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It is well known that there is a generalized Mayer Vietoris exact sequence in the context of smooth manifolds and the Rham cohomology, where a sequence involving countably many open sets is obtained. It is explained in Bott and Tu's book Differential Forms in Algebraic Topology.

Is there an analogous result for a covering by a finite family of open sets in the topological setting? Could you give me a reference?

What I exactly mean: I want a kind of Mayer-Vietoris exact sequence involving a covering of a compact topological space (such as a manifold (topological) or a finite CW-complex) by a finite family of open sets (not necessarily two sets). (Imagine I am working with locally trivial fibre bundles and Leray-Hirsch is not enough). In reference to the homology or cohomology theories involved, the singular ones would be nice. And I am working with coefficients in a field if that helps (so in the case of singular theories homology and cohomology are isomorphic).

EDIT: I want to avoid spectral sequences. Thanks in advance.

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    $\begingroup$ What does "the topological setting" mean (topological manifolds/ topological space) ? What are your objects? What cohomology theory do you want to use? If you mean "is there an analogue of de Rham theory to work in more topological spaces?" Then yes, it is called singular cohomology, and a similar Mayer-Vietoris sequence holds there (and can be found in Hatcher, Spanier or any other beginner algebraic topology book besides Bott and Tu). $\endgroup$ – PVAL-inactive May 27 '17 at 11:44
  • $\begingroup$ @PVAL-inactive sorry for the ambiguity. I have corrected it. $\endgroup$ – D1811994 May 27 '17 at 12:03
  • $\begingroup$ I don't have Bott-Tu on hand. Is there something you need that isn't in the Mayer-Vietoris in Ch. 3 of Hatcher (which is all available freely from Hatcher's webpage)? If not could you explain what "generalized" means in this context. $\endgroup$ – PVAL-inactive May 27 '17 at 12:07
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A main result, Theorem 8.1.5, of the book partially titled Nonabelian Algebraic Topology (EMS 2011) (NAT), is on open covers of a filtered space. It crucially uses the notion of the homotopically defined crossed complex $\Pi X_*$ of a filtered space $X_*= (X_i:i \geqslant 0)$, a construction considered under other names by Blakers (1948) and Whitehead (Combinatorial Homotopy II, 1949).

It also involves the notion of a connected filtered space which I won't go into here, but is satisfied if the filtered space is the skeletal filtration of a CW-complex.

Theorem Let $X_*$ be a filtered space and let $\mathcal U = \{U^\lambda, \lambda \in \Lambda \}$ be an open cover of $X$. For each $\nu \in \Lambda ^n$ let $U^\nu$ denote the intersection of the sets $U^{\nu_i}$ and let $U^\nu _*$ be the filtered space formed by intersection with the $X_i, i \geqslant 0$. Suppose that for all $n \geqslant 1$ each $U^\nu_*, \nu \in \Lambda ^n$ is a connected filtered space. Then

(Con) $X_*$ is connected, and

(Iso) the following diagram, where $a,b,c,$ are induced by inclusions, is a coequaliser diagram of crossed complexes:

$$\bigsqcup _{\nu \in \Lambda ^2} \Pi U^\nu _* \rightrightarrows^a_b \bigsqcup _{\lambda \in \Lambda } \Pi U^\lambda _* \xrightarrow{c} \Pi X_* . $$

This result is related to the cellular operator chains of a CW-complex in Section 8.4 of NAT.

Note also that the Theorem gives a precise colimit type result.

An Introduction to the background of some of these ideas is given in the paper Modeling and Computing Homotopy Types: I, to appear in Indag. Math. in 2017 in a volume in honor of L.E.J. Brouwer.

The proofs do not involve spectral sequences, or singular homology, but the relevance to "fields as coefficients" is not at all clear! The question of the relationship to Morse Theory, which of course uses noncellular filtrations, is mentioned in Problem 16.1.17 of NAT.

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