Is the following presumed folklore fact written anywhere?
Let $E^*$ a multiplicative cohomology theory. Then the coboundary map in the Mayer–Vietoris sequence of an excisive triad $(X;U,V)$ preserves the $E^*X$-module structures induced by the inclusions.
The proof follows from two other statements that also seem not to show up in introductory texts.
The first appears in a MathOverflow answer but nowhere else I'm aware of:
- The Mayer–Vietoris sequence for a pointed excisive triad $(X;U,V)$ arises as the long exact sequence of the cofibration $U \vee V \to U \cup \big((U \cap V) \wedge I\big) \cup V \simeq X.$
And this one is stated as Proposition 2.15 of Allen Hatcher's Vector Bundles and K-Theory (for complex topological K-theory):
- Let $E^*$ be a multiplicative cohomology theory. Every object in the long exact sequence of a pair $(X,A)$ is a module over $E^*X$ in such a way that all maps are $E^*X$-module homomorphisms.
The proof uses only that the external product is natural, the smash-diagonal $X \to X \wedge X$ makes $X$ into a coalgebra, and that allied maps make the Puppe sequence of $A \hookrightarrow X$ into a sequence of $X$-comodules, so it should hold in general.
Are the forms of these I want written anywhere?
All of this condenses an unanswered Math.StackExchange question.