Algebraic topology

Method of algebraic invariants

An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the CW complex). In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups, which led to the change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces.

In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statements easier to prove. Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space, but they are often nonabelian and can be difficult to work with. The fundamental group of a (finite) simplicial complex does have a finite presentation.

Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are completely classified and are particularly easy to work with.

Setting in category theory

In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond—a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings.

One of the first mathematicians to work with different types of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question. De Rham showed that all of these approaches were interrelated and that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham cohomology. This was extended in the 1950s, when Samuel Eilenberg and Norman Steenrod generalized this approach. They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms (e.g., a weak equivalence of spaces passes to an isomorphism of homology groups), verified that all existing (co)homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory.

Applications

Classic applications of algebraic topology include:

• Using that the fundamental group of the circle is ${\displaystyle \mathbb{Z}}$, one may easily prove the fundamental theorem of algebra. Underlying this proof are intuitive geometric ideas.
• The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point.
• The free rank of the nth homology group of a simplicial complex is the nth Betti number, which allows one to calculate the Euler–Poincaré characteristic.
• One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question.
• A manifold is orientable when the top-dimensional integral homology group is the integers, and is non-orientable when it is 0.
• The n-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd. (For n = 2, this is sometimes called the "hairy ball theorem".)
• Invariance of domain and invariance of dimension.
• The Borsuk–Ulam theorem: any continuous map from the n-sphere to Euclidean n-space identifies at least one pair of antipodal points.
• Jordan curve theorem and its generalizations.
• Nielsen–Schreier theorem - any subgroup of a free group is free and there is an explicit relation between its index and number of generators. This result is quite interesting, because the statement is purely algebraic yet the simplest known proof is topological. Namely, any free group G may be realized as the fundamental group of a graph X. The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some covering space Y of X; but every such Y is again a graph. Therefore, its fundamental group H is free. On the other hand, this type of application is also handled more simply by the use of covering morphisms of groupoids, and that technique has yielded subgroup theorems not yet proved by methods of algebraic topology; see Higgins (1971).
• Topological combinatorics.

Notable people

• Frank Adams
• Michael Atiyah
• Enrico Betti
• Armand Borel
• Karol Borsuk
• Raoul Bott
• Luitzen Egbertus Jan Brouwer
• William Browder
• Ronald Brown
• Henri Cartan
• Shiing-Shen Chern
• Albrecht Dold
• Charles Ehresmann
• Samuel Eilenberg
• Hans Freudenthal
• Peter Freyd
• Pierre Gabriel
• Israel Gelfand
• Alexander Grothendieck
• Allen Hatcher
• Friedrich Hirzebruch
• Heinz Hopf
• Michael J. Hopkins
• Witold Hurewicz
• Egbert van Kampen
• Daniel Kan
• Hermann Künneth
• Ruth Lawrence
• Solomon Lefschetz
• Jean Leray
• Saunders Mac Lane
• Mark Mahowald
• J. Peter May
• Barry Mazur
• John Milnor
• John Coleman Moore
• Jack Morava
• Joseph Neisendorfer
• Emmy Noether
• Sergei Novikov
• Grigori Perelman
• Henri Poincaré
• Lev Pontryagin
• Nicolae Popescu
• Mikhail Postnikov
• Daniel Quillen
• Herbert Seifert
• Jean-Pierre Serre
• Isadore Singer
• Stephen Smale
• Edwin Spanier
• Norman Steenrod
• Dennis Sullivan
• René Thom
• Hiroshi Toda
• Leopold Vietoris
• Hassler Whitney
• J. H. C. Whitehead
• Gordon Thomas Whyburn

Important theorems

• Blakers–Massey theorem
• Borsuk–Ulam theorem
• Brouwer fixed point theorem
• Cellular approximation theorem
• Dold–Thom theorem
• Eilenberg–Ganea theorem
• Eilenberg–Zilber theorem
• Freudenthal suspension theorem
• Hurewicz theorem
• Künneth theorem
• Lefschetz fixed-point theorem
• Leray–Hirsch theorem
• Poincaré duality theorem
• Seifert–van Kampen theorem
• Universal coefficient theorem
• Whitehead theorem