Spaces and maps edit

In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, 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.

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 edit

Let I denote the unit interval. A family of maps indexed by I, ${\displaystyle h_{t}:X\to Y}$  is called a homotopy from ${\displaystyle h_{0}}$  to ${\displaystyle h_{1}}$  if ${\displaystyle h:I\times X\to Y,(t,x)\mapsto h_{t}(x)}$  is a map (e.g., it must be a continuous function). When X, Y are pointed spaces, the ${\displaystyle h_{t}}$  are required to preserve the basepoints. A homotopy can be shown to be an equivalence relation. Given a pointed space X and an integer ${\displaystyle n\geq 1}$ , let ${\displaystyle \pi _{n}(X)=[S^{n},X]_{*}}$  be the homotopy classes of based maps ${\displaystyle S^{n}\to X}$  from a (pointed) n-sphere ${\displaystyle S^{n}}$  to X. As it turns out, ${\displaystyle \pi _{n}(X)}$  are groups; in particular, ${\displaystyle \pi _{1}(X)}$  is called the fundamental group of X.

If one prefers to work with a space instead of a pointed space, there is the notion of a fundamental groupoid (and higher variants): by definition, the fundamental groupoid of a space X is the category where the objects are the points of X and the morphisms are paths.

Cofibration and fibration edit

A map ${\displaystyle f:A\to X}$  is called a cofibration if given (1) a map ${\displaystyle h_{0}:X\to Z}$  and (2) a homotopy ${\displaystyle g_{t}:A\to Z}$ , there exists a homotopy ${\displaystyle h_{t}:X\to Z}$  that extends ${\displaystyle h_{0}}$  and such that ${\displaystyle h_{t}\circ f=g_{t}}$ . In some loose sense, it is an analog of the defining diagram of an injective module in abstract algebra. The most basic example is a CW pair ${\displaystyle (X,A)}$ ; since many work only with CW complexes, the notion of a cofibration is often implicit.

A fibration in the sense of Serre is the dual notion of a cofibration: that is, a map ${\displaystyle p:X\to B}$  is a fibration if given (1) a map ${\displaystyle Z\to X}$  and (2) a homotopy ${\displaystyle g_{t}:Z\to B}$ , there exists a homotopy ${\displaystyle h_{t}:Z\to X}$  such that ${\displaystyle h_{0}}$  is the given one and ${\displaystyle p\circ h_{t}=g_{t}}$ . A basic example is a covering map (in fact, a fibration is a generalization of a covering map). If ${\displaystyle E}$  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 ${\displaystyle p:E\to X}$  is an example of a fibration.

Classifying spaces and homotopy operations edit

Given a topological group G, the classifying space for principal G-bundles ("the" up to equivalence) is a space ${\displaystyle BG}$  such that, for each space X,

${\displaystyle [X,BG]=}$  {principal G-bundle on X} / ~ ${\displaystyle ,\,\,[f]\mapsto f^{*}EG}$ 

where

• the left-hand side is the set of homotopy classes of maps ${\displaystyle X\to BG}$ ,
• ~ refers isomorphism of bundles, and
• = is given by pulling-back the distinguished bundle ${\displaystyle EG}$  on ${\displaystyle BG}$  (called universal bundle) along a map ${\displaystyle X\to BG}$ .

Brown's representability theorem guarantees the existence of classifying spaces.

Spectrum and generalized cohomology edit

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 A (such as ${\displaystyle \mathbb {Z} }$ ),

${\displaystyle [X,K(A,n)]=\operatorname {H} ^{n}(X;A)}$ 

where ${\displaystyle K(A,n)}$  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 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: ${\displaystyle S^{0}\to S^{1}\to S^{2}\to \cdots }$ 

• Seifert–van Kampen theorem
• Homotopy excision theorem
• Freudenthal suspension theorem (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.
• Universal coefficient theorem

See also: Characteristic class, Postnikov tower, Whitehead torsion

There are several specific theories

• simple homotopy theory
• stable homotopy theory
• chromatic homotopy theory
• rational homotopy theory
• p-adic homotopy theory
• equivariant homotopy theory

One of the basic questions in the foundations of homotopy theory is the nature of a space. The homotopy hypothesis asks whether a space is something fundamentally algebraic.

Concepts edit

• fiber sequence
• cofiber sequence

Model categories edit

Simplicial homotopy theory edit

• Simplicial homotopy

• Highly structured ring spectrum
• Homotopy type theory
• Pursuing Stacks
• Shape theory

• May, J. A Concise Course in Algebraic Topology
• George William Whitehead (1978). Elements of homotopy theory. Graduate Texts in Mathematics. Vol. 61 (3rd ed.). New York-Berlin: Springer-Verlag. pp. xxi+744. ISBN 978-0-387-90336-1. MR 0516508. Retrieved September 6, 2011.
• Ronald Brown, Topology and groupoids (2006) Booksurge LLC ISBN 1-4196-2722-8.

• Cisinski's notes
• http://ncatlab.org/nlab/files/Abstract-Homotopy.pdf
• Math 527 - Homotopy Theory Spring 2013, Section F1, lectures by Martin Frankland

• https://ncatlab.org/nlab/show/homotopy+theory