Geometry, topology and homotopy

Topology provides a formal language for qualitative mathematics whereas geometry is mainly quantitative. Thus, in topology we study relationships of proximity or nearness, without using distances. A map between topological spaces is called continuous if it preserves the nearness structures. In algebra we study maps that preserve product structures, for example group homomorphisms between groups. Algebraic topology finds the solution of topological problems by casting them into simpler form by means of groups. Like analytical geometry and differential geometry before it, algebraic topology provides models for fundamental theories in physics.

Figure 1: Monkey saddle coloured by its mean curvature function, which is shown on the right

In differential geometry we study the embedding of curves and surfaces in three-dimensional Euclidean space, developing the concept of Gaussian curvature and mean curvature, to classify the surfaces geometrically. Thus, a plane has zero curvature, a sphere has positive mean curvature and a saddle has negative mean curvature. The Figure 1 shows a monkey saddle, which has height given by $z=x^3-3xy^2,$ coloured by the mean curvature function, shown on the right. Formally, the rate of change of a unit normal vector to the surface at a point in a given tangent direction is a linear operator on tangent vectors and its determinant is called the Gaussian curvature $K.$ The mean curvature $H$ is half the sum of the eigenvalues of $K.$ Now, some geometrical properties control the topological shape of a curve or surface: a plane curve of constant positive curvature is forced to be a circle and a surface of constant positive curvature is forced to be a sphere. There are further such interactions in higher dimensions and it is still an active area of research to discover links between geometry and topology. See White's Theorem which relates curvature of space curves to link number and applies to DNA supercoiling

You can find out more about two of the giants of modern geometry, Gauss and his pupil Riemann, by clicking about Guass and about Riemann Here's a short mpeg video sequence of a family of Thomsen's surfaces A website which contains educational articles on geometrical subjects, including collections of formulae, can be accessed by clicking Geometry website. A website which collects survey articles on topological subjects, including introductory treatments of topics in topology, can be accessed by clicking Topology website. Following the idea of continuity, the fundamental concept in topology is that of homotopy, for spaces and maps; we do not need homotopy theory for this course but it is so important in pure mathematics and you can understand what it is about quite easily through some examples. Homotopy arguments have led to some of the deepest theorems in all mathematics, particularly in the algebraic classification of topological spaces and in the solution of extension and lifting problems. The intuitive idea is very simple:

• Two spaces are of the same homotopy type if one can be continuously deformed into the other; that is, without losing any holes or introducing any cuts. For example, a circle, a cylinder and a Möbius strip have this property (cf. Figure 2), as do a disk and a point. So, coming from geometry, general topology or analysis, we notice immediately that the homotopy relationship transcends dimension, compactness and cardinality for spaces.

  Figure 2: Spaces of the same homotopy type

• Two maps are homotopic if the graph of one can be continuously deformed into that of the other. For example, the graphs of maps from a circle to itself lie on the surface of a torus (which is topologically the product space $\mathbb{S}^1\times \mathbb{S}^1)$ and circuit once the horizontal copy of $\Bbb{S}^1,$ as indicated in Figure 3. Two such maps will be homotopic if they circuit the vertical copy of $\mathbb{S}^1$ the same number of times; then they have the same degree.

Figure 3: Left: a torus and on it the graph of a map from a circle to itself. Right: a torus knot.

Thus, for spaces and maps, the classification up to homotopy equivalence precisely captures their qualitative features. Homotopy yields algebraic invariants for a topological space, the homotopy groups, which consist of homotopy classes of maps from spheres to the space. In knot theory we study the first homotopy group, or fundamental group, for maps from $\mathbb{S}^1.$ Continuous maps between spaces induce group homomorphisms between their homotopy groups; moreover, homotopic spaces have isomorphic groups and homotopic maps induce the same group homomorphisms. On-Line Materials:

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