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Home , Geometry and topology, Topology

1

Geometry and

Andrew Ranicki ([email protected]) substituting for Michael Weiss ([email protected])

SMSTC Symposium, Perth, 10th October, 2007 2

Prerequisites

I A course in spaces or topological spaces (or both). Important concepts: Open sets and neighbourhoods in metric spaces.

I Standard courses. Some knowledge of vector calculus (e.g. div, grad, curl and Green’s theorem) would be useful.

I One or two basic courses in linear . Important concepts: Abstract , quotient vector spaces.

I A course in theory. 3

Relations with other SMSTC courses

I Algebra Groups, both commutative and non-commutative, are ever-present in the /Topology course. Rings. If you are a commutative enthusiast, you might be pleased to know that many essential geometric constructions in our course can be reformulated in terms of commutative rings and their modules.

I Pure Analysis Although we don’t need Lebesgue integration theory, as developed in the Pure Analysis stream, integrals are of some importance in the Geometry/Topology course.

I Applied Analysis and PDEs The last quarter of the Geometry/Topology course, on differential geometry, is somewhat related to the first quarter of the Applied Analysis course, on dynamical systems. An important class of dynamical systems (geodesic flows) comes from differential geometry. 4

Themes of the Geometry/Topology course

I Smooth

I theory

I Vector calculus on smooth manifolds and applications to

I The small-scale and large-scale geometry of Riemannian manifolds.

I Assessment 4 written homework assignments per semester, to be marked and returned by the stream-team within one or two weeks of hand-in. 5 Smooth manifolds

I Manifolds are the topological spaces of greatest interest! I Definition A smooth n-dimensional is a M equipped with a collection A of n-dimensional coordinate charts, which coordinatise regions of M, such that

I every point of M is in at least one chart I where two charts overlap, the functions describing how the coordinates transform are smooth, i.e., differentiable to any order. The collection A is called an for M. n I Example The R is a smooth n-dimensional manifold. I Roughly speaking, an n-dimensional manifold is a space which n can be obtained by glueing together copies of R using differentiable functions. Poincar´e(ca. 1900) used manifolds to study the qualitative properties of quantitively insoluble systems of differential , such as planetary . 6 Examples of manifolds

I The of the Earth is the 2-dimensional smooth manifold S2, with coordinate charts given by latitude and longitude. I The surface of a doughnut is the 2-dimensional smooth manifold S1 × S1. Again, elements can be described by two angles. I The space of positions of a movable segment of length 1 3 in R is a 5-dimensional smooth manifold. n m I The space M = {x ∈ R | f (x) = 0 ∈ R } of the solutions of a of m simultaneous differential equations in n variables is an (n − m)-dimensional smooth manifold, provided that n > m and that for each x ∈ M the Jacobian m × n matrix (∂fi /∂xj ) has the maximal rank m. n I The unit S in (n + 1)-dimensional Euclidean space n+1 R is a compact n-dimensional smooth manifold: apply the n+1 previous example with f : R → R; x 7→ kxk − 1, n −1 n+1 S = f (0) ⊂ R . 7

Homotopy theory

I Idea: Before we can distinguish topological spaces, we must learn to distinguish continuous maps.

I Let X and Y be topological spaces. Definition. Two continuous maps f : X → Y and g : X → Y are homotopic if there exist continuous maps ht : X → Y for 0 6 t 6 1 such that

h0 = f , h1 = g : X → Y

and ht (x) depends continuously on t and x. Regard {ht } as a ‘film’ which starts at f and ends at g.

I ‘Homotopic” is an equivalence relation. I The determination of the set [X , Y ] of equivalence classes of continuous maps f : X → Y , for fixed X and Y , can often be reduced to algebra. 8

Examples of [X , Y ]

1 1 1 I Let S be the unit in C. Let f : S → S be continuous. The degree of f is the unique k ∈ Z such that f is homotopic k 1 1 to z 7→ z , and [S , S ] = Z via the degree. n n+1 I Let S = {x ∈ R | kxk = 1}. If m < n, then all continuous maps Sm → Sn are in the same homotopy class, so [Sm, Sn] = {∗}. n n I If n > 0, the homotopy classes of continuous maps S → S n n are in canonical bijection with the integers, so [S , S ] = Z. m I For m > 1 the set [S , X ] has the structure of a group, called the for m = 1. Abelian for m > 1. n I πm(S ) is finitely generated, but not well understood in the cases m > n, especially when m > n + 100. 9

Homotopy type

I Definition Two topological spaces X and Y have the same homotopy type if there exist continuous maps f : X → Y and g : Y → X such that f ◦ g is homotopic to idY and g ◦ f is homotopic to idX . n+1 n I Example R r {0} has the same homotopy type as S . 2 I Example R minus two distinct points has the same homotopy type as a “figure eight” in the .

I Example The M¨obius band is homotopy equivalent to a circle. 10 Vector calculus and homotopy types of manifolds

I In low dimensions, standard vector calculus gives some information about homotopy types. 3 I Let U be a nonempty in R . Let A be the vector space of all smooth functions from U to R. Let B be the vector space of all smooth vector fields on U. I Vector calculus provides linear maps grad A −−−−→ B −−−−→curl B −−−−→div A such that any two consecutive ones compose to zero. I Therefore im(grad) ⊂ ker(curl) and im(curl) ⊂ ker(div). If 3 U = R , these inclusions are equalities, but in general they are not! The dimensions of the vector spaces ker(curl)/im(grad) , ker(div)/im(curl) are invariants of the homotopy type of U. 11

deRham

I In our course, vector calculus will be generalised to be applicable to arbitrary smooth n-manifolds M. The above of grad, div and curl generalises to a sequence of vector spaces and linear maps

d d d dn−1 Ω0(M) −−−−→0 Ω1(M) −−−−→1 Ω2(M) −−−−→·2 · · −−−−→ Ωn(M)

where di ◦ di−1 = 0, so that im(di−1) ⊆ ker(di ).

I The dimensions of the vector spaces

i H (M) = ker(di )/im(di−1)

are invariants of the homotopy type of M.

I If M is compact, the dimensions are finite. 12

Riemannian manifolds

I A smooth manifold M becomes a Riemannian manifold through a choice of a Riemannian metric on M. This structure makes it possible to assign a length to any smooth segment in M. Following Gauss, Riemann and others, we will isolate the intrinsic aspects of in terms of length measurements. I Curvature properties of a Riemannian manifold are often related to the homotopy type of the manifold. Examples in 2 dimensions: 2 I For any Riemannian metric on S , there will be points where the curvature is positive.

I For any Riemannian metric on the surface of a smooth pretzel, there will be points where the curvature is negative.

I These statements follow from the Gauss-Bonnet theorem. We will see some generalisations to higher dimensions.