Connecting homomorphism

The connecting homomorphism is the degree-shifting map produced by the lift-and-boundary construction in a long exact sequence.

The Bockstein homomorphism is the connecting homomorphism associated with a short exact sequence of coefficient groups. For $0\to \mathbb{Z}/n\to \mathbb{Z}/n^2\to \mathbb{Z}/n\to 0$, it has degree $-1$ on homology and degree $+1$ on cohomology.

For the coefficient sequence $0\to {\mathbb{Z}}_2\to {\mathbb{Z}}_4\to {\mathbb{Z}}_2\to 0$, the Bockstein homomorphism on infinite-dimensional real projective space is zero from an even degree and an isomorphism from an odd degree. In cellular cohomology, lift the mod-two generator to $1\in {\mathbb{Z}}_4$. The coboundary is zero in even degree and two in odd degree; identifying two with the image of $1\in {\mathbb{Z}}_2$ gives the formula.

The Bockstein homomorphism for $0\to \mathbb{Z}/n\to \mathbb{Z}/n^2\to \mathbb{Z}/n\to 0$, with injection $[a]\mapsto [na]$, is coefficient reduction $\rho$ after the integral Bockstein homomorphism. The commuting maps of coefficient sequences prove this by naturality of the connecting homomorphism.

Reduction $\rho$ of the integral Bockstein homomorphism associated to $0\to \mathbb{Z}\stackrel{2}{\to }\mathbb{Z}\to {\mathbb{F}}_2\to 0$ equals the cup product square on degree-one classes. One can see this on an ordered two-simplex: lift the values of a mod-two one-cocycle to zero or one. The half-coboundary is one precisely when both consecutive edge values are one, giving the cup-square cocycle.

Exactness of the integral coefficient sequence gives $\hat{\beta }\rho =0$. The Bockstein factorization through integral cohomology therefore gives ${\beta }^2=\rho \hat{\beta }\rho \hat{\beta }=0$, for every modulus $n\ge 2$, including composite moduli.

The degree-one Bockstein homomorphism makes modulo-$n$ cohomology into a cochain complex. Its cohomology is the Bockstein cohomology. For prime $n$ this is a graded vector space; for composite $n$ it is a graded $\mathbb{Z}/n$-module. For ${\mathbb{RP}}^3$ at modulus two, its only nonzero groups are ${\mathbb{F}}_2$ in degrees zero and three. This is the cohomological counterpart of Bockstein homology.

The integral Bockstein is the connecting homomorphism of $0\to \mathbb{Z}\stackrel{n}{\to }\mathbb{Z}\to \mathbb{Z}/n\to 0$. Lift a modulo-$n$ cocycle to an integral cochain $a$; write its coboundary as $\delta a=nb$. Then $\hat{\beta }[a]=[b]$. Its image is the subgroup of integral cohomology killed by $n$.

For the three-dimensional lens space $L(p)$, the integral coefficient sequence shows that the integral connecting map $H^1(L(p);{\mathbb{F}}_p)\to H^2(L(p);\mathbb{Z})$ is an isomorphism between groups of order $p$. Reduction modulo $p$ is also an isomorphism in degree two, so their composite Bockstein is an isomorphism.

For a generator $a\in H^1(L(p);{\mathbb{F}}_p)$, Poincare duality and the Bockstein isomorphism for a three-dimensional lens space make $t(a)$ nonzero. Replacing $a$ by $na$ multiplies $t(a)$ by $n^2$. An orientation-reversing homotopy equivalence multiplies the evaluation by $-1$, so its existence forces $n^2=-1$ in ${\mathbb{F}}_p$ for some $n$.

When the homological Bockstein satisfies ${\beta }^2=0$, it makes $H_{\ast }(C\otimes \mathbb{Z}/n)$ into a chain complex. Its homology is the Bockstein homology $\beta H_{\ast }(C;n)$.

A short exact sequence of coefficient groups $0\to A\to B\to C\to 0$ induces a natural long exact sequenceThe connecting map is a Bockstein homomorphism.

For the modulo-$m$ cohomological Bockstein homomorphism,This follows by lifting cocycles to integral cochains and applying the graded Leibniz rule for the coboundary before dividing by $m$.