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Circle homology with monodromy

Statement

Give $S^{1}$ its CW structure with one vertex and one oriented edge. If an $R$ -module local system has fiber $M$ and monodromy $T \in {Aut}_{R} (M)$ around the positive loop, its cellular local chain complex is $0 ⟶ M \overset{T - 1}{\to} M ⟶ 0.$ Consequently $H_{1} (S^{1}; M) = \ker (T - 1)$ , $H_{0} (S^{1}; M) = coker (T - 1)$ , and all other homology groups vanish.

Facts & Assumptions

Given: The CW circle, $R$ -module $M$ , and automorphism $T$ in the statement.

[F1]

Cellular chains compute local homology computes local homology from lifted cellular incidences with the right chain action $c \cdot g = g^{- 1} c$ and the left fiber action $g m = T_{\overset{ˉ}{g}} m$ .

Proof

technique · direct

1.1

Let $g$ be the positive loop and lift the vertex to $\tilde{v} \in R$ . Choose the lifted edge $\tilde{e}$ from $\tilde{v}$ to $g \tilde{v}$ . Its boundary is $g \tilde{v} - \tilde{v} = \tilde{v} \cdot (g^{- 1} - 1)$ under [F1]'s right action. In the left fiber module, $g^{- 1} m = T_{\overline{g^{- 1}}} m = T_{g} m = T m$ . Tensoring therefore sends $m$ to $(T - 1) m$ .

2.1

There is one chain module $M$ in degrees one and zero and none elsewhere, so the kernel and cokernel of step 1.1 are exactly the displayed homology groups. Reversing the chosen loop yields $T^{- 1} - 1 = - T^{- 1} (T - 1)$ and hence an isomorphic complex, so the answer is independent of the orientation convention. For $M = 0$ all groups vanish; for $T = 1$ the differential is zero and ordinary circle homology with coefficients in $M$ is recovered. No AC is used.

F1step 1.1∎

Depends on

Cellular chains compute local homology

Used by

Nothing in the library uses this result yet.

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

Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §2.1, Exercise 76, pp.99–100 (standard reference, not scraped)