|
|
| Line 1: |
Line 1: |
| In [[mathematics]], particularly [[algebraic topology]] and [[homology theory]], the '''Mayer–Vietoris sequence''' is an [[algebra]]ic tool to help compute [[algebraic invariant]]s of [[topological space]]s, known as their [[Homology group|homology]] and [[cohomology group]]s. The result is due to two [[Austria]]n mathematicians, [[Walther Mayer]] and [[Leopold Vietoris]]. The method consists of splitting a space into pieces, called [[Subspace topology|subspaces]], for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a [[Natural (category theory)|natural]] [[long exact sequence]], whose entries are the (co)homology groups of the whole space, the [[direct sum of abelian groups|direct sum]] of the (co)homology groups of the subspaces, and the (co)homology groups of the [[intersection (set theory)|intersection]] of the subspaces.
| |
|
| |
|
| The Mayer–Vietoris sequence holds for a variety of [[cohomology theory|cohomology]] and [[homology theory|homology theories]], including [[singular homology]] and [[singular cohomology]]. In general, the sequence holds for those theories satisfying the [[Eilenberg–Steenrod axioms]], and it has variations for both [[Reduced homology|reduced]] and [[Relative homology|relative]] (co)homology. Because the (co)homology of most spaces cannot be computed directly from their definitions, one uses tools such as the Mayer–Vietoris sequence in the hope of obtaining partial information. Many spaces encountered in [[topology]] are constructed by piecing together very simple patches. Carefully choosing the two covering subspaces so that, together with their intersection, they have simpler (co)homology than that of the whole space may allow a complete deduction of the (co)homology of the space. In that respect, the Mayer–Vietoris sequence is analogous to the [[Seifert–van Kampen theorem]] for the [[fundamental group]], and a precise relation exists for homology of dimension one.
| |
|
| |
|
| ==Background, motivation, and history==
| | Adrianne is what you has the [http://Www.alexa.com/search?q=capability&r=topsites_index&p=bigtop capability] to call me but We all don't like when everyone use my full name. What I love doing is to play croquet coupled with now I have your time to take on issues. The job I've been occupying suitable for years is an shop clerk. My [http://Www.Dict.cc/englisch-deutsch/husband.html husband] and I opt to reside in Guam but I will own to move in the new year or two. You can find my website here: http://circuspartypanama.com<br><br>Feel free to visit my site :: [http://circuspartypanama.com clash of clans unlimited gems no survey] |
| | |
| [[Image:Vietoris4343.jpg|Right|thumb|Leopold Vietoris on his 110th birthday]]
| |
| | |
| Like the [[fundamental group]] or the higher [[homotopy group]]s of a space, homology groups are important topological invariants. Although some (co)homology theories are computable using tools of [[linear algebra]], many other important (co)homology theories, especially singular (co)homology, are not computable directly from their definition for nontrivial spaces. For singular (co)homology, the singular (co)chains and (co)cycles groups are often too big to handle directly. More subtle and indirect approaches become necessary. The Mayer–Vietoris sequence is such an approach, giving partial information about the (co)homology groups of any space by relating it to the (co)homology groups of two of its subspaces and their intersection.
| |
| | |
| The most natural and convenient way to express the relation involves the algebraic concept of [[exact sequence]]s: sequences of [[Object (category theory)|objects]] (in this case [[Group (mathematics)|groups]]) and [[morphism]]s (in this case [[group homomorphism]]s) between them such that the [[Image (mathematics)|image]] of one morphism equals the [[Kernel (algebra)|kernel]] of the next. In general, this does not allow (co)homology groups of a space to be completely computed. However, because many important spaces encountered in topology are [[topological manifold]]s, [[simplicial complex]]es, or [[CW complex]]es, which are constructed by piecing together very simple patches, a theorem such as that of Mayer and Vietoris is potentially of broad and deep applicability.
| |
| | |
| Mayer was introduced to topology by his colleague Vietoris when attending his lectures in 1926 and 1927 at a local university in [[Vienna]].<ref>{{harvnb|Hirzebruch|1999}}</ref> He was told about the conjectured result and a way to its solution, and solved the question for the [[Betti number]]s in 1929.<ref>{{harvnb|Mayer|1929}}</ref> He applied his results to the [[torus]] considered as the union of two cylinders.<ref>{{harvnb|Dieudonné|1989|p=39}}</ref><ref>{{harvnb|Mayer|1929|p=41}}</ref> Vietoris later proved the full result for the homology groups in 1930 but did not express it as an exact sequence.<ref>{{harvnb|Vietoris|1930}}</ref> The concept of an exact sequence only appeared in print in the 1952 book ''Foundations of Algebraic Topology'' by [[Samuel Eilenberg]] and [[Norman Steenrod]]<ref>{{harvnb|Corry|2004|p=345}}</ref> where the results of Mayer and Vietoris were expressed in the modern form.<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=Theorem 15.3}}</ref>
| |
| {{-}}
| |
| | |
| ==Basic versions for singular homology==
| |
| Let ''X'' be a [[topological space]] and ''A'', ''B'' be two subspaces whose [[Interior (topology)|interiors]] cover ''X''. (The interiors of ''A'' and ''B'' need not be disjoint.) The Mayer–Vietoris sequence in [[singular homology]] for the triad (''X'', ''A'', ''B'') is a [[long exact sequence]] relating the singular homology groups (with coefficient group the integers '''Z''') of the spaces ''X'', ''A'', ''B'', and the [[intersection (set theory)|intersection]] ''A''∩''B''.<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=§15}}</ref> There is an unreduced and a reduced version.
| |
| | |
| ===Unreduced version===
| |
| For unreduced homology, the Mayer–Vietoris sequence states that the following sequence is exact:<ref name="Hatcher149">{{harvnb|Hatcher|2002|p=149}}</ref>
| |
| | |
| <br /><math>\begin{align}
| |
| \cdots\rightarrow H_{n+1}(X)\,&\xrightarrow{\partial_*}\,H_{n}(A\cap B)\,\xrightarrow{(i_*,j_*)}\,H_{n}(A)\oplus H_{n}(B)\,\xrightarrow{k_* - l_*}\,H_{n}(X)\xrightarrow{\partial_*}\\
| |
| &\quad\xrightarrow{\partial_*}\,H_{n-1} (A\cap B)\rightarrow \cdots\rightarrow H_0(A)\oplus H_0(B)\,\xrightarrow{k_* - l_*}\,H_0(X)\rightarrow\,0. | |
| \end{align}</math>
| |
| | |
| Here the maps ''i'' : ''A''∩''B'' ↪ ''A'', ''j'' : ''A''∩''B'' ↪ ''B'', ''k'' : ''A'' ↪ ''X'', and ''l'' : ''B'' ↪ ''X'' are [[inclusion map]]s and <math>\oplus</math> denotes the [[direct sum of abelian groups]].
| |
| | |
| ===Boundary map===
| |
| [[Image:Mayer Vietoris sequence boundary map on torus.png|thumb|280px|right|Illustration of the boundary map ∂<sub>*</sub> on the torus where the 1-cycle ''x'' = ''u'' + ''v'' is the sum of two 1-chains whose boundary lies in the intersection of ''A'' and ''B''.]]
| |
| The boundary maps ∂<sub>*</sub> lowering the dimension may be made explicit as follows.<ref name="Hatcher 2002 150">{{harvnb|Hatcher|2002|p=150}}</ref> An element in ''H''<sub>n</sub>(''X'') is the homology class of an ''n''-cycle ''x'' which, by [[barycentric subdivision]] for example, can be written as the sum of two ''n''-chains ''u'' and ''v'' whose images lie wholly in ''A'' and ''B'', respectively. Thus ∂''x'' = ∂(''u'' + ''v'') = 0 so that ∂''u'' = −∂''v''. This implies that the images of both these boundary (''n'' − 1)-cycles are contained in the intersection ''A''∩''B''. Then ∂<sub>*</sub>([''x'']) is the class of ∂''u'' in ''H''<sub>n−1</sub>(''A''∩''B''). Choosing a different representative ''x′'' does not affect ∂''u'' since ∂''x′'' = ∂''x'' = 0; nor does choosing another decomposition ''x'' = ''u′'' + ''v′'' since then ∂''u'' + ∂''v'' − ∂''u′'' − ∂''v′'' = 0 which implies ∂''u'' = ∂''u′'' and ∂''v'' = ∂''v′''. Notice that the maps in the Mayer–Vietoris sequence depend on choosing an order for ''A'' and ''B''. In particular, the boundary map changes sign if ''A'' and ''B'' are swapped.
| |
| | |
| ===Reduced version===
| |
| For [[reduced homology]] there is also a Mayer–Vietoris sequence, under the assumption that ''A'' and ''B'' have [[non-empty]] intersection.<ref>{{harvnb|Spanier|1966|p=187}}</ref> The sequence is identical for positive dimensions and ends as:
| |
| | |
| <br /><math>\cdots\rightarrow\tilde{H}_0(A\cap B)\,\xrightarrow{(i_*,j_*)}\,\tilde{H}_0(A)\oplus\tilde{H}_0(B)\,\xrightarrow{k_* - l_*}\,\tilde{H}_0(X)\rightarrow\,0.</math>
| |
| | |
| ===Analogy with the Seifert–van Kampen theorem===
| |
| There is an analogy between the Mayer–Vietoris sequence (especially for homology groups of dimension 1) and the [[Seifert–van Kampen theorem]].<ref name="Hatcher 2002 150"/><ref>{{harvnb|Massey|1984|p=240}}</ref> Whenever ''A''∩''B'' is [[path-connected]] the reduced Mayer–Vietoris sequence yields the isomorphism
| |
| | |
| :<math>H_1(X) \cong (H_1(A)\oplus H_1(B))/\text{Ker} (k_* - l_*)</math>
| |
| | |
| where, by exactness,
| |
| | |
| :<math>\text{Ker} (k_* - l_*) \cong \text{Im} (i_*, j_*).</math>
| |
| | |
| This is precisely the [[Commutator subgroup#Abelianization|abelianized]] statement of the Seifert–van Kampen theorem. Compare with the fact that ''H''<sub>1</sub>(''X'') is the abelianization of the [[fundamental group]] π<sub>1</sub>(''X'') when ''X'' is path-connected.<ref>{{harvnb|Hatcher|2002|loc=Theorem 2A.1, p. 166}}</ref>
| |
| | |
| ==Basic applications==
| |
| | |
| ===''k''-sphere===
| |
| [[Image:SphereCoverStriped.png|thumb|250px|right|The decomposition for ''X'' = ''S''<sup>2</sup>]]
| |
| To completely compute the homology of the [[n-sphere|''k''-sphere]] ''X'' = ''S''<sup>''k''</sup>, let ''A'' and ''B'' be two hemispheres of ''X'' with intersection [[homotopy equivalent]] to a (''k'' − 1)-dimensional equatorial sphere. Since the ''k''-dimensional hemispheres are [[homeomorphic]] to ''k''-discs, which are [[contractible]], the homology groups for ''A'' and ''B'' are [[Trivial group|trivial]]. The Mayer–Vietoris sequence for [[reduced homology]] groups then yields
| |
| | |
| :<br /><math> \cdots\rightarrow 0 \rightarrow \tilde{H}_{n}\left(S^k\right) \xrightarrow{\partial_*}\, \tilde{H}_{n-1}\left(S^{k-1}\right) \rightarrow 0 \rightarrow \cdots \! </math>
| |
| | |
| Exactness immediately implies that the map ∂<sub>*</sub> is an isomorphism. Using the [[reduced homology]] of the [[0-sphere]] (two points) as a [[Mathematical induction|base case]], it follows<ref>{{harvnb|Hatcher|2002|loc=Example 2.46, p. 150}}</ref>
| |
| | |
| :<br /><math>\tilde{H}_n\left(S^k\right)\cong\delta_{kn}\,\mathbb{Z}=\left\{\begin{matrix} | |
| \mathbb{Z} & \mbox{if } n=k \\
| |
| 0 & \mbox{if } n \ne k \end{matrix}\right.</math>
| |
| | |
| where δ is the [[Kronecker delta]]. Such a complete understanding of the homology groups for spheres is in stark contrast with current knowledge of [[homotopy groups of spheres]], especially for the case ''n'' > ''k'' about which little is known.<ref>{{harvnb|Hatcher|2002|p=384}}</ref>
| |
| {{-}}
| |
| | |
| ===Klein bottle===
| |
| [[Image:KleinBottle2D covered by Möbius strips.svg|thumb|200px|right|The Klein bottle ([[fundamental polygon]] with appropriate edge identifications) decomposed as two Möbius strips ''A'' (in blue) and ''B'' (in red).]]
| |
| A slightly more difficult application of the Mayer–Vietoris sequence is the calculation of the homology groups of the [[Klein bottle]] ''X''. One uses the decomposition of ''X'' as the union of two [[Möbius strip]]s ''A'' and ''B'' [[Quotient space|glued]] along their boundary circle (see illustration on the right). Then ''A'', ''B'' and their intersection ''A''∩''B'' are [[Homotopy#Homotopy equivalence and null-homotopy|homotopy equivalent]] to circles, so the nontrivial part of the sequence yields<ref>{{harvnb|Hatcher|2002|p=151}}</ref>
| |
| | |
| :<br /><math> 0 \rightarrow H_{2}(X) \rightarrow\, \mathbb{Z}\ \xrightarrow{\alpha} \ \mathbb{Z} \oplus \mathbb{Z} \rightarrow \, H_1(X) \rightarrow 0 \! </math>
| |
| | |
| and the trivial part implies vanishing homology for dimensions greater than 2. The central map α sends 1 to (2, −2) since the boundary circle of a Möbius band wraps twice around the core circle. In particular α is [[Injective function|injective]] so homology of dimension 2 also vanishes. Finally, choosing (1, 0) and (1, −1) as a basis for '''Z'''<sup>2</sup>, it follows
| |
| | |
| :<br /><math>\tilde{H}_n\left(X\right)\cong\delta_{1n}\,(\mathbb{Z}\oplus\mathbb{Z}_2)=\left\{\begin{matrix}
| |
| \mathbb{Z}\oplus\mathbb{Z}_2 & \mbox{if } n=1\\
| |
| 0 & \mbox{if } n\ne1 \end{matrix}\right.
| |
| </math>
| |
| {{-}}
| |
| | |
| ===Wedge sums===
| |
| [[Image:WedgeSumSpheres.png|right|300px|thumb|This decomposition of the wedge sum ''X'' of two 2-spheres ''K'' and ''L'' yields all the homology groups of ''X''.]]
| |
| Let ''X'' be the [[wedge sum]] of two spaces ''K'' and ''L'', and suppose furthermore that the identified [[basepoint]] is a [[deformation retract]] of [[Neighbourhood (mathematics)|open neighborhoods]] ''U'' ⊂ ''K'' and ''V'' ⊂ ''L''. Letting ''A'' = ''K''∪''V'' and ''B'' = ''U''∪''L'' it follows that ''A''∪''B'' = ''X'' and ''A''∩''B'' = ''U''∪''V'', which is [[contractible]] by construction. The reduced version of the sequence then yields (by exactness)<ref>{{harvnb|Hatcher|2002|loc=Exercise 31}}</ref>
| |
| :<math>\tilde{H}_n(K\vee L)\cong \tilde{H}_n(K)\oplus\tilde{H}_n(L)</math>
| |
| for all dimensions ''n''. The illustration on the right shows ''X'' as the sum of two 2-spheres ''K'' and ''L''. For this specific case, using the result [[Mayer–Vietoris sequence#k-sphere|from above]] for 2-spheres, one has
| |
| :<math>\tilde{H}_n\left(S^2\vee S^2\right)\cong\delta_{2n}\,(\mathbb{Z}\oplus\mathbb{Z})=\left\{\begin{matrix}
| |
| \mathbb{Z}\oplus\mathbb{Z} & \mbox{if } n=2 \\
| |
| 0 & \mbox{if } n \ne 2 \end{matrix}\right.</math>
| |
| {{-}}
| |
| | |
| ===Suspensions===
| |
| [[Image:0-Sphere Suspension - Mayer-Vietoris Cover.svg|right|500px|thumb|This decomposition of the suspension ''X'' of the 0-sphere ''Y'' yields all the homology groups of ''X''.]]
| |
| If ''X'' is the [[Suspension (topology)|suspension]] ''SY'' of a space ''Y'', let ''A'' and ''B'' be the [[Complement (set theory)|complements]] in ''X'' of the top and bottom 'vertices' of the double cone, respectively. Then ''X'' is the union ''A''∪''B'', with ''A'' and ''B'' contractible. Also, the intersection ''A''∩''B'' is homotopy equivalent to ''Y''. Hence the Mayer–Vietoris sequence yields, for all ''n'',<ref>{{harvnb|Hatcher|2002|loc=Exercise 32}}</ref>
| |
| :<math>\tilde{H}_n(SY)\cong \tilde{H}_{n-1}(Y)</math>
| |
| | |
| The illustration on the right shows the 1-sphere ''X'' as the suspension of the 0-sphere ''Y''. Noting in general that the ''k''-sphere is the suspension of the (''k'' − 1)-sphere, it is easy to derive the homology groups of the ''k''-sphere by induction, [[Mayer–Vietoris sequence#k-sphere|as above]].
| |
| {{-}}
| |
| | |
| ==Further discussion==
| |
| | |
| ===Relative form===
| |
| A [[relative homology|relative]] form of the Mayer–Vietoris sequence also exists. If ''Y'' ⊂ ''X'' and is the union of ''C'' ⊂ ''A'' and ''D'' ⊂ ''B'', then the exact sequence is:<ref>{{harvnb|Hatcher|2002|p=152}}</ref>
| |
| | |
| <br /><math>\cdots\rightarrow H_{n}(A\cap B,C\cap D)\,\xrightarrow{(i_*,j_*)}\,H_{n}(A,C)\oplus H_{n}(B,D)\,\xrightarrow{k_* - l_*}\,H_{n}(X,Y)\,\xrightarrow{\partial_*}\,H_{n-1}(A\cap B,C\cap D)\rightarrow\cdots</math>
| |
| | |
| ===Naturality===
| |
| The homology groups are [[Natural (category theory)|natural]] in the sense that if ''ƒ'' is a [[Continuous function (topology)|continuous]] map from ''X''<sub>1</sub> to ''X''<sub>2</sub>, then there is a canonical [[pushforward (homology)|pushforward]] map ''ƒ''<sub>∗</sub> of homology groups ''ƒ''<sub>∗</sub> : ''H''<sub>''k''</sub>(''X''<sub>1</sub>) → ''H''<sub>''k''</sub>(''X''<sub>2</sub>), such that the composition of pushforwards is the pushforward of a composition: that is, <math>(g\circ h)_* = g_*\circ h_*</math>. The Mayer–Vietoris sequence is also natural in the sense that if ''X''<sub>1</sub> = ''A''<sub>1</sub>∪''B''<sub>1</sub> to ''X''<sub>2</sub> = ''A''<sub>2</sub>∪''B''<sub>2</sub> and the mapping ''ƒ'' satisfies ''ƒ''(''A''<sub>1</sub>) ⊂ ''A''<sub>2</sub> and ''ƒ''(''B''<sub>1</sub>) ⊂ ''B''<sub>2</sub>, then the connecting morphism ∂<sub>∗</sub> of the Mayer–Vietoris sequence commutes with ''ƒ''<sub>∗</sub>.<ref>{{harvnb|Massey|1984|p=208}}</ref> That is,<ref>{{harvnb|Eilenberg|Steenrod|1952|loc=Theorem 15.4}}</ref> the following diagram [[Commutative diagram|commutes]] (the horizontal maps are the usual ones):
| |
| [[Image:Mayer-Vietoris naturality.png|center|740px]]
| |
| | |
| ===Cohomological versions===
| |
| | |
| The Mayer–Vietoris long exact sequence for [[singular cohomology]] groups with coefficient [[group (mathematics)|group]] ''G'' is [[Duality (mathematics)|dual]] to the homological version. It is the following:<ref>{{harvnb|Hatcher|2002|p=203}}</ref>
| |
| | |
| <br /><math>\cdots\rightarrow H^{n}(X;G)\rightarrow H^{n}(A;G)\oplus H^{n}(B;G)\rightarrow H^{n}(A\cap B;G)\rightarrow H^{n+1}(X;G)\rightarrow\cdots</math> | |
| | |
| where the dimension preserving maps are restriction maps induced from inclusions, and the (co-)boundary maps are defined in a similar fashion to the homological version. There is also a relative formulation.
| |
| | |
| As an important special case when ''G'' is the group of [[real number]]s '''R''' and the underlying topological space has the additional structure of a [[smooth manifold]], the Mayer–Vietoris sequence for [[de Rham cohomology]] is
| |
| | |
| <br /><math>\cdots\rightarrow H^{n}(X)\,\xrightarrow{\rho}\,H^{n}(U)\oplus H^{n}(V)\,\xrightarrow{\Delta}\,H^{n}(U\cap V)\,\xrightarrow{d^*}\,H^{n+1}(X)\rightarrow\cdots</math> | |
| | |
| where {''U'', ''V''} is an [[open cover]] of ''X'', ''ρ'' denotes the restriction map, and Δ is the difference. The map ''d*'' is defined similarly as the map ''∂''<sub>*</sub> from above. It can be briefly described as follows. For a cohomology class [''ω''] represented by [[closed and exact differential forms|closed form]] ''ω'' in ''U''∩''V'', express ''ω'' as a difference of forms ''ω<sub>U</sub>'' - ''ω<sub>V</sub>'' via a [[partition of unity]] subordinate to the open cover {''U'', ''V''}, for example. The exterior derivative ''dω<sub>U</sub>'' and ''dω<sub>V</sub>'' agree on ''U''∩''V'' and therefore together define an ''n'' + 1 form ''σ'' on ''X''. One then has ''d*''([''ω'']) = [''σ''].
| |
| | |
| ===Derivation===
| |
| Consider the [[Homological algebra#Functoriality|long exact sequence associated to]] the [[short exact sequence]]s of [[chain group]]s (constituent groups of [[chain complex]]es)
| |
| | |
| :<math>0 \rightarrow C_n(A\cap B)\,\xrightarrow{\alpha}\,C_n(A) \oplus C_n(B)\,\xrightarrow{\beta}\,C_n(A+B) \rightarrow 0 </math>
| |
| | |
| where α(''x'') = (''x'', −''x''), β(''x'', ''y'') = ''x'' + ''y'', and ''C''<sub>''n''</sub>(''A'' + ''B'') is the chain group consisting of sums of chains in ''A'' and chains in ''B''.<ref name="Hatcher149"/> It is a fact that the singular ''n''-simplices of ''X'' whose images are contained in either ''A'' or ''B'' generate all of the homology group ''H''<sub>''n''</sub>(''X'').<ref>{{harvnb|Hatcher|2002|loc=Proposition 2.21, p. 119}}</ref> In other words, ''H''<sub>''n''</sub>(''A'' + ''B'') is isomorphic to ''H''<sub>''n''</sub>(''X''). This gives the Mayer–Vietoris sequence for singular homology.
| |
| | |
| The same computation applied to the short exact sequences of vector spaces of [[differential form]]s
| |
| | |
| :<math>
| |
| 0\rightarrow\Omega^{n}(X)\rightarrow\Omega^{n}(U)\oplus\Omega^{n}(V)\rightarrow\Omega^{n}(U\cap V)\rightarrow0
| |
| </math>
| |
| | |
| yields the Mayer–Vietoris sequence for de Rham cohomology.<ref>{{harvnb|Bott|Tu|1982|loc=§I.2}}</ref>
| |
| | |
| From a formal point of view, the Mayer–Vietoris sequence can be derived from the [[Eilenberg–Steenrod axioms]] for [[homology theory|homology theories]] using the [[long exact sequence in homology]], at least for CW complexes.<ref>{{harvnb|Hatcher|2002|p=162}}</ref>
| |
| | |
| ===Other homology theories===
| |
| The derivation of the Mayer–Vietoris sequence from the Eilenberg–Steenrod axioms does not require the [[dimension axiom]],<ref>{{harvnb|Kōno|Tamaki|2006|pp=25–26}}</ref> so in addition to existing in [[List of cohomology theories#Ordinary homology theories|ordinary cohomology theories]], it holds in [[extraordinary cohomology theories]] (such as [[topological K-theory]] and [[cobordism]]).
| |
| | |
| ===Sheaf cohomology===
| |
| From the point of view of [[sheaf cohomology]], the Mayer–Vietoris sequence is related to [[Čech cohomology]]. Specifically, it arises from the [[Spectral sequence|degeneration]] of the [[spectral sequence]] that relates Čech cohomology to sheaf cohomology (sometimes called the [[Mayer–Vietoris spectral sequence]]) in the case where the open cover used to compute the Čech cohomology consists of two open sets.<ref>{{harvnb|Dimca|2004|pp=35–36}}</ref> This spectral sequence exists in arbitrary [[Topos|topoi]].<ref>{{harvnb|Verdier|1972}} (SGA 4.V.3)</ref>
| |
| | |
| ==See also==
| |
| *[[Excision theorem]]
| |
| *[[Zig-zag lemma]]
| |
| | |
| ==Notes==
| |
| {{reflist|colwidth=30em}}
| |
| | |
| ==References==
| |
| | |
| *{{citation
| |
| |last1=Bott
| |
| |first1=Raoul
| |
| |author1-link=Raoul Bott
| |
| |last2=Tu
| |
| |first2=Loring W.
| |
| |title=Differential Forms in Algebraic Topology
| |
| |publisher=[[Springer Science+Business Media|Springer-Verlag]]
| |
| |location=Berlin, New York
| |
| |isbn=978-0-387-90613-3
| |
| |year=1982}}.
| |
| | |
| *{{citation
| |
| |first= Leo
| |
| |last= Corry
| |
| |authorlink= Leo Corry
| |
| |title= Modern Algebra and the Rise of Mathematical Structures
| |
| |publisher= Birkhäuser
| |
| |year= 2004
| |
| |page= 345
| |
| |isbn= 3-7643-7002-5
| |
| }}.
| |
| | |
| *{{citation
| |
| |first= Jean
| |
| |last= Dieudonné
| |
| |authorlink= Jean Dieudonné
| |
| |title= A History of Algebraic and Differential Topology 1900–1960
| |
| |publisher= Birkhäuser
| |
| |year= 1989
| |
| |page= 39
| |
| |isbn= 0-8176-3388-X
| |
| }}.
| |
| | |
| *{{citation
| |
| | last=Dimca
| |
| | first=Alexandru
| |
| | title=Sheaves in topology
| |
| | publisher=[[Springer-Verlag]]
| |
| | year=2004
| |
| | location=Berlin
| |
| | series=Universitext
| |
| | isbn=978-3-540-20665-1
| |
| | mr=2050072
| |
| }}
| |
| | |
| * {{citation
| |
| |last1=Eilenberg
| |
| |first1=Samuel
| |
| |authorlink1=Samuel Eilenberg
| |
| |last2=Steenrod
| |
| |first2=Norman
| |
| |authorlink2=Norman Steenrod
| |
| |title=Foundations of Algebraic Topology
| |
| |year=1952
| |
| |isbn=978-0-691-07965-3
| |
| |publisher= [[Princeton University Press]]
| |
| }}.
| |
| | |
| *{{citation
| |
| |first= Allen
| |
| |last= Hatcher
| |
| |author-link= Allen Hatcher
| |
| |title= Algebraic Topology
| |
| |url= http://www.math.cornell.edu/%7Ehatcher/AT/ATpage.html
| |
| |year= 2002
| |
| |publisher= [[Cambridge University Press]]
| |
| |isbn= 978-0-521-79540-1
| |
| |mr= 1867354
| |
| }}.
| |
| | |
| *{{citation
| |
| |title= The Heritage of Emmy Noether
| |
| |last=Hirzebruch
| |
| |first=Friedrich
| |
| |authorlink=Friedrich Hirzebruch
| |
| |contribution=Emmy Noether and Topology
| |
| |pages=61–63
| |
| |editor= Teicher, M.
| |
| |series= Israel Mathematical Conference Proceedings
| |
| |publisher= [[Bar-Ilan University]]/[[American Mathematical Society]]/[[Oxford University Press]]
| |
| |year= 1999
| |
| |isbn= 978-0-19-851045-1
| |
| |oclc= 223099225
| |
| }}.
| |
| | |
| *{{citation
| |
| |last=Kōno
| |
| |first=Akira
| |
| |last2=Tamaki
| |
| |first2=Dai
| |
| |title=Generalized cohomology
| |
| |publisher=[[American Mathematical Society]]
| |
| |location=Providence, RI
| |
| |series=Iwanami Series in Modern Mathematics, Translations of Mathematical Monographs
| |
| |volume=230
| |
| |year=2006
| |
| |origyear=2002
| |
| |edition=Translated from the 2002 Japanese edition by Tamaki
| |
| |isbn=978-0-8218-3514-2
| |
| |mr=2225848
| |
| }}
| |
| | |
| *{{citation
| |
| |first= William
| |
| |last= Massey
| |
| |author-link= William S. Massey
| |
| |title= Algebraic Topology: An Introduction
| |
| |year= 1984
| |
| |publisher= [[Springer Science+Business Media|Springer-Verlag]]
| |
| |isbn= 978-0-387-90271-5
| |
| }}.
| |
| | |
| *{{citation
| |
| |first= Walther
| |
| |last= Mayer
| |
| |author-link= Walther Mayer
| |
| |title= Über abstrakte Topologie
| |
| |year= 1929
| |
| |journal= [[Monatshefte für Mathematik]]
| |
| |url= http://www.springerlink.com/content/x33611021p942518/
| |
| |doi= 10.1007/BF02307601
| |
| |issn= 0026-9255
| |
| |volume= 36
| |
| |issue= 1
| |
| |pages= 1–42
| |
| }}. {{de icon}}
| |
| | |
| *{{citation
| |
| |first= Edwin
| |
| |last= Spanier
| |
| |author-link= Edwin Spanier
| |
| |title= Algebraic Topology
| |
| |year= 1966
| |
| |publisher= [[Springer Science+Business Media|Springer-Verlag]]
| |
| |isbn= 0-387-94426-5
| |
| }}.
| |
| | |
| *{{citation
| |
| |first=Jean-Louis
| |
| |last=Verdier
| |
| |author-link=Jean-Louis Verdier
| |
| |contribution=Cohomologie dans les topos
| |
| |editor1-first=Michael
| |
| |editor1-last=Artin
| |
| |editor1-link=Michael Artin
| |
| |editor2-first=Alexander
| |
| |editor2-last=Grothendieck
| |
| |editor2-link=Alexander Grothendieck
| |
| |editor3-first=Jean-Louis
| |
| |editor3-last=Verdier
| |
| |editor3-link=Jean-Louis Verdier
| |
| |title=Séminaire de Géométrie Algébrique du Bois Marie – 1963–64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – Tome 2
| |
| |year=1972
| |
| |publisher = [[Springer Science+Business Media|Springer-Verlag]]
| |
| |location = Berlin; Heidelberg
| |
| |language = French
| |
| |series=[[Lecture Notes in Mathematics]]
| |
| |volume=270
| |
| |isbn=978-3-540-06012-3
| |
| |doi=10.1007/BFb0061320
| |
| |pages=1
| |
| }}
| |
| | |
| *{{citation
| |
| |first= Leopold
| |
| |last= Vietoris
| |
| |author-link= Leopold Vietoris
| |
| |title= Über die Homologiegruppen der Vereinigung zweier Komplexe
| |
| |year= 1930
| |
| |journal= [[Monatshefte für Mathematik]]
| |
| |volume= 37
| |
| |pages= 159–62
| |
| |doi=10.1007/BF01696765
| |
| }}. {{de icon}}
| |
| | |
| ==Further reading==
| |
| * {{citation
| |
| |last1=Reitberger
| |
| |first1=Heinrich
| |
| |title=Leopold Vietoris (1891–2002)
| |
| |url=http://www.ams.org/notices/200210/fea-vietoris.pdf
| |
| |format=PDF|year=2002
| |
| |journal=[[Notices of the American Mathematical Society]]
| |
| |issn=0002-9920
| |
| |volume=49
| |
| |issue=20
| |
| }}.
| |
| | |
| {{good article}}
| |
| | |
| {{DEFAULTSORT:Mayer-Vietoris Sequence}}
| |
| [[Category:Homology theory]]
| |