Editing Mayer–Vietoris sequence (section)

==Background, motivation, and history==

Similarly to 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>
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