Homotopy Theory

Homotopy theory is a branch of mathematics that studies spaces and transformations between them, focusing primarily on the idea of deforming one space into another. In this field, we are interested in properties that remain unchanged even when a space is continuously transformed or bent but not torn. Imagine you have a rubber sheet; you can bend and stretch it into various shapes, but unless you tear or glue it, the fundamental properties of the rubber sheet do not change.

Basic concepts

To understand homotopy theory, we must first understand some basic concepts:

Topological spaces

A topological space is a set of points, each of which has a neighborhood structure that defines how the points are spatially related to each other. Formally, it is a set ( X ) containing a collection of open subsets ( tau ) that satisfy the following:

1. The empty set and ( X ) itself are in ( tau ). 
2. Any union of elements of ( tau ) is in ( tau ). 
3. Any finite intersection of elements of ( tau ) lies in ( tau ).

Continuous maps

A function ( f: X rightarrow Y ) between two topological spaces ( X ) and ( Y ) is continuous if the preimage of every open set in ( Y ) is an open set in ( X ). This concept mirrors continuity in calculus, where small changes in the input lead to small changes in the output.

Homotopy

An isomorphism between two continuous maps ( f, g: X rightarrow Y ) is a continuous function ( H: X times [0, 1] rightarrow Y ) such that:

H(x, 0) = f(x) quad text{and} quad H(x, 1) = g(x) quad text{for all} , x in X.

Here, ( [0, 1] ) is the unit interval, and ( H ) continuously deforms ( f ) into ( g ).

Visualization of homotopy

Visualizing homotopy can help you understand it better. Let's consider two paths on a plane. Let's say one path, A, is a straight line from the point ( (0, 0) ) to ( (1, 1) ). The other path, B, is a more curved path from ( (0, 0) ) to ( (1, 1) ). To visualize their homotopy, gradually change A to B.

<svg width="100" height="100"> <line x1="0" y1="100" x2="100" y2="0" stroke="black" stroke-width="2"/> <path d="M 0 100 Q 50 50, 100 0" stroke="red" fill="transparent"/> </svg>

In this example, both paths lie in the same topological space. They are homotopic because one can be continuously transformed into the other without breaking apart or adding new points.

Homotopy equivalence

Two topological spaces ( X ) and ( Y ) are homotopy equivalent if there exist continuous maps ( f: X rightarrow Y ) and ( g: Y rightarrow X ) such that:

g circ f simeq text{id}_X quad text{and} quad f circ g simeq text{id}_Y,

where ( simeq ) denotes that the maps are isotopic, and (text{id}) is the identity map. Essentially, this means that these spaces can be deformed into each other, preserving their shape in a 'flexible' way.

Examples of homotopy equivalence

Consider a solid disk and a point. They are not homotopy equivalent, because the disk cannot be continuously compressed into a point without tearing it. However, a solid disk and a solid ring without a hole are homotopy equivalent, because you can compress the ring into a disk.

Fundamental group

The fundamental group is an essential concept in homotopy theory that reflects the topological properties of a space. It is represented by ( pi_1(X, x_0) ), where ( X ) is a topological space and ( x_0 ) is a base point in ( X ).

The fundamental group contains information about loops in space: how they can be stretched or contracted. It is defined as the set of loop homotopy classes with base point (x_0). The operation is composition of loops (combination), which satisfies the group axioms.

Basic examples of fundamental groups

Let's look at a simple example:

Circle ( S^1 ): The fundamental group of a circle is ( mathbb{Z} ), which is the group of integers. It represents the number of times the loop goes around the circle. Positive integers indicate counterclockwise rotations, zero means no rotation, and negative integers indicate clockwise rotations.

<svg width="100" height="100"> <circle cx="50" cy="50" r="40" stroke="black" fill="transparent"/> <path d="M 50 10 A 40 40, 0, 1, 1, 100 50" stroke="red" fill="transparent"/> </svg>

These loops are considered equal if one can be deformed into the other without cutting the circle.

Higher symmetry groups

Beyond the fundamental group, we have higher homotopy groups ( pi_n(X)) for ( n geq 2), which consider higher-dimensional analogues of loops. These groups give us more profound insight into the structure of topological spaces. For example, ( pi_2(X)) consists of mapping spheres (like the surface of a balloon) into ( X).

Simple applications of homotopy theory

Homotopy theory is used in a variety of fields, from string theory in physics to computer graphics and data analysis. In computer science, analyzing the shape of data through continuous transformations can help identify key features and relationships within the data.

Visualizing in higher dimensions

It is challenging to visualise homotopy in more than three dimensions, as our physical intuition is limited. However, the use of projection and analogy can help express higher-dimensional concepts. For example, imagine the 2D shadow of a 4D object to gain insight into its structure.

Conclusion

Homotopy theory enriches topology by focusing on the continuous deformation of spaces. By expanding our understanding through fundamental groups and homotopy classes, it provides important insights into the properties that define the shape and structure of a space. By considering spaces up to homotopy equivalence, homotopy theory not only simplifies the often complex scenarios of geometry and topology, but also provides powerful tools for a variety of applications in mathematics and beyond.

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