Mayer–Vietoris sequence

Mayer–Vietoris sequence
Name	Mayer–Vietoris sequence
Field	Algebraic topology
Introduced	1920s
Contributors	Walther Mayer, Leopold Vietoris

Contents

Mayer–Vietoris sequence The Mayer–Vietoris sequence is a fundamental tool in algebraic topology that provides long exact sequences relating the homology or cohomology of a space to that of two overlapping subspaces. It plays a central role in computations and structural arguments in the work of many mathematicians and institutions, and it connects to classical topics studied at University of Göttingen, University of Vienna, École Normale Supérieure, Princeton University, and Massachusetts Institute of Technology. Developed in an era contemporary with contributions from figures associated with Hilbert, Noether, Emmy Noether, Poincaré, Alexander Graham Bell (historical contemporaries and institutional networks), the sequence has become standard in courses and texts by authors at Cambridge University, Harvard University, University of Chicago, Stanford University, and California Institute of Technology.

Introduction

The sequence arises when a topological space is expressed as the union of two subspaces, a setting that features prominently in work at University of Paris (Sorbonne), Imperial College London, Birkbeck, University of London, University of Oxford, and King's College London. In practical terms it enables computations that are routine in expositions by educators associated with Princeton Lectures, Cambridge Tracts, Bourbaki-influenced seminars, or graduate courses taught at Yale University, Columbia University, University of Michigan, University of California, Berkeley, and ETH Zurich. The conceptual lineage intersects with developments in homological algebra linked to scholars at Cartan Institute, Moscow State University, École Polytechnique, and University of Göttingen.

Given a space X written as the union of subspaces A and B, the construction uses inclusion maps from A ∩ B into A and B and from A and B into X, bringing together ideas familiar from lectures at University of Cambridge, University of Oxford, Princeton University, Harvard University, and Mendelson Seminar Series. One forms chain complexes associated to singular chains, cellular chains, or Čech chains; these procedures are discussed in canonical texts linked to Springer-Verlag, Cambridge University Press, Elsevier, American Mathematical Society, and Society for Industrial and Applied Mathematics. The algebraic mechanism uses short exact sequences of chain complexes and the connecting homomorphism from homological algebra developed in the tradition of Emmy Noether, David Hilbert, Emil Artin, Stefan Banach, and Hermann Weyl. The resulting long exact sequence connects homology groups H_n(A ∩ B), H_n(A), H_n(B), and H_n(X), mirroring constructions taught at Massachusetts Institute of Technology, University of Chicago, Stanford University, and Columbia University.

Standard computations include those for spheres, tori, and surfaces studied historically at University of Göttingen, Sainte-Geneviève, University of Vienna, ETH Zurich, and University of Padua. For example, decomposing an n-sphere into two hemispheres recovers homology calculations appearing in lectures by faculty at Princeton University, Cambridge University, Harvard University, and University of California, Berkeley. The torus calculation via union of cylinders or via CW-complex decompositions is a staple in courses at Imperial College London, University of Oxford, King's College London, and University College London. More elaborate examples include wedge sums and connected sums examined in seminars at University of Michigan, McGill University, University of Toronto, and Australian National University, while explicit chain-level manipulations appear in notes from Institute for Advanced Study, Max Planck Institute, Clay Mathematics Institute, and Fields Institute.

Applications range from proofs of the Seifert–van Kampen theorem in contexts discussed at Princeton University, Cambridge University, and Harvard University, to calculations in cohomology theories relevant to work at Institute for Advanced Study and Mathematical Sciences Research Institute. It is used in classification problems connected to vector bundles and K-theory pursued at University of Chicago, Stanford University, and Columbia University, and in linking topological invariants that appear in studies by researchers affiliated with Max Planck Institute, CNRS, Sorbonne University, and University of Bonn. The sequence also underpins spectral sequence arguments found in research linked to IHÉS, Institut des Hautes Études Scientifiques, Princeton, and Rutgers University.

Variants include versions for cohomology, reduced homology, relative homology, and Mayer–Vietoris for Čech cohomology often taught at Cambridge University, University of Oxford, Harvard University, Yale University, and Columbia University. Generalizations appear in contexts of sheaf cohomology and derived categories developed at IHÉS, École Normale Supérieure, Institut des Hautes Études Scientifiques, University of Paris, and Princeton University. Higher-categorical and homotopical analogues are studied in programs at Massachusetts Institute of Technology, Stanford University, University of California, Berkeley, Caltech, and research centers such as Simons Foundation and Clay Mathematics Institute.

Proofs proceed by constructing short exact sequences of chain complexes and invoking the snake lemma or long exact sequence in homology, methods rooted in homological algebra taught at University of Göttingen, École Normale Supérieure, Princeton University, Harvard University, and University of Chicago. Chain-homotopy arguments and excision principles used in rigorous proofs appear in expositions associated with Cambridge University Press, Springer-Verlag, American Mathematical Society, and graduate courses at Stanford University and Massachusetts Institute of Technology. The exactness at each position in the long sequence is verified using boundary maps and connecting morphisms, techniques that pervade modern algebraic topology research at Institute for Advanced Study, Max Planck Institute, Clay Mathematics Institute, and Fields Institute.