Mayer–Vietoris Sequence
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
and
homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
, the Mayer–Vietoris sequence is an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
ic tool to help compute
algebraic invariant Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descrip ...
s of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s, known as their
homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
and
cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
s. The result is due to two
Austria Austria, , bar, Östareich officially the Republic of Austria, is a country in the southern part of Central Europe, lying in the Eastern Alps. It is a federation of nine states, one of which is the capital, Vienna, the most populous ...
n mathematicians,
Walther Mayer Walther Mayer (11 March 1887 – 10 September 1948) was an Austrian mathematician, born in Graz, Austria-Hungary. With Leopold Vietoris he is the namesake of the Mayer–Vietoris sequence in topology.. He served as an assistant to Albert Einstei ...
and Leopold Vietoris. The method consists of splitting a space into 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 Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
long exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
, whose entries are the (co)homology groups of the whole space, the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces. The Mayer–Vietoris sequence holds for a variety of
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
and homology theories, including
simplicial homology In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case ...
and
singular cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
. In general, the sequence holds for those theories satisfying the
Eilenberg–Steenrod axioms In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homo ...
, and it has variations for both reduced and
relative Relative may refer to: General use *Kinship and family, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be ''relatives'' Philosophy *Relativism, the concept that ...
(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 In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
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 In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
, and a precise relation exists for homology of dimension one.


Background, motivation, and history

Like the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
or the higher
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
s of a space, homology groups are important topological invariants. Although some (co)homology theories are computable using tools of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, 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 An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...
s: sequences of
objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
(in this case
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
) and
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s (in this case
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
s) between them such that the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of one morphism equals the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
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 In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
s,
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
es, or
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
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 en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = Postal code , postal_code = , timezone = CET , utc_offset = +1 , timezone_DST ...
. He was told about the conjectured result and a way to its solution, and solved the question for the
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s in 1929. He applied his results to the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
considered as the union of two cylinders. Vietoris later proved the full result for the homology groups in 1930 but did not express it as an exact sequence. The concept of an exact sequence only appeared in print in the 1952 book ''Foundations of Algebraic Topology'' by
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to a ...
and
Norman Steenrod Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University of ...
where the results of Mayer and Vietoris were expressed in the modern form.


Basic versions for singular homology

Let ''X'' be a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
and ''A'', ''B'' be two subspaces whose
interiors ''Interiors'' is a 1978 American drama film written and directed by Woody Allen. It stars Kristin Griffith, Mary Beth Hurt, Richard Jordan, Diane Keaton, E. G. Marshall, Geraldine Page, Maureen Stapleton, and Sam Waterston. Allen's first ful ...
cover ''X''. (The interiors of ''A'' and ''B'' need not be disjoint.) The Mayer–Vietoris sequence in
singular homology In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
for the triad (''X'', ''A'', ''B'') is a
long exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
relating the singular homology groups (with coefficient group the integers Z) of the spaces ''X'', ''A'', ''B'', and the intersection ''A''∩''B''. There is an unreduced and a reduced version.


Unreduced version

For unreduced homology, the Mayer–Vietoris sequence states that the following sequence is exact: :\cdots\to H_(X)\,\xrightarrow\,H_(A\cap B)\,\xrightarrow\,H_(A)\oplus H_(B) \, \xrightarrow\, H_(X)\, \xrightarrow\, H_ (A\cap B)\to \cdots : \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \cdots \to H_0(A)\oplus H_0(B)\,\xrightarrow\,H_0(X)\to 0. Here ''i'' : ''A''∩''B'' ↪ ''A'', ''j'' : ''A''∩''B'' ↪ ''B'', ''k'' : ''A'' ↪ ''X'', and ''l'' : ''B'' ↪ ''X'' are
inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iot ...
s and \oplus denotes the
direct sum of abelian groups The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
.


Boundary map

The boundary maps ∂ lowering the dimension may be defined as follows. An element in ''Hn''(''X'') is the homology class of an ''n''-cycle ''x'' which, by
barycentric subdivision In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension on simplicial complexes is a canonical method to refine them. Therefore, the barycentric subdivision is an important tool i ...
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 ∂( 'x'' can be defined to be the class of ∂''u'' in ''H''''n''−1(''A''∩''B''). Choosing another decomposition ''x'' = ''u′'' + ''v′'' does not affect ''u'' since ∂''u'' + ∂''v'' = ∂''x'' = ∂''u′'' + ∂''v′'', which implies ∂''u'' − ∂''u′'' = ∂(''v′'' − ''v''), and therefore ∂''u'' and ∂''u′'' lie in the same homology class; nor does choosing a different representative ''x′'', since then ∂''x′'' = ∂''x'' = 0. 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 In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise stat ...
there is also a Mayer–Vietoris sequence, under the assumption that ''A'' and ''B'' have
non-empty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
intersection. The sequence is identical for positive dimensions and ends as: :\cdots\to\tilde_0(A\cap B)\,\xrightarrow\,\tilde_0(A)\oplus\tilde_0(B)\,\xrightarrow\,\tilde_0(X)\to 0.


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. Whenever A\cap B is
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
, the reduced Mayer–Vietoris sequence yields the isomorphism :H_1(X) \cong (H_1(A)\oplus H_1(B))/\text (k_* - l_*) where, by exactness, :\text (k_* - l_*) \cong \text (i_*, j_*). This is precisely the abelianized statement of the Seifert–van Kampen theorem. Compare with the fact that H_1(X) is the abelianization of the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
\pi_1(X) when X is path-connected.


Basic applications


''k''-sphere

To completely compute the homology of the ''k''-sphere ''X'' = ''S''''k'', let ''A'' and ''B'' be two hemispheres of ''X'' with intersection
homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to a (''k'' − 1)-dimensional equatorial sphere. Since the ''k''-dimensional hemispheres are
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to ''k''-discs, which are
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that ...
, the homology groups for ''A'' and ''B'' are
trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
. The Mayer–Vietoris sequence for
reduced homology In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise stat ...
groups then yields : \cdots \longrightarrow 0 \longrightarrow \tilde_\!\left(S^k\right)\, \xrightarrow\,\tilde_\!\left(S^\right) \longrightarrow 0 \longrightarrow \cdots Exactness immediately implies that the map ∂* is an isomorphism. Using the
reduced homology In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise stat ...
of the
0-sphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, ca ...
(two points) as a base case, it follows :\tilde_n\!\left(S^k\right)\cong\delta_\,\mathbb= \begin \mathbb & \mbox n=k \\ 0 & \mbox n \ne k \end where δ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. Such a complete understanding of the homology groups for spheres is in stark contrast with current knowledge of
homotopy groups of spheres In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure o ...
, especially for the case ''n'' > ''k'' about which little is known.


Klein bottle

A slightly more difficult application of the Mayer–Vietoris sequence is the calculation of the homology groups of the
Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
''X''. One uses the decomposition of ''X'' as the union of two
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Augu ...
s ''A'' and ''B'' glued along their boundary circle (see illustration on the right). Then ''A'', ''B'' and their intersection ''A''∩''B'' are
homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to circles, so the nontrivial part of the sequence yields : 0 \rightarrow \tilde_(X) \rightarrow \mathbb\ \xrightarrow \ \mathbb \oplus \mathbb \rightarrow \, \tilde_1(X) \rightarrow 0 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 In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
so homology of dimension 2 also vanishes. Finally, choosing (1, 0) and (1, −1) as a basis for Z2, it follows :\tilde_n\left(X\right)\cong\delta_\,(\mathbb\oplus\mathbb_2)= \begin \mathbb\oplus\mathbb_2 & \mbox n=1\\ 0 & \mbox n\ne1 \end


Wedge sums

Let ''X'' be the
wedge sum In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the qu ...
of two spaces ''K'' and ''L'', and suppose furthermore that the identified basepoint is a
deformation retract In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deforma ...
of 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 In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that ...
by construction. The reduced version of the sequence then yields (by exactness) :\tilde_n(K\vee L)\cong \tilde_n(K)\oplus\tilde_n(L) 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
from above "From Above" is a song with music by Ben Folds and lyrics by Nick Hornby. It was the lead single from their 2010 collaboration album Lonely Avenue. The song features guest vocals from Australian singer-songwriter Kate Miller-Heidke. Music video ...
for 2-spheres, one has :\tilde_n\left(S^2\vee S^2\right)\cong\delta_\,(\mathbb\oplus\mathbb)=\left\{\begin{matrix} \mathbb{Z}\oplus\mathbb{Z} & \mbox{if } n=2 \\ 0 & \mbox{if } n \ne 2 \end{matrix}\right.


Suspensions

If ''X'' is the
suspension Suspension or suspended may refer to: Science and engineering * Suspension (topology), in mathematics * Suspension (dynamical systems), in mathematics * Suspension of a ring, in mathematics * Suspension (chemistry), small solid particles suspend ...
''SY'' of a space ''Y'', let ''A'' and ''B'' be the 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'', :\tilde{H}_n(SY)\cong \tilde{H}_{n-1}(Y) 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,
as above ''As Above...'' was an album released in 1982 by Þeyr, an Icelandic new wave and rock group. It was issued through the Shout record label on a 12" vinyl record. Consisting of 12 tracks, ''As above...'' contained English versions of the band' ...
.


Further discussion


Relative form

A
relative Relative may refer to: General use *Kinship and family, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be ''relatives'' Philosophy *Relativism, the concept that ...
form of the Mayer–Vietoris sequence also exists. If ''Y'' ⊂ ''X'' and is the union of the interiors of ''C'' ⊂ ''A'' and ''D'' ⊂ ''B'', then the exact sequence is: :\cdots\to 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)\to\cdots


Naturality

The homology groups are
natural Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
in the sense that if f:X_1 \to X_2 is a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
map, then there is a canonical
pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
map of homology groups f_*: H_k(X_1) \to H_k(X_2) such that the composition of pushforwards is the pushforward of a composition: that is, (g\circ h)_* = g_*\circ h_*. The Mayer–Vietoris sequence is also natural in the sense that if :\begin{matrix} X_1 = A_1 \cup B_1 \\ X_2 = A_2 \cup B_2 \end{matrix} \qquad \text{and} \qquad \begin{matrix} f(A_1) \subset A_2 \\f(B_1) \subset B_2\end{matrix} then the connecting morphism of the Mayer–Vietoris sequence, \partial_*, commutes with f_*. That is, the following diagram commutes (the horizontal maps are the usual ones): :\begin{matrix} \cdots & H_{n+1}(X_1) & \longrightarrow & H_n(A_1\cap B_1) & \longrightarrow & H_n(A_1)\oplus H_n(B_1) & \longrightarrow & H_n(X_1) & \longrightarrow &H_{n-1}(A_1\cap B_1) & \longrightarrow & \cdots\\ & f_* \Bigg\downarrow & & f_* \Bigg\downarrow & & f_* \Bigg\downarrow & & f_* \Bigg\downarrow & & f_* \Bigg\downarrow\\ \cdots & H_{n+1}(X_2) & \longrightarrow & H_n(A_2\cap B_2) & \longrightarrow & H_n(A_2)\oplus H_n(B_2) & \longrightarrow & H_n(X_2) & \longrightarrow &H_{n-1}(A_2\cap B_2) & \longrightarrow & \cdots\\ \end{matrix}


Cohomological versions

The Mayer–Vietoris long exact sequence for
singular cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
groups with coefficient
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
''G'' is dual to the homological version. It is the following: :\cdots\to H^{n}(X;G)\to H^{n}(A;G)\oplus H^{n}(B;G)\to H^{n}(A\cap B;G)\to H^{n+1}(X;G)\to\cdots 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 In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s R and the underlying topological space has the additional structure of a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, the Mayer–Vietoris sequence for de Rham cohomology is :\cdots\to 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) \to \cdots where is an
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...
of denotes the restriction map, and is the difference. The map d^* is defined similarly as the map \partial_* from above. It can be briefly described as follows. For a cohomology class represented by closed form in , express as a difference of forms \omega_U - \omega_V via a
partition of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0, ...
subordinate to the open cover , for example. The exterior derivative and agree on and therefore together define an form on . One then has . For de Rham cohomology with compact supports, there exists a "flipped" version of the above sequence: \cdots\to H_{c}^{n}(U\cap V)\,\xrightarrow{\delta}\,H_{c}^{n}(U)\oplus H_{c}^{n}(V)\,\xrightarrow{\Sigma}\,H_{c}^{n}(X)\, \xrightarrow{d^*}\, H_{c}^{n+1}(U\cap V) \to \cdots where U,V,X are as above, \delta is the signed inclusion map \delta : \omega \mapsto (i^U_*\omega,-i^V_*\omega) where i^U extends a form with compact support to a form on U by zero, and \Sigma is the sum.


Derivation

Consider the long exact sequence associated to the
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
s of chain groups (constituent groups of
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
es) :0 \to C_n(A\cap B)\,\xrightarrow{\alpha}\,C_n(A) \oplus C_n(B)\,\xrightarrow{\beta}\,C_n(A+B) \to 0 where α(''x'') = (''x'', −''x''), β(''x'', ''y'') = ''x'' + ''y'', and ''C''''n''(''A'' + ''B'') is the chain group consisting of sums of chains in ''A'' and chains in ''B''. 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''''n''(''X''). In other words, ''H''''n''(''A'' + ''B'') is isomorphic to ''H''''n''(''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 In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s :0\to\Omega^{n}(X)\to\Omega^{n}(U)\oplus\Omega^{n}(V)\to\Omega^{n}(U\cap V)\to 0 yields the Mayer–Vietoris sequence for de Rham cohomology. From a formal point of view, the Mayer–Vietoris sequence can be derived from the
Eilenberg–Steenrod axioms In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homo ...
for homology theories using the
long exact sequence in homology Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology In mathematics, combinatorial to ...
.


Other homology theories

The derivation of the Mayer–Vietoris sequence from the Eilenberg–Steenrod axioms does not require the dimension axiom, so in addition to existing in ordinary cohomology theories, it holds in
extraordinary cohomology theories In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
(such as
topological K-theory In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early ...
and
cobordism In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dim ...
).


Sheaf cohomology

From the point of view of
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...
, the Mayer–Vietoris sequence is related to
Čech cohomology In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech. Motivation Let ''X'' be a topolo ...
. Specifically, it arises from the degeneration of the
spectral sequence In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...
that relates Čech cohomology to sheaf cohomology (sometimes called the
Mayer–Vietoris spectral sequence Mayer-Vietoris may refer to: * Mayer–Vietoris axiom * Mayer–Vietoris sequence In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of top ...
) in the case where the open cover used to compute the Čech cohomology consists of two open sets. This spectral sequence exists in arbitrary
topoi In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a noti ...
. (SGA 4.V.3)


See also

*
Excision theorem In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space X and subspaces A and U such that U is also a subspace of A, the theorem ...
*
Zig-zag lemma In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category. Statement In an abel ...


Notes


References

*. *. *. * * . *. *. * *. *. *. * *.


Further reading

* . {{DEFAULTSORT:Mayer-Vietoris Sequence Homology theory