Bocksteins are natural and stable

The Bockstein is natural in spaces. [given, F1, F2] For $f\text{\hspace{0.17em}⁣}:X\to Y$, if $q_{\ast }b=c$ and $i_{\ast }a=\delta b$ on $Y$, then $q_{\ast }{f}^{\ast }b=f^{\ast }c$ and $i_{\ast }{f}^{\ast }a=\delta f^{\ast }b$. Therefore ${\beta }_X(f^{\ast }[c])=f^{\ast }{\beta }_Y[c]$; [F2] removes dependence on the displayed representatives.

The Bockstein is natural in the coefficient sequence. [given, F1, F2] For a commutative morphism of short exact sequences with vertical maps $u\text{\hspace{0.17em}⁣}:A\to A^{\mathrm{\prime }}$, $v\text{\hspace{0.17em}⁣}:B\to B^{\mathrm{\prime }}$, and $w\text{\hspace{0.17em}⁣}:C\to C^{\mathrm{\prime }}$, the cochain $v_{\ast }b$ lifts $w_{\ast }c$, and $\delta (v_{\ast }b)=v_{\ast }{i}_{\ast }a=i_{\ast }^{\mathrm{\prime }}{u}_{\ast }a$. Hence ${\beta }^{\mathrm{\prime }}[w_{\ast }c]=u_{\ast }\beta [c]$.

The cone quotient comparison defines the signed suspension. [F3, F4, F6, F7, F8] Take a based CW complex $X$. If its chosen basepoint lies inside a positive-dimensional open cell, subdivide that one characteristic disk radially at the point and retain the same attaching maps for higher cells; this finite refinement makes the basepoint a vertex without changing the based space or choosing any new data. Form $CX=(X\times I)\mathrm{/}(X\times \{1\}\cup \{x_0\}\times I)$ and $\mathrm{\Sigma }X=CX\mathrm{/}X$. For each nonbasepoint $n$-cell of $X$, its product with the open height interval is an $(n+1)$-cell of $CX$; the height-zero cells form the cone base $X$, while the height-one face and basepoint track collapse to one vertex. The product characteristic maps have finite boundary-cell support, so their quotient map-out and CW weak-topology tests make $CX$ a CW complex with $X$ a closed subcomplex. A cellwise radial collar of this subcomplex, extended over the characteristic disks and assembled by the weak topology, gives an open neighborhood $V$ that strongly deformation retracts onto $X$. Since $V$ contains the whole fibre $X$ collapsed by $q:CX\to \mathrm{\Sigma }X$, it is saturated; $q(V)$ is open and its descended flow retracts onto the quotient vertex.

The coefficient connector anticommutes with the unsigned cone-pair connector. [F1, F5, step 1.3] Take $q_{\ast }b=c$ on $X$, with $i_{\ast }a=\delta b$. Extend $c,b,a$ by zero on the singular simplices of $CX$ not lying in $X$, writing the extensions with bars. Then $d:=\delta \bar{b}-i_{\ast }\bar{a}$ is a relative $B$-cochain lifting the relative cocycle $\delta \bar{c}$, and

The signed suspension commutes with the Bockstein. [F4, step 1.1, step 1.2, step 1.3, step 2.1] For $x\in {\tilde{H}}^n(X;C)$,