In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, homotopy theory is a systematic study of situations in which
maps can come with
homotopies between them. It originated as a topic in
algebraic topology
Algebraic topology is a branch of mathematics that 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 ...
but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
(e.g.,
A1 homotopy theory) and
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
(specifically the study of
higher categories).
Concepts
Spaces and maps
In homotopy theory and algebraic topology, the word "space" denotes a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
. In order to avoid
pathologies
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in ...
, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being
compactly generated, or
Hausdorff, or a
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
.
In the same vein as above, a "
map" is a continuous function, possibly with some extra constraints.
Often, one works with a
pointed space -- that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves basepoints; that is, it sends the basepoint of the domain to that of the codomain. In contrast, a free map is one which needn't preserve basepoints.
Homotopy
Let ''I'' denote the unit interval. A family of maps indexed by ''I'',
is called a homotopy from
to
if
is a map (e.g., it must be a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
). When ''X'', ''Y'' are pointed spaces, the
are required to preserve the basepoints. A homotopy can be shown to be an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
. Given a pointed space ''X'' and an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
, let
be the homotopy classes of based maps
from a (pointed) ''n''-sphere
to ''X''. As it turns out,
are
groups; in particular,
is called the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
of ''X''.
If one prefers to work with a space instead of a pointed space, there is the notion of a
fundamental groupoid In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a ...
(and higher variants): by definition, the fundamental groupoid of a space ''X'' is the
category where the
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
are the points of ''X'' and the
morphisms are paths.
Cofibration and fibration
A map
is called a
cofibration if given (1) a map
and (2) a homotopy
, there exists a homotopy
that extends
and such that
. To some loose sense, it is an analog of the defining diagram of an
injective module in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
. The most basic example is a
CW pair
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
; since many work only with CW complexes, the notion of a cofibration is often implicit.
A
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all ma ...
in the sense of Serre is the dual notion of a cofibration: that is, a map
is a fibration if given (1) a map
and (2) a homotopy
, there exists a homotopy
such that
is the given one and
. A basic example is a covering map (in fact, a fibration is a generalization of a covering map). If
is a
principal ''G''-bundle, that is, a space with a
free and transitive (topological)
group action of a (
topological) group, then the projection map
is an example of a fibration.
Classifying spaces and homotopy operations
Given a topological group ''G'', the
classifying space for
principal ''G''-bundles ("the" up to equivalence) is a space
such that, for each space ''X'',
:
/ ~
where
*the left-hand side is the set of homotopy classes of maps
,
*~ refers isomorphism of bundles, and
*= is given by pulling-back the distinguished bundle
on
(called universal bundle) along a map
.
Brown's representability theorem
In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor ''F'' on the homotopy category ''Hotc'' of pointed connected CW complexes, to the category of sets Set, to be ...
guarantees the existence of classifying spaces.
Spectrum and generalized cohomology
The idea that a classifying space classifies principal bundles can be pushed further. For example, one might try to classify cohomology classes: given an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
''A'' (such as
),
:
where
is the
Eilenberg–MacLane space. The above equation leads to the notion of a generalized cohomology theory; i.e., a
contravariant functor from the category of spaces to the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object of ...
that satisfies the axioms generalizing ordinary cohomology theory. As it turns out, such a functor may not be
representable by a space but it can always be represented by a sequence of (pointed) spaces with structure maps called a spectrum. In other words, to give a generalized cohomology theory is to give a spectrum.
A basic example of a spectrum is a
sphere spectrum In stable homotopy theory, a branch of mathematics, the sphere spectrum ''S'' is the monoidal unit in the category of spectra. It is the suspension spectrum of ''S''0, i.e., a set of two points. Explicitly, the ''n''th space in the sphere spectru ...
:
Key theorems
*
Seifert–van Kampen theorem
In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space X in t ...
*
Homotopy excision theorem
*
Freudenthal suspension theorem
In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the b ...
(a corollary of the excision theorem)
*
Landweber exact functor theorem
*
Dold–Kan correspondence
*
Eckmann–Hilton argument - this shows for instance higher homotopy groups are
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
.
*
Universal coefficient theorem
Obstruction theory and characteristic class
See also:
Characteristic class,
Postnikov tower
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with t ...
,
Whitehead torsion In geometric topology, a field within mathematics, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau(f) which is an element in the Whitehead group \op ...
Localization and completion of a space
Specific theories
There are several specific theories
*
simple homotopy theory
In mathematics, simple homotopy theory is a homotopy theory (a branch of algebraic topology) that concerns with the simple-homotopy type of a space. It was originated by Whitehead in his 1950 paper "Simple homotopy type".
See also
*Whitehead to ...
*
stable homotopy theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the F ...
*
chromatic homotopy theory
*
rational homotopy theory
In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homoto ...
*
p-adic homotopy theory
*
equivariant homotopy theory
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...
Homotopy hypothesis
One of the basic questions in the foundations of homotopy theory is the nature of a space. The
homotopy hypothesis
In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states that the ∞-groupoids are spaces. If we model our ∞-groupoids as Kan complexes, then the homotopy types of the geometric realizations of these sets give m ...
asks whether a space is something fundamentally algebraic.
Abstract homotopy theory
Concepts
*
fiber sequence
*
cofiber sequence
Model categories
Simplicial homotopy theory
*
Simplicial homotopy In algebraic topology, a simplicial homotopypg 23 is an analog of a homotopy between topological spaces for simplicial sets. If
:f, g: X \to Y
are maps between simplicial sets, a simplicial homotopy from ''f'' to ''g'' is a map
:h: X \times \Delta^ ...
See also
*
Highly structured ring spectrum
In mathematics, a highly structured ring spectrum or A_\infty-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an A_\infty-ring is called an E_\infty-ring. Wh ...
*
Homotopy type theory
*
Pursuing Stacks
References
*May, J
A Concise Course in Algebraic Topology*
*Ronald Brown,
' (2006) Booksurge LLC {{ISBN, 1-4196-2722-8.
Further reading
Cisinski's notes*http://ncatlab.org/nlab/files/Abstract-Homotopy.pdf
lectures by Martin Frankland
External links
*https://ncatlab.org/nlab/show/homotopy+theory