Algebraic topology is a branch of mathematics which 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 up to homotopy equivalence.
Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
The 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). The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants by mapping spaces, for example, to groups which have a great deal of manageable structure in a way 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 are often 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.
Results on homology
Several useful results follow immediately from working with finitely generated abelian groups. The free rank of the n-th homology group of a simplicial complex is equal to the n-th Betti number, so one can use the homology groups of a simplicial complex to calculate its Euler-Poincaré characteristic. As another example, the top-dimensional integral homology group of a closed manifold detects orientability: this group is isomorphic to either the integers or 0, according as the manifold is orientable or not. Thus, a great deal of topological information is encoded in the homology of a given topological space.
Beyond simplicial homology, which is defined only for simplicial complexes, 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 Eilenberg and 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.
A new approach uses a functor from filtered spaces to crossed complexes defined directly and homotopically using relative homotopy groups; a higher homotopy van Kampen theorem proved for this functor enables basic results in algebraic topology, especially on the border between homology and homotopy, to be obtained without using singular homology or simplicial approximation. This approach is also called nonabelian algebraic topology, and generalises to higher dimensions ideas coming from the fundamental group.
Applications of algebraic topology
Classic applications of algebraic topology include:
- The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point.
- 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".)
- The Borsuk–Ulam theorem: any continuous map from the n-sphere to Euclidean n-space identifies at least one pair of antipodal points.
- Any subgroup of a free group is free. This result is quite interesting, because the statement is purely algebraic yet the simplest 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 the book by Higgins listed under groupoids.)
- Topological combinatorics
Notable algebraic topologists
- Frank Adams
- Enrico Betti
- Armand Borel
- Karol Borsuk
- Luitzen Egbertus Jan Brouwer
- William Browder
- Ronald Brown (mathematician)
- Henri Cartan
- Charles Ehresmann
- Samuel Eilenberg
- Hans Freudenthal
- Peter Freyd
- Pierre Gabriel
- Alexander Grothendieck
- Friedrich Hirzebruch
- Heinz Hopf
- Michael J. Hopkins
- Witold Hurewicz
- Egbert van Kampen
- Daniel Kan
- Hermann Künneth
- Solomon Lefschetz
- Jean Leray
- Saunders Mac Lane
- Mark Mahowald
- J. Peter May
- Barry Mazur
- John Milnor
- John Coleman Moore
- Jack Morava
- Emmy Noether
- Sergei Novikov
- Grigori Perelman
- Lev Pontryagin
- Nicolae Popescu
- Mikhail Postnikov
- Daniel Quillen
- Jean-Pierre Serre
- Stephen Smale
- Edwin Spanier
- Norman Steenrod
- Dennis Sullivan
- René Thom
- Hiroshi Toda
- Leopold Vietoris
- Hassler Whitney
- J. H. C. Whitehead
Important theorems in algebraic topology
- Borsuk–Ulam theorem
- Brouwer fixed point theorem
- Cellular approximation theorem
- Eilenberg–Zilber theorem
- Freudenthal suspension theorem
- Hurewicz theorem
- Künneth theorem
- Poincaré duality theorem
- Universal coefficient theorem
- Van Kampen's theorem
- Generalized van Kampen's theorems
- Higher homotopy, generalized van Kampen's theorem
- Whitehead's theorem
- R. Brown, K. A. Hardie, K. H. Kamps and T. Porter (2002), "A homotopy double groupoid of a Hausdorff space", Theory and Applications of Categories 10: 71–93.
- R. Brown, K. H. Kamps and T. Porter (2005), Theory and Applications of Categories 14 (9): 200–220.
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- Greenberg, Marvin J. and John R. Harper. (1981), Algebraic Topology: A First Course, Revised edition, Mathematics Lecture Note Series, Westview/Perseus, ISBN 9780805335576. A functorial, algebraic approach originally by Greenberg with geometric flavoring added by Harper.
- Bredon, Glen E. (1993), Topology and Geometry, Graduate Texts in Mathematics 139, Springer, ISBN 0-387-97926-3, retrieved 2008-04-01.
- Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0. A modern, geometrically flavoured introduction to algebraic topology.
- Maunder, C. R. F. (1970), Algebraic Topology, London: Van Nostrand Reinhold, ISBN 0-486-69131-4.
- tom Dieck, T., Algebraic topology. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Z\"urich (2008).
- R. Brown and A. Razak, A van Kampen theorem for unions of non-connected spaces, Archiv. Math. 42 (1984) 85–88. "Gives a general theorem on the fundamental groupoid with a set of base points of a space which is the union of open sets."
- P. J. Higgins, Categories and groupoids (1971) Van Nostrand-Reinhold.
- Ronald Brown, Higher dimensional group theory (2007) (Gives a broad view of higher dimensional van Kampen theorems involving multiple groupoids).
- E. R. van Kampen. On the connection between the fundamental groups of some related spaces. American Journal of Mathematics, vol. 55 (1933), pp. 261–267.
- R. Brown and P.J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978) 193–212. "The first 2-dimensional version of van Kampen's theorem."
- R. Brown, P.J. Higgins, and R. Sivera. Non-Abelian Algebraic Topology: filtered spaces, crossed complexes, cubical higher homotopy groupoids; European Mathematical Society Tracts in Mathematics Vol. 15, 2011,  This provides a homotopy theoretic approach to basic algebraic topology, without needing a basis in singular homology, or the method of simplicial approximation. It contains a lot of material on crossed modules.
- Van Kampen's theorem, PlanetMath.org.
- Van Kampen's theorem result, PlanetMath.org.
- R. Brown, K. Hardie, H. Kamps, T. Porter: The homotopy double groupoid of a Hausdorff space., Theory Appl. Categories, 10:71–-93 (2002).
- Dylan G. L. Allegretti, Simplicial Sets and van Kampen's Theorem (Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets).
- Allen Hatcher, Algebraic topology. (2002) Cambridge University Press, Cambridge, xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0.
- May, J. P. (1999), A Concise Course in Algebraic Topology, U. Chicago Press, Chicago, retrieved 2008-09-27.(Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids).
- Higher dimensional algebra