The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory.

His conjectural theory of motives has been a driving force behind modern developments in algebraic K-theory, motivic homotopy theory, and motivic integration.

This letter and successive parts were distributed from Bangor ( see External Links below ): in an informal manner, as a kind of diary, Grothendieck explained and developed his ideas on the relationship between algebraic homotopy theory and algebraic geometry and prospects for a noncommutative theory of stacks.

Much of this work anticipated the subsequent development of the motivic homotopy theory of Fabien Morel and V. Voevodsky in the mid 1990s.

A successful mathematical classification method for physical lattice defects, which works not only with the theory of dislocations and other defects in crystals but also, e. g., for disclinations in liquid crystals and for excitations in superfluid < sup > 3 </ sup > He, is the topological homotopy theory.

Since the fundamental group is a homotopy invariant, the theory of the winding number for the complex plane minus one point is the same as for the circle.

In mathematics, especially in category theory and homotopy theory, a groupoid ( less often Brandt groupoid or virtual group ) generalises the notion of group in several equivalent ways.

* Spectrum ( homotopy theory )

Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups.

In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology.

Sometimes, the term " unit interval " is used to refer to objects that play a role in various branches of mathematics analogous to the role that plays in homotopy theory.

Subsequently Milnor worked on the topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fibre has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number.

His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002.

Along with Fabien Morel, Voevodsky introduced a homotopy theory for schemes.

There are others, coming from stable homotopy theory.

* A topological defect, a generalization of the idea of a soliton to any system which is stable against decay due to homotopy theory

** S-duality ( homotopy theory )

Finally, two useful and equivalent definitions were given by Daniel Quillen using homotopy theory in 1969 and 1972.

Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology.

The fundamental group is the first and simplest of the homotopy groups.

Now the fundamental group of X with base point x is this set modulo homotopy

With the above product, the set of all homotopy classes of loops with base point x < sub > 0 </ sub > forms the fundamental group of X at the point x < sub > 0 </ sub > and is denoted

In Euclidean space R < sup > n </ sup >, or any convex subset of R < sup > n </ sup >, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element.

Unlike the homology groups and higher homotopy groups associated to a topological space, the fundamental group need not be abelian.

The condition that U be simply connected means that U has no " holes " or, in homotopy terms, that the fundamental group of U is trivial ; for instance, every open disk < math > U =

Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group.

The first and simplest homotopy group is the fundamental group, which records information about loops in a space.

The set of homotopy classes of maps from a sphere to a path connected space is not the homotopy group, but is essentially the set of orbits of the fundamental group on the homotopy group, and in general has no natural group structure.

Unlike the Seifert – van Kampen theorem for the fundamental group and the Excision theorem for singular homology and cohomology, there is no simple way to calculate the homotopy groups of a space by breaking it up into smaller spaces.

The homotopy theory of defects uses the fundamental group of the order parameter space of a medium to discuss the existence, stability and classifications of topological defects in that medium.

When the theory was put on an organised basis around 1950 ( with the definitions reduced to homotopy theory ) it became clear that the most fundamental characteristic classes known at that time ( the Stiefel-Whitney class, the Chern class, and the Pontryagin classes ) were reflections of the classical linear groups and their maximal torus structure.

Like the fundamental group or the higher homotopy groups of a space, homology groups are important topological invariants.

Then the intersection of A and B is an annulus, which is known to be homotopy equivalent to ( and so has the same fundamental group as ) a circle.

In particular, this theorem says that the abelianization of the first homotopy group ( the fundamental group ) is isomorphic to the first homology group:

The proof of this theorem uses a higher homotopy van Kampen type theorem for triadic homotopy groups, which requires a notion of the fundamental cat < sup > n </ sup >- group of an n-cube of spaces.

* There is a natural, surjective group homomorphism π < sub > 1 </ sub >( M ) → Hol (∇)/ Hol < sup > 0 </ sup >(∇), where π < sub > 1 </ sub >( M ) is the fundamental group of M, which sends the homotopy class to the coset P < sub > γ </ sub >· Hol < sup > 0 </ sup >(∇).

The fundamental group is trivial and all other higher homotopy groups are those of the 5-sphere, i. e. torsion.

They were the first known examples of 3-manifolds which were not determined by their homology and fundamental group alone, and the simplest examples of closed manifolds whose homeomorphism type is not determined by their homotopy type.

This is the group of homotopy classes of loops based at x < sub > 0 </ sub >.

This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphism from π ( X, x < sub > 0 </ sub >) to π ( Y, y < sub > 0 </ sub >).

The constraints arise almost always because the differential equations must obey a set of boundary conditions, and the boundary has a non-trivial homotopy group, preserved by the differential equations.

* Invariant ( mathematics ), something unaltered by a transformation, for example: taking a homotopy group functor on the category of topological spaces.

As a homotopy theory, the notion of covering spaces works well when the deck transformation group is discrete, or, equivalently, when the space is locally path-connected.

Since the homotopy group

Since the homotopy group, this model predicts monopoles.

For a Yang – Mills theory these inequivalent sectors can be ( in an appropriate gauge ) classified by the third homotopy group of SU ( 2 ) ( whose group manifold is the 3-sphere ).

As the third homotopy group of has been found to be the set of integers,

To define the n-th homotopy group, the base point preserving maps from an n-dimensional sphere ( with base point ) into a given space ( with base point ) are collected into equivalence classes, called homotopy classes.