In Sheaf theory one can obtain the Mayer Vietoris spectral sequence for cohomology. For $\mathcal{U}$ an open cover of $X$ we get the convergence

$E_2^{pq} = \check H^p(\mathcal{U},H^q(-,F)) \Longrightarrow H^{p+q}(X,F)$.

In topological K theory it is a fact (see for exemple Karoubi's book II.4.18) that we have the classical Mayer Vietoris long exact sequence for a cover of $X$ by two open (or closed) sets. This can be seen as the collapsing of the previous spectral sequence adapted to K theory at the stage 1.

My question is : is there a way to build a MV spectral sequence for topological K theory ?

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    $\begingroup$ I believe this can be realized as the spectral sequence of a simplicial space - given a cover by $(U_i)_{i\in I}$ let $Y=\coprod_{i\in I}U_i$ with the canonical map $Y\to X$, and consider the simplicial space $$Y\leftleftarrows Y\times_XY\begin{smallmatrix}\leftarrow\\\leftarrow\\\leftarrow\end{smallmatrix}Y\times_XY\times_XY\begin{smallmatrix}\leftarrow\\[-1ex]\vdots\\[.4ex]\leftarrow\end{smallmatrix}\cdots;$$ in good cases its geometric realization is equivalent to $X$, and its $h^*$-spectral sequence converges to $h^*$ of the geometric realization for decent cohomology theories $h^*$. $\endgroup$ Jan 17 '17 at 19:20
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    $\begingroup$ @მამუკაჯიბლაძე You should post it as an answer $\endgroup$ Jan 18 '17 at 9:53
  • $\begingroup$ Thanks, do you have any reference for it ? I only know how to get spectral sequences for bicomplexes or so. $\endgroup$ Jan 18 '17 at 9:55
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    $\begingroup$ @Bleuderk The spectral sequence of a (co)simplicial abelian group/space/whatever is called the Bousfield-Kan spectral sequence, since it was developed by Bousfield and Kan in their book Homotopy limits, completion and localization. In your case, you should consider the cosimplicial spectrum $F(Y\times_XY\times_X\cdots\times_XY,KU)$ obtained by applying $F(-,KU)$ to the simplicial diagram above. $\endgroup$ Jan 18 '17 at 11:06
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    $\begingroup$ If anyone cares, this spectral sequence appears much earlier, for any cohomology theory, in Segal's "classifying spaces and spectral sequences". This is one of his explicit examples. $\endgroup$ Mar 6 '18 at 18:53

As suggested by Denis Nardin I am moving my comment here. However I don't know details well enough, so I am making this cw in case somebody can fill them in.

So, choose a cohomology theory $h^*$ like e. g. $K$-theory, and, given a cover $(U_i)_{i\in I}$ of $X$, let $p:Y\to X$ be the canonical map $\coprod_{i\in I}U_i\to X$. Then consider the associated simplicial space $$ \check C(p):=\left(Y\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix} Y\times_XY\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix}Y\times_XY\times_XY\begin{smallmatrix}\leftarrow\\\vdots\\\leftarrow\end{smallmatrix}\cdots\right), $$ so that $\check C(p)_n=\coprod_{i_0,...,i_n}U_{i_0}\times_X\cdots\times_XU_{i_n}$.

A result of Dugger (found in "Topological Hypercovers") shows that geometric realization of this simplicial space is weakly homotopy equivalent to $X$, and so is the homotopy colimit of it. This induces isomorphisms on $h^*$ between the geometric realization of $\check C(p)$ to $X$, and between the homotopy colimit of $\check C(p)$ to $X$.

If $\check C(p)$ were Reedy cofibrant, Theorem 5.83, on page 163 of "Generalized Cohomology" by Kōno and Tamaki, would say there is a spectral sequence converging (again under some mild conditions) to $h^*$ of the geometric realization, with the second page given by cohomology of the cochain complex corresponding to the cosimplicial abelian group $h^*(\check C(p))$. This would be the Moore normalization of the latter complex (given by restricting to nondegenerate simplices of $\check C(p)$) has $\prod_{\{i_0,...,i_n\}\subseteq I}h^*(U_{i_0}\cap\cdots\cap U_{i_n})$ in the $n$th degree, with understandable differentials.

But $\check C(p)$ is not in general Reedy cofibrant (See "Topological Hypercovers" again; the discussion in the beginning of Secion 3). The idea is that Reedy cofibrancy would imply that the finite intersections are cofibrant, which isn't necessarily the case.

One way to recover the result would be to take a cofibrant replacement of $\check C(p)$ first and then take the cohomology spectral sequence of that resulting simplicial space. This is precisely the homotopy colimit spectral sequence, but unfortunately does not have as nice a description at the second page; it involves taking the homology of the Moore normalization of the simplicial replacement.

This generalizes (Dugger's Primer, Prop. 18.17): given a spectrum $\mathscr E$, and a diagram $F : \mathbf C \to \textbf{Top}$, there is a spectral sequence describd by taking the left derived colimit $$ E_{s, t}^2 = H_s(\mathbf C, \mathscr E_t(F)) = \operatorname{colim}_s \mathscr E_t(F) \Rightarrow \mathscr E_{s + t}(\operatorname{hocolim}_{\mathbf C}(F)) \cong \mathscr E_{s + t}(X). $$

Here, we would choose the diagram to be the above simplicial space, and the desired K-Theory spectrum.

  • $\begingroup$ The theorem you reference presupposes the simplicial space is Reedy cofibrant, which I don't think this simplicial space you define is, generally. You might be able to recover something with a cofibrant replacement functor. Also, Dugger shows that the geometric realization of this simplicial space is always weakly equivalent to $X$, which should give the desired isomorphism. $\endgroup$ Mar 2 '18 at 20:21
  • $\begingroup$ Here's an alternate approach: given a spectrum, we can construct the homotopy colimit spectral sequence associated to a diagram, which in this case we choose to be this simplicial space you defined above. Since the homotopy colimit of that simplicial space is weakly equivalent to $X$, we get the convergence to the homology of the space $X$. The second page can then be described in terms of the derived colimit functors, and the spectrum. Definitely not as concise a description as yours, at the second page. Actually, this is exactly @DenisNardin s answer. I may edit this wiki later. $\endgroup$ Mar 2 '18 at 20:59
  • $\begingroup$ Please do, I am by no means an expert. Would you still have cohomologies of tuplewise intersections visible? $\endgroup$ Mar 3 '18 at 5:24
  • $\begingroup$ Possibly. Each term in the first page should be able to be described as the homology of the simplicial replacement of this simplicial space composed with the cohomology functors, which in some sense contains these tuplewise intersections; but it doesn't look as nice as this description. It contains too much information, and it is not obvious (except for maybe the two cover) that on passing to the second page that extra information gets wiped out and your left with the nicer description above. $\endgroup$ Mar 3 '18 at 19:43
  • $\begingroup$ Could you add links to the references? I have problems finding the second one. Also, I believe it would help a reader like me if you would explain a little why one should not expect cofibrancy (or Dugger's weakening that he calls free degeneracies). $\endgroup$ Mar 5 '18 at 20:28

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