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Mayer-Vietoris sequence

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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\mathcal{C} be an (∞,1)-category with finite (∞,1)-limits and let X,Y,BX, Y, B be pointed objects and

f:XB f : X \to B

and

g:YB g : Y \to B

be any two morphisms with common codomain preserving the base points. Let X× BYX \times_B Y be the (∞,1)-pullback

X× BY Y g X f B. \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× BYX×YX \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\mathcal{C} be a presentable (∞,1)-category.

Then X× BYX \times_B Y is equivalently given by the (∞,1)-pullback

X× BY B Δ B X×Y (f,g) B×B, \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× BYX×YX \times_B Y \to X \times Y is the loop space object ΩB\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 (,1)(\infty,1)-pullbacks it follows that with the left square in

ΩB X× BY B * X×Y (f,g) B×B \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 (,1)(\infty,1)-pullback, so is the total outer rectangle. But again by the first statement, this is equivalent to the (,1)(\infty,1)-pullback

ΩB * * B, \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

ΩBX× BYX×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

π n+1Bπ nX× BY(f *,g *)π nXπ nYf *g *π nB \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
π 2Bπ 1X× BY(f +,g *)π 1X×π 1Yf *g * 1π 1Bπ 0(X× BY)π 0(X×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\mathcal{C} is presented by a category of fibrant objects CC (for instance the subcategory on the fibrant objects of a model category).

Then the (,1)(\infty,1)-pullback X× BYX \times_B Y is presented by a homotopy pullback, and by the factorization lemma, this is given by the ordinary limit

X× B hY Y g B I B X f B, \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 BB IB×BB \stackrel{\simeq}{\to} B^I \to B \times B is a path object for BB. This limit coincides, up to isomorphism, with the pullback

X× B hY B I X×Y (f,g) B×B. \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 hYX×YX \times_B^h Y \to X \times Y is the loop space object ΩB\Omega B, being the fiber of the path space object projection.

Over an \infty-group

We consider the special case where BB is an abelian ∞-group in a presentable (∞,1)-category 𝒞\mathcal{C}.

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

B * Δ B 0 B×B B, \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(id,() 1)B×B+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 SS be a small site and let 𝒞=Sh (,1)(S)\mathcal{C} = Sh_{(\infty,1)}(S) be the (∞,1)-category of (∞,1)-sheaves on SS.

This is presented by the projective model structure on simplicial presheaves

𝒞([S op,sSet] proj,loc) . \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

Ξ:[S op,Ch 0(Ab)] proj[S op,sAb] proj[S op,sSet] proj. \Xi : [S^{op}, Ch_{\bullet \leq 0}(Ab)]_{proj} \to [S^{op}, sAb]_{proj} \to [S^{op}, sSet]_{proj} \,.

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

We claim now that such BB satisfies the above assumption.

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

Therefore the ordiary pullback of presheaves of chain complexes

B˜ * Δ B˜ 0 B˜×B˜ B˜ \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 op,Ch 0(Ab)] proj[S^{op}, Ch_{\bullet \geq 0}(Ab)]_{proj}, as is the ordinary pullback of simplicial presheaves

B * Δ B 0 B×B B \array{ B &\to& * \\ \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{\mathrlap{0}} \\ B \times B &\stackrel{-}{\to}& B }

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

Since ∞-stackification preserves finite (∞,1)-limits, this presents an (∞,1)-pullback also in 𝒞\mathcal{C}.

More generally:

Example

Let 𝒞\mathcal{C} be an (∞,1)-topos with a 1-site SS of definition (a 1-localic (∞,1)-topos).

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

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

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

Let USU \in S be any test object. A diagram

Λ[k] i (ha,hb) B(U)×B(U) j Δ[k] σ B(U) \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 kk-cell σB(U)\sigma \in B(U) together with a choice of decomposition of the iith horn j *σj^* \sigma as a difference

(j *σ) l=ha lhb l 1. (j^* \sigma)_l = ha_l \cdot hb_l^{-1} \,.

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

a:=σb. a := \sigma \cdot b \,.

Since all the face maps are group homomorphisms, this is indeed a filler of haha:

δ l(a) =δ l(σb) =δ l(σ)δ l(b) =δ l(σ)hb l =ha l. \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)(a,b) is a filler in

Λ[k] i (ha,hb) B(U)×B(U) i (a,b) Δ[k] σ B(U). \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×BB- : B \times B \to B is a projective fibration, it follows as before that the ordinary pullback

B * Δ B e B×B B \array{ B &\to& * \\ \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{e} \\ B \times B &\stackrel{-}{\to}& B }

is a homotopy pullback.

Observation

For BB an ∞-group object as above, the (∞,1)-pullback X× BYX \times_B Y is equivalently given by the (,1)(\infty,1)-pullback

X× BY * 0 X×Y fg B. \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× BYX \times_B Y is the (,1)(\infty,1)-pullback in

X× BY B Δ B X×Y (f,g) B×B. \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

X× BY B * Δ B 0 X×Y (f,g) B×B B. \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 (fg)(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 XX equipped with an open cover {U 1,U 2X}\{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

UV U V X \array{ U \cap V &\hookrightarrow& U \\ \downarrow && \downarrow \\ V &\to& X }

in the (∞,1)-topos H=\mathbf{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 AGrpdTopA \in \infty Grpd \simeq Top be some coefficient object, for instance an Eilenberg-MacLane object B nG\mathbf{B}^n G (Eilenberg-MacLane space K(G,n)\cdots \simeq K(G,n)) for the definition of ordinary singular cohomology with coefficients in an abelian group GG.

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

H(X,A) H(U,A) H(V,A) H(UV,A) \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 nG)\mathbf{H}(X,\mathbf{B}^n G) are the cohomology groups of XX with coefficients in GG

π kH(X,B nG)H nk(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)H(U,A)×H(V,A)H(UV,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 XX by UU and VV in AA-cohomology.

By duality (see universal coefficient theorem) an analogous statement holds for the homology of XX, UU and VV.

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 January 27, 2014 13:23:28 by David Roberts (129.127.252.10)