# nLab Mayer-Vietoris sequence

cohomology

### Theorems

#### Limits and colimits

limits and colimits

# Contents

## Idea

Mayer-Vietoris sequence is the term for the fiber sequence – or often for the corresponding long exact sequence of homotopy groups – induced from an (∞,1)-pullback (or for a homotopy pullback presenting it).

## Definition

Let $𝒞$ be an (∞,1)-category with finite (∞,1)-limits and let $X,Y,B$ be pointed objects and

$f:X\to B$f : X \to B

and

$g:Y\to B$g : Y \to B

be any two morphisms with common codomain preserving the base points. Let $X{×}_{B}Y$ be the (∞,1)-pullback

$\begin{array}{ccc}X{×}_{B}Y& \to & Y\\ ↓& {⇙}_{\simeq }& {↓}^{g}\\ X& \stackrel{f}{\to }& B\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ X \times_B Y &\to& Y \\ \downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{g}} \\ X &\stackrel{f}{\to}& B } \,.

The corresponding Mayer-Vietoris sequence is the fiber sequence of the induced morphism $X{×}_{B}Y\to X×Y$. Often the term is used (only) for the corresponding long exact sequence of homotopy groups.

## Properties

### General

###### Proposition

Let $𝒞$ be a presentable (∞,1)-category.

Then $X{×}_{B}Y$ is equivalently given by the (∞,1)-pullback

$\begin{array}{ccc}X{×}_{B}Y& \to & B\\ ↓& {⇙}_{\simeq }& {↓}^{{\Delta }_{B}}\\ X×Y& \stackrel{\left(f,g\right)}{\to }& B×B\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ X \times_B Y &\to& B \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\Delta_B}} \\ X \times Y &\stackrel{(f,g)}{\to}& B \times B } \,,

where the right vertical morphism is the diagonal.

Moreover, the homotopy fiber of $X{×}_{B}Y\to X×Y$ is the loop space object $\Omega B$.

###### Proof

The first statement one checks for instance by choosing a presentation by a combinatorial model category and then proceeding as below in the discussion Presentation by fibrant objects. Then by the pasting law for $\left(\infty ,1\right)$-pullbacks it follows that with the left square in

$\begin{array}{ccccc}\Omega B& \to & X{×}_{B}Y& \to & B\\ ↓& {⇙}_{\simeq }& ↓& {⇙}_{\simeq }& ↓\\ *& \to & X×Y& \stackrel{\left(f,g\right)}{\to }& B×B\end{array}$\array{ \Omega B &\to& X \times_B Y &\to & B \\ \downarrow &\swArrow_{\simeq}& \downarrow &\swArrow_{\simeq}& \downarrow \\ * &\to& X \times Y &\stackrel{(f,g)}{\to}& B \times B }

an $\left(\infty ,1\right)$-pullback, so is the total outer rectangle. But again by the first statement, this is equivalent to the $\left(\infty ,1\right)$-pullback

$\begin{array}{ccc}\Omega B& \to & *\\ ↓& {⇙}_{\simeq }& ↓\\ *& \to & B\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ \Omega B &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ * &\to& B } \,,

which is the defining pullback for the loop space object.

Therefore the Mayer-Vietoris fiber sequence is of the form

$\Omega B\to X{×}_{B}Y\to X×Y\phantom{\rule{thinmathspace}{0ex}}.$\Omega B \to X \times_B Y \to X \times Y \,.
###### Corollary

The corresponding long exact sequence of homotopy groups is of the form

$\cdots \to {\pi }_{n+1}B\to {\pi }_{n}X{×}_{B}Y\stackrel{\left({f}_{*},{g}_{*}\right)}{\to }{\pi }_{n}X\oplus {\pi }_{n}Y\stackrel{{f}_{*}-{g}_{*}}{\to }{\pi }_{n}B\to \cdots$\cdots \to \pi_{n+1} B \to \pi_n X \times_B Y \stackrel{(f_*, g_*)}{\to} \pi_n X \oplus \pi_n Y \stackrel{f_* - g_*}{\to} \pi_n B \to \cdots
$\cdots \to {\pi }_{2}B\to {\pi }_{1}X{×}_{B}Y\stackrel{\left({f}_{+},{g}_{*}\right)}{\to }{\pi }_{1}X×{\pi }_{1}Y\stackrel{{f}_{*}\cdot {g}_{*}^{-1}}{\to }{\pi }_{1}B\to {\pi }_{0}\left(X{×}_{B}Y\right)\to {\pi }_{0}\left(X×Y\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \to \pi_2 B \to \pi_1 X \times_B Y \stackrel{(f_+, g_*)}{\to} \pi_1 X \times \pi_1 Y \stackrel{f_* \cdot g_*^{-1}}{\to} \pi_1 B \to \pi_0 (X \times_B Y) \to \pi_0 (X \times Y) \,.

This has historically been the definition of Mayer-Vietories sequences (Eckmann-Hilton).

### Presentation by fibrant objects

Suppose that the (∞,1)-category $𝒞$ is presented by a category of fibrant objects $C$ (for instance the subcategory on the fibrant objects of a model category).

Then the $\left(\infty ,1\right)$-pullback $X{×}_{B}Y$ is presented by a homotopy pullback, and by the factorization lemma, this is given by the ordinary limit

$\begin{array}{ccccc}X{×}_{B}^{h}Y& \to & & \to & Y\\ ↓& & & & {↓}^{g}\\ & & {B}^{I}& \to & B\\ ↓& & ↓\\ X& \stackrel{f}{\to }& B\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ X \times^h_B Y &\to& &\to& Y \\ \downarrow && && \downarrow^{\mathrlap{g}} \\ && B^I &\to& B \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& B } \,,

where $B\stackrel{\simeq }{\to }{B}^{I}\to B×B$ is a path object for $B$. This limit coincides, up to isomorphism, with the pullback

$\begin{array}{ccc}X{×}_{B}^{h}Y& \to & {B}^{I}\\ ↓& & ↓\\ X×Y& \stackrel{\left(f,g\right)}{\to }& B×B\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ X \times_B^h Y &\to& B^I \\ \downarrow && \downarrow \\ X \times Y &\stackrel{(f,g)}{\to}& B \times B } \,.

This implies in particular that the homotopy fiber of $X{×}_{B}^{h}Y\to X×Y$ is the loop space object $\Omega B$, being the fiber of the path space object projection.

### Over an $\infty$-group

We consider the special case where $B$ is an abelian ∞-group in a presentable (∞,1)-category $𝒞$.

In this case we have an (∞,1)-pullback

$\begin{array}{ccc}B& \to & *\\ {↓}^{{\Delta }_{B}}& {⇙}_{\simeq }& {↓}^{0}\\ B×B& \stackrel{-}{\to }& B\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ B &\to& * \\ \downarrow^{\mathrlap{\Delta_B}} &\swArrow_{\simeq}& \downarrow^\mathrlap{0} \\ B \times B &\stackrel{-}{\to}& B } \,,

where the bottom horizontal morphism is the composite

$-:B×B\stackrel{\left(\mathrm{id},\left(-{\right)}^{-1}\right)}{\to }B×B\stackrel{+}{\to }B$- : B \times B \stackrel{(id, (-)^{-1})}{\to} B \times B \stackrel{+}{\to} B

of a morphism that sends the second argument to its inverse with the group composition operation.

###### Example

Let $S$ be a small site and let $𝒞={\mathrm{Sh}}_{\left(\infty ,1\right)}\left(S\right)$ be the (∞,1)-category of (∞,1)-sheaves on $S$.

This is presented by the projective model structure on simplicial presheaves

$𝒞\simeq \left(\left[{S}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj},\mathrm{loc}}{\right)}^{\circ }\phantom{\rule{thinmathspace}{0ex}}.$\mathcal{C} \simeq ([S^{op}, sSet]_{proj, loc})^\circ \,.

As discussed there, the Dold-Kan correspondence prolongs to a Quillen adjunction on presheaves whose right adjoint is

$\Xi :\left[{S}^{\mathrm{op}},{\mathrm{Ch}}_{•\le 0}\left(\mathrm{Ab}\right){\right]}_{\mathrm{proj}}\to \left[{S}^{\mathrm{op}},\mathrm{sAb}{\right]}_{\mathrm{proj}}\to \left[{S}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj}}\phantom{\rule{thinmathspace}{0ex}}.$\Xi : [S^{op}, Ch_{\bullet \leq 0}(Ab)]_{proj} \to [S^{op}, sAb]_{proj} \to [S^{op}, sSet]_{proj} \,.

Let then $B\in 𝒞$ be an object with a presentation in $\left[{S}^{\mathrm{op}},\mathrm{sSet}\right]$ in the image of this $\Xi$. We write $B$ also for this presentation, and hence $B=\Xi \left(\stackrel{˜}{B}\right)$ for some presheaf of chain complexes $\stackrel{˜}{B}$.

We claim now that such $B$ satisfies the above assumption.

To see this, first notice that the evident morphism $-:\stackrel{˜}{B}×\stackrel{˜}{B}\to \stackrel{˜}{B}$ is degreewise an epimorphism, hence it is a fibration in $\left[{S}^{\mathrm{op}},{\mathrm{Ch}}_{•\ge 0}\left(\mathrm{Ab}\right){\right]}_{\mathrm{proj}}$, and since $\Xi$ is right Quillen, so is the corresponding morphism $-:B×B\to B$ in $\left[{S}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj}}$.

Therefore the ordiary pullback of presheaves of chain complexes

$\begin{array}{ccc}\stackrel{˜}{B}& \to & *\\ {↓}^{{\Delta }_{\stackrel{˜}{B}}}& & {↓}^{0}\\ \stackrel{˜}{B}×\stackrel{˜}{B}& \stackrel{-}{\to }& \stackrel{˜}{B}\end{array}$\array{ \tilde B &\to& * \\ \downarrow^{\mathrlap{\Delta_{\tilde B}}} && \downarrow^{\mathrlap{0}} \\ \tilde B \times \tilde B &\stackrel{-}{\to}& \tilde B }

is a homotopy pullback in $\left[{S}^{\mathrm{op}},{\mathrm{Ch}}_{•\ge 0}\left(\mathrm{Ab}\right){\right]}_{\mathrm{proj}}$, as is the ordinary pullback of simplicial presheaves

$\begin{array}{ccc}B& \to & *\\ {↓}^{{\Delta }_{B}}& & {↓}^{0}\\ B×B& \stackrel{-}{\to }& B\end{array}$\array{ B &\to& * \\ \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{\mathrlap{0}} \\ B \times B &\stackrel{-}{\to}& B }

in $\left[{S}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj}}$.

Since ∞-stackification preserves finite (∞,1)-limits, this presents an (∞,1)-pullback also in $𝒞$.

More generally:

###### Example

Let $𝒞$ be an (∞,1)-topos with a 1-site $S$ of definition (a 1-localic (∞,1)-topos).

Then (as discussed there) every ∞-group object in $𝒞$ has a presentation by a presheaf of simplicial groups

$B\in \left[{S}^{\mathrm{op}},\mathrm{sGrp}{\right]}_{\mathrm{proj}}\to \left[{S}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj}}\phantom{\rule{thinmathspace}{0ex}}.$B \in [S^{op}, sGrp]_{proj} \to [S^{op}, sSet]_{proj} \,.

We claim that the canonical morphism $-:B×B\to B$ is objectwise a Kan fibration and hence a fibration in the projective model structure on simplicial presheaves.

Let $U\in S$ be any test object. A diagram

$\begin{array}{ccc}\Lambda \left[k{\right]}^{i}& \stackrel{\left(\mathrm{ha},\mathrm{hb}\right)}{\to }& B\left(U\right)×B\left(U\right)\\ {↓}^{j}& & ↓\\ \Delta \left[k\right]& \stackrel{\sigma }{\to }& B\left(U\right)\end{array}$\array{ \Lambda[k]^i &\stackrel{(ha, hb)}{\to}& B(U) \times B(U) \\ \downarrow^{\mathrlap{j}} && \downarrow \\ \Delta[k] &\stackrel{\sigma}{\to}& B(U) }

corresponds to a $k$-cell $\sigma \in B\left(U\right)$ together with a choice of decomposition of the $i$th horn ${j}^{*}\sigma$ as a difference

$\left({j}^{*}\sigma {\right)}_{l}={\mathrm{ha}}_{l}\cdot {\mathrm{hb}}_{l}^{-1}\phantom{\rule{thinmathspace}{0ex}}.$(j^* \sigma)_l = ha_l \cdot hb_l^{-1} \,.

Since $B\left(U\right)$ itself is a Kan complex (being a simplicial group, as discussed there) there is a filler $b:\Delta \left[k\right]\to B\left(U\right)$ of the horn $\mathrm{hb}:\Lambda \left[k{\right]}^{i}\to B\left(U\right)$. Define then

$a:=\sigma \cdot b\phantom{\rule{thinmathspace}{0ex}}.$a := \sigma \cdot b \,.

Since all the face maps are group homomorphisms, this is indeed a filler of $\mathrm{ha}$:

$\begin{array}{rl}{\delta }_{l}\left(a\right)& ={\delta }_{l}\left(\sigma \cdot b\right)\\ & ={\delta }_{l}\left(\sigma \right)\cdot {\delta }_{l}\left(b\right)\\ & ={\delta }_{l}\left(\sigma \right)\cdot {\mathrm{hb}}_{l}\\ & ={\mathrm{ha}}_{l}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} \delta_l(a) & = \delta_l(\sigma \cdot b) \\ & = \delta_l(\sigma) \cdot \delta_l(b) \\ & = \delta_l(\sigma) \cdot hb_l \\ & = ha_l \end{aligned} \,.

Moreover, by construction, $\left(a,b\right)$ is a filler in

$\begin{array}{ccc}\Lambda \left[k{\right]}^{i}& \stackrel{\left(\mathrm{ha},\mathrm{hb}\right)}{\to }& B\left(U\right)×B\left(U\right)\\ {↓}^{i}& {}^{\left(a,b\right)}↗& ↓\\ \Delta \left[k\right]& \stackrel{\sigma }{\to }& B\left(U\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \Lambda[k]^i &\stackrel{(ha, hb)}{\to}& B(U) \times B(U) \\ \downarrow^{\mathrlap{i}} &{}^{(a,b)}\nearrow& \downarrow \\ \Delta[k] &\stackrel{\sigma}{\to}& B(U) } \,.

Since therefore $-:B×B\to B$ is a projective fibration, it follows as before that the ordinary pullback

$\begin{array}{ccc}B& \to & *\\ {↓}^{{\Delta }_{B}}& & {↓}^{e}\\ B×B& \stackrel{-}{\to }& B\end{array}$\array{ B &\to& * \\ \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{e} \\ B \times B &\stackrel{-}{\to}& B }

is a homotopy pullback.

###### Observation

For $B$ an ∞-group object as above, the (∞,1)-pullback $X{×}_{B}Y$ is equivalently given by the $\left(\infty ,1\right)$-pullback

$\begin{array}{ccc}X{×}_{B}Y& \to & *\\ ↓& {⇙}_{\simeq }& {↓}^{0}\\ X×Y& \stackrel{f-g}{\to }& B\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ X \times_B Y &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{0}} \\ X \times Y &\stackrel{f-g}{\to}& B } \,.
###### Proof

By prop. 1 the object $X{×}_{B}Y$ is the $\left(\infty ,1\right)$-pullback in

$\begin{array}{ccc}X{×}_{B}Y& \to & B\\ ↓& {⇙}_{\simeq }& {↓}^{{\Delta }_{B}}\\ X×Y& \stackrel{\left(f,g\right)}{\to }& B×B\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ X \times_B Y &\to& B \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\Delta_B}} \\ X \times Y &\stackrel{(f,g)}{\to}& B \times B } \,.

By the pasting law this is equivalently given by the composite pullback of

$\begin{array}{ccccc}X{×}_{B}Y& \to & B& \to & *\\ ↓& {⇙}_{\simeq }& {↓}^{{\Delta }_{B}}& {⇙}_{\simeq }& {↓}^{0}\\ X×Y& \stackrel{\left(f,g\right)}{\to }& B×B& \stackrel{-}{\to }& B\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ X \times_B Y &\to& B &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\Delta_B}} &\swArrow_{\simeq}& \downarrow^{\mathrlap{0}} \\ X \times Y &\stackrel{(f,g)}{\to}& B \times B &\stackrel{-}{\to}& B } \,.

Here the composite bottom morphism is $\left(f-g\right)$.

## Examples

### (Co)Homology of a cover

A special case of the general Mayer-Vietoris sequence, corollary 1 – which historically was the first case considered – applies to the cohomology/homology of a topological space $X$ equipped with an open cover $\left\{{U}_{1},{U}_{2}\to X\right\}$.

Being a cover means (see effective epimorphism in an (∞,1)-category) that there is a homotopy pushout diagram of the form

$\begin{array}{ccc}U\cap V& ↪& U\\ ↓& & ↓\\ V& \to & X\end{array}$\array{ U \cap V &\hookrightarrow& U \\ \downarrow && \downarrow \\ V &\to& X }

in the (∞,1)-topos $H=$ ∞Grpd/Top.

When this is presented by the standard model structure on simplicial sets or in terms of CW-complexes by the model structure on topological spaces, it is given by an ordinary pushout.

Let then $A\in \infty \mathrm{Grpd}\simeq \mathrm{Top}$ be some coefficient object, for instance an Eilenberg-MacLane object ${B}^{n}G$ (Eilenberg-MacLane space $\cdots \simeq K\left(G,n\right)$) for the definition of ordinary singular cohomology with coefficients in an abelian group $G$.

Then applying the derived hom space functor $H\left(-,A\right):{H}^{\mathrm{op}}\to \infty \mathrm{Grpd}$ yields the (∞,1)-pullback diagram

$\begin{array}{ccc}H\left(X,A\right)& \to & H\left(U,A\right)\\ ↓& & ↓\\ H\left(V,A\right)& \to & H\left(U\cap V,A\right)\end{array}$\array{ \mathbf{H}(X, A) &\to& \mathbf{H}(U,A) \\ \downarrow && \downarrow \\ \mathbf{H}(V,A) &\to& \mathbf{H}(U \cap V, A) }

to which we can apply the homotopical Mayer-Vietoris sequence.

Notice that (as discussed in detail at cohomology) the homotopy groups of the ∞-groupoid $H\left(X,{B}^{n}G\right)$ are the cohomology groups of $X$ with coefficients in $G$

${\pi }_{k}H\left(X,{B}^{n}G\right)\simeq {H}^{n-k}\left(X,G\right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_k \mathbf{H}(X, \mathbf{B}^n G) \simeq H^{n-k}(X, G) \,.

By the above general properties the above homotopy pullback is equivalent to

$H\left(X,A\right)\to H\left(U,A\right)×H\left(V,A\right)\to H\left(U\cap V,A\right)$\mathbf{H}(X,A) \to \mathbf{H}(U,A) \times \mathbf{H}(V,A) \to \mathbf{H}(U \cap V, A)

being a fiber sequence. The corresponding long exact sequence in cohomology (as discussed above) is what is traditionally called the Mayer-Vietoris sequence of the cover of $X$ by $U$ and $V$ in $A$-cohomology.

By duality (see universal coefficient theorem) an analogous statement holds for the homology of $X$, $U$ and $V$.

## References

An original reference is

A more modern review that emphasizes the role of fiber sequences is in

• Eldon Dyer, Joseph Roitberg, Note on sequence of Mayer-Vietoris type, Proceedings of the AMS, volume 80, number 4 (1980) (pdf)

Revised on August 26, 2012 18:47:51 by Urs Schreiber (89.204.137.239)