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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
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 ...
ic tool to help compute
algebraic invariants 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 poin ...
s, known as their
homology and
cohomology groups. 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 and
Leopold Vietoris
Leopold Vietoris (; ; 4 June 1891 – 9 April 2002) was an Austrian mathematician, World War I veteran and supercentenarian. He was born in Radkersburg and died in Innsbruck.
He was known for his contributions to topology—notably the Mayer– ...
. 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 ar ...
long exact sequence, 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 mor ...
of the (co)homology groups of the subspaces, and the (co)homology groups of the
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
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 view ...
and
homology theories, including
simplicial homology and
singular cohomology. 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 (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 words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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
In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space X in t ...
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, o ...
, 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, o ...
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 matrice ...
, 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 sequences: 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 ai ...
(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 ...
s (in this case
group homomorphisms) 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-dimensio ...
of one morphism equals the
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 manifolds,
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 ...
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 cl ...
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 numbers 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 n ...
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 and
Norman Steenrod 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 poin ...
and ''A'', ''B'' be two subspaces whose
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
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
''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:
:
:
Here ''i'' : ''A''∩''B'' ↪ ''A'', ''j'' : ''A''∩''B'' ↪ ''B'', ''k'' : ''A'' ↪ ''X'', and ''l'' : ''B'' ↪ ''X'' are
inclusion maps and
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 mor ...
.
Boundary map
The boundary maps ∂
∗ lowering the dimension may be defined as follows.
An element in ''H
n''(''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 there is also a Mayer–Vietoris sequence, under the assumption that ''A'' and ''B'' have
non-empty intersection. The sequence is identical for positive dimensions and ends as:
:
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
In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space X in t ...
.
Whenever
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 ...
, the reduced Mayer–Vietoris sequence yields the isomorphism
:
where, by exactness,
:
This is precisely the
abelianized statement of the Seifert–van Kampen theorem. Compare with the fact that
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, o ...
when
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 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 isomor ...
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 th ...
, 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 fork ...
. The Mayer–Vietoris sequence for
reduced homology groups then yields
:
Exactness immediately implies that the map ∂
* is an isomorphism. Using the
reduced homology 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, c ...
(two points) as a
base case, it follows
:
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, 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 ...
''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 A ...
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 to circles, so the nontrivial part of the sequence yields
:
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 contrapositi ...
so homology of dimension 2 also vanishes. Finally, choosing (1, 0) and (1, −1) as a basis for Z
2, it follows
:
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 q ...
of two spaces ''K'' and ''L'', and suppose furthermore that the identified
basepoint is a
deformation retract 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 th ...
by construction. The reduced version of the sequence then yields (by exactness)
:
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 for 2-spheres, one has
:
Suspensions
If ''X'' is the
suspension ''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'',
:
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.
Further discussion
Relative form
A
relative 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:
:
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 ar ...
in the sense that if
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 g ...
map, then there is a canonical
pushforward map of homology groups
such that the composition of pushforwards is the pushforward of a composition: that is,
The Mayer–Vietoris sequence is also natural in the sense that if
:
then the connecting morphism of the Mayer–Vietoris sequence,
commutes with
. That is, the following diagram
commutes (the horizontal maps are the usual ones):
:
Cohomological versions
The Mayer–Vietoris long exact sequence for
singular cohomology groups with coefficient
group ''G'' is
dual to the homological version. It is the following:
:
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 ...
s R and the underlying topological space has the additional structure of a
smooth manifold, the Mayer–Vietoris sequence for
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adap ...
is
:
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_\alp ...
of denotes the restriction map, and is the difference. The map
is defined similarly as the map
from above. It can be briefly described as follows. For a cohomology class represented by
closed form in , express as a difference of forms
via a
partition of unity 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:
where
,
,
are as above,
is the signed inclusion map
where
extends a form with compact support to a form on
by zero, and
is the sum.
Derivation
Consider the
long exact sequence associated to the
short exact sequences of
chain groups (constituent groups of
chain complexes)
:
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 application ...
s
:
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 (a precursor to algebraic topolo ...
.
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 (such as
topological K-theory 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 di ...
).
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 whe ...
, the Mayer–Vietoris sequence is related to
ÄŒech cohomology. 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 h ...
that relates ÄŒech 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. 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
*
Zig-zag lemma
Notes
References
*.
*.
*.
*
* .
*.
*.
*
*.
*.
*.
*
*.
Further reading
* .
{{DEFAULTSORT:Mayer-Vietoris Sequence
Homology theory