4
$\begingroup$

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.

$\endgroup$
4
  • 1
    $\begingroup$ I think you can think of the Mayer-Vietoris SS as a Bousfield-Kan SS. Think of all the (finitary) intersections of the members of the cover as a big poset, ordered by inclusion. Then the homotopy colimit of that poset of spaces recovers $X$. IIRC, the BKSS converging to the homology groups of that homotopy colimit coincides with the Mayer-Vietoris SS. The BKSS approach generalizes so widely that maybe it applies in whatever context you have in mind--certainly it doesn't require you use any details of any particular construction of homology. $\endgroup$
    – user509184
    Feb 29 at 2:06
  • 2
    $\begingroup$ The best (OK, only) reference I know for the Mayer-Vietoris SS is Bott and Tu. They are mainly working in the smooth category, but the treatment is in terms of sheaves and good covers so should apply in greater generality. $\endgroup$
    – Mark Grant
    Feb 29 at 7:46
  • 1
    $\begingroup$ You can check chapter 2 section 5 of Godemont's Topologie Algebrique et Theorie des Faisceaux for a sheaf-theoretic treatment in the topological category. $\endgroup$
    – Tyrone
    Feb 29 at 13:59
  • $\begingroup$ See section 2.8 of Kashiwara and Schapira Sheaves on manifolds section 4 of these notes www3.nd.edu/~lnicolae/sheaves_coh.pdf $\endgroup$ Feb 29 at 15:20

1 Answer 1

3
$\begingroup$

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 $$ E_1^{pq} = \bigoplus_{\substack{J \subseteq I \\ |J|=p+1}} H^{q}(U_J,F) \implies H^{p+q}(X,F).$$ Note that the $q$th row of the spectral sequence is by definition the Cech 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 Cech-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 $$ 0 \to F \to \bigoplus_{|J|=1} F_J \to \bigoplus_{|J|=2} F_J \to \bigoplus_{|J|=3} F_J \to \dots$$ The stalk of this complex at a point $x \in X$ is given by tensoring $F_x$ with the acyclic complex $$ 0 \to \mathbf Z \to \bigoplus_{\substack{|J|=1 \\ x \in U_J}} \mathbf Z \to \bigoplus_{\substack{|J|=2 \\ x \in U_J}} \mathbf Z \to \dots $$ 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 \dots $ 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.

$\endgroup$
1
  • $\begingroup$ I guess I cheated a bit when saying $DDF \cong F$ -- some finiteness hypotheses are necessary here. We want $F$ and $D\mathbf Z$ to both be perfect. $\endgroup$ Feb 29 at 10:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.