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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.

See also at homotopy pullback this corollary.

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 homotopy fiber sequence is of the form

ΩBX× BYX×Y. \Omega B \to X \times_B Y \to X \times Y \,.

For 𝒞=\mathcal{C} = ∞Grpd L whe\simeq L_{whe} Top, this point of view is amplified in (Dyer-Roitberg 80).

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 is what has historically been the definition of Mayer-Vietories sequences (Eckmann-Hilton 64).

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 now the case where BB carries the structure of an ∞-group (or just a grouplike H-space object) in a presentable (∞,1)-category or locally Cartesian closed (∞,1)-category 𝒞\mathcal{C}.

In this case (as discussed in a moment), we have an (∞,1)-pullback

B * Δ B e B×B ()() 1 B, \array{ B &\to& * \\ \downarrow^{\mathrlap{\Delta_B}} &\swArrow_{\simeq}& \downarrow^{\mathrlap{e}} \\ B \times B &\stackrel{(-)\cdot (-)^{-1}}{\to}& B } \,,

where the bottom horizontal morphism is the composite

()() 1:B×B(id,() 1)B×BB (-)\cdot (-)^{-1} : B \times B \stackrel{(id, (-)^{-1})}{\to} B \times B \stackrel{\cdot}{\to} B

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

It then follows by the pasting law and prop. 1 that in this case the morphism X× BYX×YX \times_B Y \to X \times Y in the Mayer-Vietoris sequence is itself the homotopy fiber of X×Yfg 1BX \times Y \stackrel{f \cdot g^{-1}}{\longrightarrow} B, hence that we have a long homotopy fiber sequence of the form

ΩBX× BYX×Yfg 1B. \Omega B \longrightarrow X \times_B Y \longrightarrow X \times Y \stackrel{f \cdot g^{-1}}{\longrightarrow} B \,.

First consider two more concrete special cases.

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 ordinary 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}.

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 \colon \Delta[k] \to B(U) of the horn hb:Λ[k] iB(U)hb \colon \Lambda[k]^i \to B(U). Define then

aσb. a \coloneqq \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- \colon 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.

Proposition

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 1 B. \array{ X \times_B Y &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{0}} \\ X \times Y &\stackrel{f \cdot g^{-1}}{\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).

Summing this up:

Proposition

For H\mathbf{H} an (∞,1)-sheaf (∞,1)-topos, BB an ∞-group-object in H\mathbf{H} and f:XBf\colon X \to B and g:YBg \colon Y\to B two morphisms, then there is a long homotopy fiber sequence of the form

ΩBX× BYX×Yfg 1B. \Omega B \longrightarrow X \times_B Y \longrightarrow X \times Y \stackrel{f \cdot g^{-1}}{\longrightarrow} B \,.
Proof

For 𝒞\mathcal{C} an (∞,1)-site of definition, there is a reflection

H[C op,Grpd] \mathbf{H} \stackrel{\longleftarrow}{\hookrightarrow} [C^{op},\infty Grpd]

of H\mathbf{H} into an (∞,1)-category of (∞,1)-presheaves.

By prop. 2 the statement holds in [C op,Grpd][C^{op},\infty Grpd]. Since embedding and reflection both preserve finite (∞,1)-limits, it hence also holds in H\mathbf{H}.

Still more generally and more simply:

Proposition

Let 𝒞\mathcal{C} be a locally Cartesian closed (∞,1)-category. Let GG be an ∞-group object (or just a grouplike H-space-object). Then for ϕ:DG\phi \colon D \longrightarrow G any morphism we have a homotopy pullback square of the form

G×D D ϕ G×G ()() 1 G. \array{ G \times D &\longrightarrow& D \\ \downarrow && \downarrow^{\mathrlap{\phi}} \\ G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G } \,.

(nForum discussion)

Proof

By this discussion we may use homotopy type theory reasoning. Starting out with the discussion at homotopy pullback – In homotopy type theory we obtain

D× G(G×G) = d:D g 1:G g 2:G(g 1g 2 1=ϕ(d)) = d:D g 1:G g 2:G(g 1=ϕ(d)g 2) = d:D g 2:G g 1:G(g 1=ϕ(d)g 2) = d:D g 2:G1 =D×G, \begin{aligned} D\times_G (G\times G) &= \sum_{d:D} \sum_{g_1:G} \sum_{g_2:G} (g_1\cdot g_2^{-1} = \phi(d)) \\ &= \sum_{d:D} \sum_{g_1:G} \sum_{g_2:G} (g_1 = \phi(d)\cdot g_2) \\ &= \sum_{d:D} \sum_{g_2:G} \sum_{g_1:G} (g_1 = \phi(d)\cdot g_2) \\ &= \sum_{d:D} \sum_{g_2:G} \mathbf{1}\\ &= D\times G \end{aligned} \,,

where the second but last step consists of observing a contractible based path space object (see the discussion at factorization lemma).

Corollary

Let 𝒞\mathcal{C} be a locally Cartesian closed (∞,1)-category. Let GG be an ∞-group object (or just a grouplike H-space-object).

Then for f:XGf \colon X \to G and g:YGg \colon Y \to G two morphisms, there is a Mayer-Vietoris-type homotopy fiber sequence

ΩGX× GYX×Yf(g 1)G. \cdots \to \Omega G \longrightarrow X \times_G Y \longrightarrow X \times Y \stackrel{f \cdot (g^{-1})}{\longrightarrow} G \,.
Proof

Use prop. 4 with ϕ\phi being the canonical point, i.e. the inclusion e:*Ge \colon \ast \to G of the neutral element to find the homotopy pullback

G * Δ e G×G ()() 1 G. \array{ G &\longrightarrow& \ast \\ \downarrow^{\mathrlap{\Delta}} && \downarrow^{\mathrlap{e}} \\ G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G } \,.

Then use the pasting law as above.

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 homotopy 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)

Discussion in the context of stable model categories includes

  • Peter May, lemma 5.7 of The additivity of traces in triangulated categories, Adv. Math., 163(1):34-73, 2001 (pdf)

Discussion in the context of homotopy type theory includes

  • E Cavallo et al, Exactness of the Mayer-Vietoris Sequence in Homotopy Type Theory (pdf)

Revised on June 10, 2015 19:28:25 by Urs Schreiber (62.48.248.82)