),\mathrm{\Sigma }X=CX\mathrm{/}(X\times \{0\})\mathrm{.}$ The common collapsed set is the basepoint; the copy of X at height zero is the cone base. All products and mapping conventions are those of Compactly generated conventions for based homotopy, and well-pointedness means Cofibration and homotopy extension property.

For a based $f:X\to Y$ put $C_f=Y{\cup }_{f}CX$, the reduced homotopy cofiber, with $i:Y\to C_f$ the inclusion and $q:C_f\to \mathrm{\Sigma }X$ collapsing Y. This reduced notation is used in this item and its based consumers; it differs from the unreduced cone of Mapping cylinder and mapping cone. A reduced cylinder also collapses the basepoint track. Maps on all these quotients are defined by For a quotient map $q:X\to Y$, a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$; and a composite of quotient maps is a quotient map.

Write ${\rho }_X([x,t])=[x,1-t]$ and $-\mathrm{\Sigma }f={\rho }_Y\mathrm{\Sigma }f$. The cofiber sequence convention is $X\stackrel{f}{\to }Y\stackrel{i}{\to }{C}_f\stackrel{q}{\to }\mathrm{\Sigma }X\stackrel{-\mathrm{\Sigma }f}{\to }\mathrm{\Sigma }Y\stackrel{-\mathrm{\Sigma }i}{\to }\mathrm{\Sigma }{C}_f\stackrel{-\mathrm{\Sigma }q}{\to }{\mathrm{\Sigma }}^{2}X\stackrel{{\mathrm{\Sigma }}^{2}f}{\to }{\mathrm{\Sigma }}^{2}Y⟶\cdots \mathrm{.}$ The homotopy equivalences and mapping-set exactness behind this notation are proved below; this definition does not assert a covariant exact sequence of homotopy groups.