nLab
Mayer-Vietoris sequence

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 $X{\times }_{B}Y$ be the (∞,1)-pullback

$$\begin{array}{ccc}X{\times }_{B}Y& \to & Y\\ \downarrow & {⇙}_{\simeq }& {\downarrow }^g\\ X& \stackrel{f}{\to }& B\end{array}.$$\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 $X{\times }_{B}Y\to X\times Y$. 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 $X{\times }_{B}Y$ is equivalently given by the (∞,1)-pullback

$$\begin{array}{ccc}X{\times }_{B}Y& \to & B\\ \downarrow & {⇙}_{\simeq }& {\downarrow }^{{\Delta }_B}\\ X\times Y& \stackrel{(f,g)}{\to }& B\times B\end{array},$$\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 $X{\times }_{B}Y\to X\times Y$ 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 $(\mathrm{\infty },1)$-pullbacks it follows that with the left square in

$$\begin{array}{ccccc}\Omega B& \to & X{\times }_{B}Y& \to & B\\ \downarrow & {⇙}_{\simeq }& \downarrow & {⇙}_{\simeq }& \downarrow \\ *& \to & X\times Y& \stackrel{(f,g)}{\to }& B\times B\end{array}$$\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 $(\mathrm{\infty },1)$-pullback, so is the total outer rectangle. But again by the first statement, this is equivalent to the $(\mathrm{\infty },1)$-pullback

$$\begin{array}{ccc}\Omega B& \to & *\\ \downarrow & {⇙}_{\simeq }& \downarrow \\ *& \to & B\end{array},$$\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

$$\Omega B\to X{\times }_{B}Y\to X\times Y.$$\Omega B \to X \times_B Y \to X \times Y \,.

Corollary

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

$$\cdots \to {\pi }_{n+1}B\to {\pi }_{n}X{\times }_{B}Y\stackrel{(f_{*},g_{*})}{\to }{\pi }_{n}X\oplus {\pi }_{n}Y\stackrel{f_{*}-g_{*}}{\to }{\pi }_{n}B\to \cdots$$\cdots \to \pi_{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

$$\cdots \to {\pi }_{2}B\to {\pi }_{1}X{\times }_{B}Y\stackrel{(f_{+},g_{*})}{\to }{\pi }_{1}X\times {\pi }_{1}Y\stackrel{f_{*}\cdot g_{*}^{-1}}{\to }{\pi }_{1}B\to {\pi }_0(X{\times }_{B}Y)\to {\pi }_0(X\times Y).$$\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 $(\mathrm{\infty },1)$-pullback $X{\times }_{B}Y$ is presented by a homotopy pullback, and by the factorization lemma, this is given by the ordinary limit

$$\begin{array}{ccccc}X{\times }_B^{h}Y& \to & & \to & Y\\ \downarrow & & & & {\downarrow }^g\\ & & B^I& \to & B\\ \downarrow & & \downarrow \\ X& \stackrel{f}{\to }& B\end{array},$$\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 $B\stackrel{\simeq }{\to }{B}^I\to B\times B$ is a path object for $B$. This limit coincides, up to isomorphism, with the pullback

$$\begin{array}{ccc}X{\times }_B^{h}Y& \to & B^I\\ \downarrow & & \downarrow \\ X\times Y& \stackrel{(f,g)}{\to }& B\times B\end{array}.$$\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 $X{\times }_B^{h}Y\to X\times Y$ is the loop space object $\Omega B$, being the fiber of the path space object projection.

Over an $\mathrm{\infty }$-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

$$\begin{array}{ccc}B& \to & *\\ {\downarrow }^{{\Delta }_B}& {⇙}_{\simeq }& {\downarrow }^0\\ B\times B& \stackrel{-}{\to }& B\end{array},$$\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

$$-:B\times B\stackrel{(\mathrm{id},(-{)}^{-1})}{\to }B\times B\stackrel{+}{\to }B$$- : 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 $𝒞={\mathrm{Sh}}_{(\mathrm{\infty },1)}(S)$ be the (∞,1)-category of (∞,1)-sheaves on $S$.

This is presented by the projective model structure on simplicial presheaves

$$𝒞\simeq ([S^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}{)}^{\circ }.$$\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

$$\Xi :[S^{\mathrm{op}},{\mathrm{Ch}}_{•\le 0}(\mathrm{Ab}){]}_{\mathrm{proj}}\to [S^{\mathrm{op}},\mathrm{sAb}{]}_{\mathrm{proj}}\to [S^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}.$$\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 $[S^{\mathrm{op}},\mathrm{sSet}]$ in the image of this $\Xi$. We write $B$ also for this presentation, and hence $B=\Xi (\tilde{B})$ for some presheaf of chain complexes $\tilde{B}$.

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

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

Therefore the ordiary pullback of presheaves of chain complexes

$$\begin{array}{ccc}\tilde{B}& \to & *\\ {\downarrow }^{{\Delta }_{\tilde{B}}}& & {\downarrow }^0\\ \tilde{B}\times \tilde{B}& \stackrel{-}{\to }& \tilde{B}\end{array}$$\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 $[S^{\mathrm{op}},{\mathrm{Ch}}_{•\ge 0}(\mathrm{Ab}){]}_{\mathrm{proj}}$, as is the ordinary pullback of simplicial presheaves

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

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

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

$$B\in [S^{\mathrm{op}},\mathrm{sGrp}{]}_{\mathrm{proj}}\to [S^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}.$$B \in [S^{op}, sGrp]_{proj} \to [S^{op}, sSet]_{proj} \,.

We claim that the canonical morphism $-:B\times B\to B$ 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

$$\begin{array}{ccc}\Lambda [k{]}^i& \stackrel{(\mathrm{ha},\mathrm{hb})}{\to }& B(U)\times B(U)\\ {\downarrow }^j& & \downarrow \\ \Delta [k]& \stackrel{\sigma }{\to }& B(U)\end{array}$$\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 $\sigma \in B(U)$ together with a choice of decomposition of the $i$th horn $j^{*}\sigma$ as a difference

$$(j^{*}\sigma {)}_l={\mathrm{ha}}_l\cdot {\mathrm{hb}}_l^{-1}.$$(j^* \sigma)_l = ha_l \cdot hb_l^{-1} \,.

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

$$a:=\sigma \cdot b.$$a := \sigma \cdot b \,.

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

$$\begin{array}{rl}{\delta }_l(a)& ={\delta }_l(\sigma \cdot b)\\ & ={\delta }_l(\sigma )\cdot {\delta }_l(b)\\ & ={\delta }_l(\sigma )\cdot {\mathrm{hb}}_l\\ & ={\mathrm{ha}}_l\end{array}.$$\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, $(a,b)$ is a filler in

$$\begin{array}{ccc}\Lambda [k{]}^i& \stackrel{(\mathrm{ha},\mathrm{hb})}{\to }& B(U)\times B(U)\\ {\downarrow }^i& {}^{(a,b)}\nearrow & \downarrow \\ \Delta [k]& \stackrel{\sigma }{\to }& B(U)\end{array}.$$\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 $-:B\times B\to B$ is a projective fibration, it follows as before that the ordinary pullback

$$\begin{array}{ccc}B& \to & *\\ {\downarrow }^{{\Delta }_B}& & {\downarrow }^e\\ B\times B& \stackrel{-}{\to }& B\end{array}$$\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 $X{\times }_{B}Y$ is equivalently given by the $(\mathrm{\infty },1)$-pullback

$$\begin{array}{ccc}X{\times }_{B}Y& \to & *\\ \downarrow & {⇙}_{\simeq }& {\downarrow }^0\\ X\times Y& \stackrel{f-g}{\to }& B\end{array}.$$\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 $X{\times }_{B}Y$ is the $(\mathrm{\infty },1)$-pullback in

$$\begin{array}{ccc}X{\times }_{B}Y& \to & B\\ \downarrow & {⇙}_{\simeq }& {\downarrow }^{{\Delta }_B}\\ X\times Y& \stackrel{(f,g)}{\to }& B\times B\end{array}.$$\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

$$\begin{array}{ccccc}X{\times }_{B}Y& \to & B& \to & *\\ \downarrow & {⇙}_{\simeq }& {\downarrow }^{{\Delta }_B}& {⇙}_{\simeq }& {\downarrow }^0\\ X\times Y& \stackrel{(f,g)}{\to }& B\times B& \stackrel{-}{\to }& B\end{array}.$$\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 $(f-g)$.

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 $\{U_1,U_2\to X\}$.

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

$$\begin{array}{ccc}U\cap V& \hookrightarrow & U\\ \downarrow & & \downarrow \\ V& \to & X\end{array}$$\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 $A\in \mathrm{\infty }\mathrm{Grpd}\simeq \mathrm{Top}$ be some coefficient object, for instance an Eilenberg-MacLane object $B^{n}G$ (Eilenberg-MacLane space $\cdots \simeq K(G,n)$) for the definition of ordinary singular cohomology with coefficients in an abelian group $G$.

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

$$\begin{array}{ccc}H(X,A)& \to & H(U,A)\\ \downarrow & & \downarrow \\ H(V,A)& \to & H(U\cap V,A)\end{array}$$\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 $H(X,B^{n}G)$ are the cohomology groups of $X$ with coefficients in $G$

$${\pi }_{k}H(X,B^{n}G)\simeq H^{n-k}(X,G).$$\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

$$H(X,A)\to H(U,A)\times H(V,A)\to H(U\cap V,A)$$\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

• Beno Eckmann and Peter Hilton, Unions and intersections in homotopy theory, Comment. Math. Helv. 3 (1964),2 93-307.

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)