Construction of the Mayer-Vietoris spectral sequence

Question

Given subspaces $\{U_i\}_{i \in I}$ of a topological space $X$ with $X = \bigcup_i U_i$ satisfying some conditions, there is a Mayer-Vietoris spectral sequence converging to the homology of $X$. Here are my questions:

1. What are the appropriate conditions on the $U_i$? The only references I know about are for $X$ a CW complex and the $U_i$ subcomplexes. However, for the Mayer-Vietoris exact sequence (corresponding to the case of a cover with two sets) you can also prove that the exact sequence holds if $X$ is the union of the interiors of the $U_i$. Is this enough for the general case, or are more conditions needed?

2. Related to the previous question, what is a good reference for the Mayer-Vietoris spectral sequence that does not just cover CW complexes and use cellular homology? Preferably it would just use the Eilenberg-Steenrod axioms and not the details of any particular construction of homology.

I would prefer the construction to not use spectra since what I really want is to use this is a slightly different context where that wouldn't make sense.

Answer 1

My favorite way to think about Mayer–Vietoris is via sheaf cohomology.

So let $F$ be a sheaf of abelian groups on a locally compact Hausdorff space $X$. Let $X=\bigcup_{i\in I} U_i$ be either an open cover or a closed cover. For $\varnothing \neq J \subseteq I$ write $U_J = \bigcap_{j\in J} U_j$. Then there is a spectral sequence $$ E_1^{pq} = \bigoplus_{\substack{J \subseteq I \\ |J|=p+1}} H^{q}(U_J,F) \Rightarrow H^{p+q}(X,F).$$ Note that the $q$th row of the spectral sequence is by definition the Čech complex of the cover $X=\bigcup_{i\in I} U_i$, with respect to the presheaf $\mathscr H^q(-,F)$. So the $E_2$-page of the spectral sequence is the usual $E_2$-page of the Čech-to-derived-functor spectral sequence.

To construct the spectral sequence, consider first the case of a closed cover. Define $F_J = (f_J)_\ast(f_J)^\ast F$, where $f_J : U_J \to X$ is the natural inclusion. There is a complex of sheaves $$ 0 \to F \to \bigoplus_{|J|=1} F_J \to \bigoplus_{|J|=2} F_J \to \bigoplus_{|J|=3} F_J \to \dotsb.$$ The stalk of this complex at a point $x \in X$ is given by tensoring $F_x$ with the acyclic complex $$ 0 \to \mathbf Z \to \bigoplus_{\substack{|J|=1 \\ x \in U_J}} \mathbf Z \to \bigoplus_{\substack{|J|=2 \\ x \in U_J}} \mathbf Z \to \dotsb $$ so the complex itself is acyclic. Hence $F$ is quasi-isomorphic to the complex $0\to \bigoplus_{|J|=1} F_J \to \bigoplus_{|J|=2} F_J \to \dotsb $ and the hypercohomology spectral sequence (that is, the spectral sequence associated to the filtration by degree of this complex of sheaves) has the $E_1$-term we are looking for.

Note that the construction would work equally well if $F$ were a complex of sheaves (in which case all complexes above would be double complexes).

For an open cover, set instead $F^J = (f_J)_! (f_J)^! F$. We get a complex of sheaves $$ \dots \to \bigoplus_{|J|=3} F^J \to \bigoplus_{|J|=2} F^J \to \bigoplus_{|J|=1} F^J\to F\to 0 $$ which is acyclic by the same argument as before. Now apply the construction to the Verdier dual $DF$. We get that $DF$ is quasi-isomorphic to a filtered object with graded pieces of the form $\bigoplus_{|J|=p} (DF)^J$. Then the Verdier dual of this filtered object is quasi-isomorphic to $DDF\cong F$, with graded pieces quasi-isomorphic to $\bigoplus_{|J|=p} D(DF)^J \cong \bigoplus_{|J|=p} (Rf_J)_! (f_J)^! F$. So the spectral sequence associated to this filtered object (or rather, the filtered object obtained by applying $R\Gamma$ to this complex of sheaves) again gives us the correct $E_1$-term.

What made the construction tick for closed covers is that $(f_J)_\ast$ and $(f_J)^\ast$ are both $t$-exact functors when $f_J$ is a closed embedding. Pushforward is not exact for open embeddings, but we have instead that both $(f_J)_!$ and $(f_J)^!$ are $t$-exact (whereas $(f_J)^!$ is not exact for closed embeddings). Hence why the two constructions of the spectral sequence differ.

Answer 2

Here is a purely algebro-topological way of obtaining the Mayer–Vietoris spectral sequence for any homology theory $h_{\bullet}$ (the cohomological version is analogous). It is obtained by combining the following facts.

1. For any simplicial space $X_{\bullet}\colon\Delta^\text{op}\rightarrow\mathbf{Top}$, its geometric realization $|X_{\bullet}|$ comes equipped with a skeletal filtration, which gives rise to a spectral sequence. For $X_{\bullet}$ Reedy-cofibrant w.r.t. the Strøm model structure (i.e. the inclusion of the subspace of degenerate simplices in each dimension is a cofibration), the spectral sequence was identified in Graeme Segal, Classifying Spaces and Spectral Sequences and it is a convergent right-half plane spectral sequence $$E^2_{pq}=H_p(h_q(X_{\bullet}))\Rightarrow h_{p+q}(|X_{\bullet}|).$$ [The simplicial abelian group $h_q(X_{\bullet})$ is made a chain complex via the alternating face maps.]

2. For any topological space $X$ and cover $\mathfrak{U}=(U_i)_{i\in I}$, we can form a simplicial space $X_{\mathfrak{U}}$ whose space of $n$-simplices is $\coprod_{i_0,\dotsc,i_n\in I}U_{i_0\dotsc i_n}$ (the face maps omit indices, the degeneracy maps duplicate them). There is a canonical map $|X_{\mathfrak{U}}|\rightarrow X$ and, if $X=\bigcup_{i\in I}U_i^{\circ}$, this map is always a weak equivalence. This is explained in Section 2 of Daniel Dugger, Daniel C. Isaksen, Topological hypercovers and $\mathbb{A}^1$-realizations (they only state it for open covers, but the weaker assumption suffices for the proof).

Together, this means under the assumptions of 2., there is a convergent right-half plane spectral sequence $$E_{pq}^2=H_p(\mathfrak{U},h_q)\Rightarrow h_{p+q}(X).$$

Two remarks:

• There are variations of $X_{\mathfrak{U}}$. You can fix a total order on $I$ and only consider $U_{i_0\dotsc i_n}$ for which $i_0\le\dotsb\le i_n$, which corresponds to using the ordered complex computing Čech cohomology. Furthermore, there is also a version obtained by a 'barycentric subdivision' of $\mathfrak{U}$. The latter model is the one used by Segal and their comparison is explained in the Dugger–Isaksen paper.

• I implicitly assumed that $h_{\bullet}$ satisfies the weak equivalence axiom. In case $\mathfrak{U}$ is numerable, the canonical map $|X_{\mathfrak{U}}|\rightarrow X$ is a homotopy equivalence on the nose, so the condition can be dropped then.