TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-12

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Naturality of singular mayer vietoris connectors

Statement

Suppose $f : X \to Y$ is continuous, $X = U \cup V$ and $Y = U^{'} \cup V^{'}$ are ordered open covers, and $f (U) \subseteq U^{'}$ , $f (V) \subseteq V^{'}$ . For the $V - U$ convention, the Mayer–Vietoris connectors satisfy $f^{*} Δ_{Y} = Δ_{X} (f ∣_{U \cap V})^{*} : H^{k} (U^{'} \cap V^{'}; R) \to H^{k + 1} (X; R) .$ The entire long exact sequences commute with these pullbacks. The ordered-cover hypotheses are part of the assertion.

Facts & Assumptions

Given: The continuous map and both ordered covers as in the statement.

[F1]

The real Mayer–Vietoris sequence uses restriction, difference $V - U$ , and the positive lift-differential connector transported along the canonical small-chain inclusion (Mayer vietoris sequence in real singular cohomology).

[F2]

A morphism of cochain short exact sequences commutes with the connecting maps (Naturality of the cohomology connecting morphism).

Proof

1.1givenF1

Postcomposition by $f$ takes an $X$ -simplex in $U$ to a $Y$ -simplex in $U^{'}$ , and likewise for $V$ and the intersections. It therefore induces chain maps on all three terms of the small-chain sequence. Composition commutes with face restriction, the signed overlap inclusion and sum. Dualizing gives cochain maps $D_{Y} \to D_{X}$ , $E_{Y} \to E_{X}$ , $F_{Y} \to F_{X}$ commuting with the two arrows in [F1].

2.1F1F2step 1.1

By [F2] this cochain diagram commutes with the connectors into $H (D)$ . Explicitly, if $b e = c$ and $δ e = a (d)$ in the $Y$ row, the images satisfy the same equations in the $X$ row, so the image of $d$ is the connecting representative of the image of $c$ . This uses an image of a supplied lift; it does not claim that zero extension is natural.

3.1F1step 1.1step 2.1algebra

Write $I_{X}, I_{Y}$ for the small-chain inclusions and $f_{s m, #}$ for the small-chain map. The equation $f_{#} I_{X} = I_{Y} f_{s m, #}$ holds on every generator. Hence the canonical isomorphisms $θ = H (I^{*})$ of [F1] commute with pullback. Transporting step 2.1 by their inverses proves the displayed connector identity. The restriction and difference squares already commute by step 1.1, giving the full sequence claim.

4.1F1step 1.1step 2.1step 3.1∎

Empty intersections yield zero connector domains; empty spaces yield zero groups. In degree zero the same lift calculation applies, with negative primitives zero. On one-point spaces or identical covers the connector is zero by [F1]. Constant or degenerate simplices still satisfy the generator equation in step 3.1. All maps are supplied by postcomposition; neither natural choices of chain inverses nor AC are required.

Depends on

Dependency tree · 2 levels

Sources

DG-16 design; Hatcher/Park control