MayerâVietoris sequence Information
In mathematics, particularly algebraic topology and homology theory, the MayerâVietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer 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 long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.
The MayerâVietoris sequence holds for a variety of cohomology and homology theories, including simplicial homology and singular cohomology. In general, the sequence holds for those theories satisfying the EilenbergâSteenrod axioms, 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 MayerâVietoris 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 MayerâVietoris sequence is analogous to the Seifertâvan Kampen theorem for the fundamental group, and a precise relation exists for homology of dimension one.
Background, motivation, and history
Like the fundamental group or the higher homotopy groups 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 MayerâVietoris 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 (in this case groups) and morphisms (in this case group homomorphisms) between them such that the image 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 complexes, or CW complexes, 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.^{ [1]} He was told about the conjectured result and a way to its solution, and solved the question for the Betti numbers in 1929.^{ [2]} He applied his results to the torus considered as the union of two cylinders.^{ [3]}^{ [4]} Vietoris later proved the full result for the homology groups in 1930 but did not express it as an exact sequence.^{ [5]} The concept of an exact sequence only appeared in print in the 1952 book Foundations of Algebraic Topology by Samuel Eilenberg and Norman Steenrod^{ [6]} where the results of Mayer and Vietoris were expressed in the modern form.^{ [7]}
Basic versions for singular homology
Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The MayerâVietoris 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 Aâ©B.^{ [8]} There is an unreduced and a reduced version.
Unreduced version
For unreduced homology, the MayerâVietoris sequence states that the following sequence is exact:^{ [9]}
Here i : Aâ©B âȘ A, j : Aâ©B âȘ B, k : A âȘ X, and l : B âȘ X are inclusion maps and denotes the direct sum of abelian groups.
Boundary map
The boundary maps â_{∗} lowering the dimension may be defined as follows.^{ [10]} An element in H_{n}(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 âx = â(u + v) = 0 so that âu = ââv. This implies that the images of both these boundary (n â 1)-cycles are contained in the intersection Aâ©B. Then â_{∗}([x]) can be defined to be the class of âu in H_{n−1}(Aâ©B). Choosing another decomposition x = uâČ + vâČ does not affect [âu], since âu + âv = âx = âuâČ + âvâČ, which implies âu − âuâČ = â(vâČ − v), and therefore âu and âuâČ lie in the same homology class; nor does choosing a different representative xâČ, since then âxâČ = âx = 0. Notice that the maps in the MayerâVietoris 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 MayerâVietoris sequence, under the assumption that A and B have non-empty intersection.^{ [11]} The sequence is identical for positive dimensions and ends as:
Analogy with the Seifertâvan Kampen theorem
There is an analogy between the MayerâVietoris sequence (especially for homology groups of dimension 1) and the Seifertâvan Kampen theorem.^{ [10]}^{ [12]} Whenever is path-connected, the reduced MayerâVietoris sequence yields the isomorphism
where, by exactness,
This is precisely the abelianized statement of the Seifertâvan Kampen theorem. Compare with the fact that is the abelianization of the fundamental group when is path-connected.^{ [13]}
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 to k-discs, which are contractible, the homology groups for A and B are trivial. The MayerâVietoris sequence for reduced homology groups then yields
Exactness immediately implies that the map â_{*} is an isomorphism. Using the reduced homology of the 0-sphere (two points) as a base case, it follows^{ [14]}
where ÎŽ 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.^{ [15]}
Klein bottle
A slightly more difficult application of the MayerâVietoris sequence is the calculation of the homology groups of the Klein bottle X. One uses the decomposition of X as the union of two MĂ¶bius strips A and B glued along their boundary circle (see illustration on the right). Then A, B and their intersection Aâ©B are homotopy equivalent to circles, so the nontrivial part of the sequence yields^{ [16]}
and the trivial part implies vanishing homology for dimensions greater than 2. The central map Î± sends 1 to (2, −2) since the boundary circle of a MĂ¶bius band wraps twice around the core circle. In particular Î± is injective 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 of two spaces K and L, and suppose furthermore that the identified basepoint is a deformation retract of open neighborhoods U â K and V â L. Letting A = K âȘ V and B = U âȘ L it follows that A âȘ B = X and A â© B = U âȘ V, which is contractible by construction. The reduced version of the sequence then yields (by exactness)^{ [17]}
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 AâȘB, with A and B contractible. Also, the intersection Aâ©B is homotopy equivalent to Y. Hence the MayerâVietoris sequence yields, for all n,^{ [18]}
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 MayerâVietoris sequence also exists. If Y â X and is the union of C â A and D â B, then the exact sequence is:^{ [19]}
Naturality
The homology groups are natural in the sense that if is a continuous 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 MayerâVietoris sequence is also natural in the sense that if
then the connecting morphism of the MayerâVietoris sequence, commutes with .^{ [20]} That is, the following diagram commutes^{ [21]} (the horizontal maps are the usual ones):
Cohomological versions
The MayerâVietoris long exact sequence for singular cohomology groups with coefficient group G is dual to the homological version. It is the following:^{ [22]}
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 numbers R and the underlying topological space has the additional structure of a smooth manifold, the MayerâVietoris sequence for de Rham cohomology is
where {U, V} is an open cover of X, Ï denotes the restriction map, and Î is the difference. The map is defined similarly as the map from above. It can be briefly described as follows. For a cohomology class [Ï] represented by closed form Ï in Uâ©V, express Ï as a difference of forms via a partition of unity subordinate to the open cover {U, V}, for example. The exterior derivative dÏ_{U} and dÏ_{V} agree on Uâ©V and therefore together define an n + 1 form Ï on X. One then has d^{∗}([Ï]) = [Ï].
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.^{ [23]}
Derivation
Consider the long exact sequence associated to the short exact sequences of chain groups (constituent groups of chain complexes)
where Î±(x) = (x, âx), ÎČ(x, y) = x + y, and C_{n}(A + B) is the chain group consisting of sums of chains in A and chains in B.^{ [9]} 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).^{ [24]} In other words, H_{n}(A + B) is isomorphic to H_{n}(X). This gives the MayerâVietoris sequence for singular homology.
The same computation applied to the short exact sequences of vector spaces of differential forms
yields the MayerâVietoris sequence for de Rham cohomology.^{ [25]}
From a formal point of view, the MayerâVietoris sequence can be derived from the EilenbergâSteenrod axioms for homology theories using the long exact sequence in homology.^{ [26]}
Other homology theories
The derivation of the MayerâVietoris sequence from the EilenbergâSteenrod axioms does not require the dimension axiom,^{ [27]} so in addition to existing in 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 MayerâVietoris sequence is related to Äech cohomology. Specifically, it arises from the degeneration of the spectral sequence that relates Äech cohomology to sheaf cohomology (sometimes called the MayerâVietoris spectral sequence) in the case where the open cover used to compute the Äech cohomology consists of two open sets.^{ [28]} This spectral sequence exists in arbitrary topoi.^{ [29]}
See also
Notes
- ^ Hirzebruch 1999
- ^ Mayer 1929
- ^ DieudonnĂ© 1989, p. 39
- ^ Mayer 1929, p. 41
- ^ Vietoris 1930
- ^ Corry 2004, p. 345
- ^ Eilenberg & Steenrod 1952, Theorem 15.3
- ^ Eilenberg & Steenrod 1952, Â§15
- ^ ^{a} ^{b} Hatcher 2002, p. 149
- ^ ^{a} ^{b} Hatcher 2002, p. 150
- ^ Spanier 1966, p. 187
- ^ Massey 1984, p. 240
- ^ Hatcher 2002, Theorem 2A.1, p. 166
- ^ Hatcher 2002, Example 2.46, p. 150
- ^ Hatcher 2002, p. 384
- ^ Hatcher 2002, p. 151
- ^ Hatcher 2002, Exercise 31 on page 158
- ^ Hatcher 2002, Exercise 32 on page 158
- ^ Hatcher 2002, p. 152
- ^ Massey 1984, p. 208
- ^ Eilenberg & Steenrod 1952, Theorem 15.4
- ^ Hatcher 2002, p. 203
- ^ Bott, Raoul. Differential forms in algebraic topology. Tu, Loring W. New York. ISBN 978-0-387-90613-3. OCLC 7597142.
- ^ Hatcher 2002, Proposition 2.21, p. 119
- ^ Bott & Tu 1982, Â§I.2
- ^ Hatcher 2002, p. 162
- ^ KĆno & Tamaki 2006, pp. 25â26
- ^ Dimca 2004, pp. 35â36
- ^ Verdier 1972 (SGA 4.V.3)
References
- Bott, Raoul; Tu, Loring W. (1982), Differential Forms in Algebraic Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90613-3.
- Corry, Leo (2004), Modern Algebra and the Rise of Mathematical Structures, BirkhĂ€user, p. 345, ISBN 3-7643-7002-5.
- DieudonnĂ©, Jean (1989), A History of Algebraic and Differential Topology 1900â1960, BirkhĂ€user, p. 39, ISBN 0-8176-3388-X.
- Dimca, Alexandru (2004), Sheaves in topology, Universitext, Berlin: Springer-Verlag, doi: 10.1007/978-3-642-18868-8, ISBN 978-3-540-20665-1, MR 2050072
- Eilenberg, Samuel; Steenrod, Norman (1952), Foundations of Algebraic Topology, Princeton University Press, ISBN 978-0-691-07965-3.
- Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79540-1, MR 1867354.
- Hirzebruch, Friedrich (1999), "Emmy Noether and Topology", in Teicher, M. (ed.), The Heritage of Emmy Noether, Israel Mathematical Conference Proceedings, Bar-Ilan University/ American Mathematical Society/ Oxford University Press, pp. 61â63, ISBN 978-0-19-851045-1, OCLC 223099225.
- KĆno, Akira; Tamaki, Dai (2006) [2002], Generalized cohomology, Iwanami Series in Modern Mathematics, Translations of Mathematical Monographs, vol. 230 (Translated from the 2002 Japanese edition by Tamaki ed.), Providence, RI: American Mathematical Society, ISBN 978-0-8218-3514-2, MR 2225848
- Massey, William (1984), Algebraic Topology: An Introduction, Springer-Verlag, ISBN 978-0-387-90271-5.
- Mayer, Walther (1929), "Ăber abstrakte Topologie", Monatshefte fĂŒr Mathematik, 36 (1): 1â42, doi: 10.1007/BF02307601, ISSN 0026-9255. (in German)
- Spanier, Edwin (1966), Algebraic Topology, Springer-Verlag, ISBN 0-387-94426-5.
- Verdier, Jean-Louis (1972), "Cohomologie dans les topos", in Artin, Michael; Grothendieck, Alexander; Verdier, Jean-Louis (eds.), SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois Marie â 1963â64 â ThĂ©orie des topos et cohomologie Ă©tale des schĂ©mas â (SGA 4) â Tome 2, Lecture Notes in Mathematics (in French), vol. 270, Berlin; Heidelberg: Springer-Verlag, p. 1, doi: 10.1007/BFb0061320, ISBN 978-3-540-06012-3
- Vietoris, Leopold (1930), "Ăber die Homologiegruppen der Vereinigung zweier Komplexe", Monatshefte fĂŒr Mathematik, 37: 159â62, doi: 10.1007/BF01696765. (in German)
Further reading
- Reitberger, Heinrich (2002), "Leopold Vietoris (1891â2002)" (PDF), Notices of the American Mathematical Society, 49 (20), ISSN 0002-9920.