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The Bockstein is independent of lift and representative

Statement

Assume AC and the hypotheses and notation of Bockstein connecting operation. The class $β [c]$ is independent of the chosen $B$ -cochain lift of $c$ and of the cocycle representing $[c]$ .

Facts & Assumptions

Given: A short exact sequence $0 \to A \overset{i}{\to} B \overset{q}{\to} C \to 0$ and a cocycle $c \in C^{n} (X; C)$ .

[F1]

The Bockstein construction chooses $b$ with $q_{*} b = c$ , uniquely solves $i_{*} a = δ b$ , and proposes $β [c] = [a]$ (Bockstein connecting operation).

[F2]

AC supplies a simultaneous choice from any set-indexed family of nonempty fibres (The Axiom of Choice).

Proof

technique · direct cochain comparison

1.1givenF1

The cochain $a$ in [F1] is a cocycle. Indeed, $i_{*} (δ a) = δ (i_{*} a) = δ^{2} b = 0$ ; injectivity of $i_{*}$ gives $δ a = 0$ .

2.1F1step 1.1

Changing only the lift changes $a$ by a coboundary. If $b^{'}$ is another lift of $c$ , then $q_{*} (b^{'} - b) = 0$ . Exactness gives a unique $h \in C^{n} (X; A)$ with $b^{'} - b = i_{*} h$ . If $a^{'}$ is defined from $b^{'}$ , then $i_{*} (a^{'} - a) = δ (b^{'} - b) = i_{*} (δ h)$ , hence $a^{'} - a = δ h$ .

3.1F1F2step 2.1

Changing the cocycle representative also changes $a$ by a coboundary. Write $c^{'} = c + δ u$ . Use the lifting choice of [F2] to take $v \in C^{n - 1} (X; B)$ with $q_{*} v = u$ . For an arbitrary lift $b^{'}$ of $c^{'}$ ,

$q_{*} (b^{'} - b - δ v) = c^{'} - c - δ u = 0.$

Thus $b^{'} = b + δ v + i_{*} h$ for some $h \in C^{n} (X; A)$ . Applying $δ$ and using $δ^{2} v = 0$ gives $a^{'} = a + δ h$ .

4.1step 2.1step 3.1∎

Steps 2.1 and 3.1 show that $[a^{'}] = [a]$ for either permitted change, so $β [c]$ is well defined.

Depends on

Used by

Mod-p reduced power operations Definition
Bockstein detects integral two-torsion in real projective space Example
Spin-c extensions over a rational ball form a metabolizer coset Lemma
Bocksteins are natural and stable Proposition

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hatcher, Algebraic Topology (standard reference, not scraped)