1 Geometry and Topology Andrew Ranicki (a.ranicki@ed.ac.uk) substituting for Michael Weiss (m.weiss@maths.abdn.ac.uk) SMSTC Symposium, Perth, 10th October, 2007
2 Prerequisites ◮ A course in metric spaces or topological spaces (or both). Important concepts: Open sets and neighbourhoods in metric spaces. ◮ Standard calculus courses. Some knowledge of vector calculus (e.g. div, grad, curl and Green’s theorem) would be useful. ◮ One or two basic courses in linear algebra. Important concepts: Abstract vector space, quotient vector spaces. ◮ A course in group theory.
3 Relations with other SMSTC courses ◮ Algebra Groups, both commutative and non-commutative, are ever-present in the Geometry/Topology course. Rings. If you are a commutative ring 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. ◮ 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. ◮ 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 ◮ Smooth manifolds ◮ Homotopy theory ◮ Vector calculus on smooth manifolds and applications to homotopy theory ◮ The small-scale and large-scale geometry of Riemannian manifolds. ◮ 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 ◮ Manifolds are the topological spaces of greatest interest! ◮ Definition A smooth n -dimensional manifold is a topological space M equipped with a collection A of n -dimensional coordinate charts, which coordinatise regions of M , such that ◮ every point of M is in at least one chart ◮ 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 atlas for M . ◮ Example The Euclidean space R n is a smooth n -dimensional manifold. ◮ Roughly speaking, an n -dimensional manifold is a space which can be obtained by glueing together copies of R n using differentiable functions. Poincar´ e (ca. 1900) used manifolds to study the qualitative properties of quantitively insoluble systems of differential equations, such as planetary motion.
6 Examples of manifolds ◮ The surface of the Earth is the 2-dimensional smooth manifold S 2 , with coordinate charts given by latitude and longitude. ◮ The surface of a doughnut is the 2-dimensional smooth manifold S 1 × S 1 . Again, elements can be described by two angles. ◮ The space of positions of a movable line segment of length 1 in R 3 is a 5-dimensional smooth manifold. ◮ The space M = { x ∈ R n | f ( x ) = 0 ∈ R m } of the solutions of a set 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 ( ∂ f i /∂ x j ) has the maximal rank m . ◮ The unit sphere S n in ( n + 1)-dimensional Euclidean space R n +1 is a compact n -dimensional smooth manifold: apply the previous example with f : R n +1 → R ; x �→ � x � − 1, S n = f − 1 (0) ⊂ R n +1 .
7 Homotopy theory ◮ Idea: Before we can distinguish topological spaces, we must learn to distinguish continuous maps. ◮ 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 h t : X → Y for 0 � t � 1 such that h 0 = f , h 1 = g : X → Y and h t ( x ) depends continuously on t and x . Regard { h t } as a ‘film’ which starts at f and ends at g . ◮ ‘Homotopic” is an equivalence relation. ◮ 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 ] ◮ Let S 1 be the unit circle in C . Let f : S 1 → S 1 be continuous. The degree of f is the unique k ∈ Z such that f is homotopic to z �→ z k , and [ S 1 , S 1 ] = Z via the degree. ◮ Let S n = { x ∈ R n +1 | � x � = 1 } . If m < n , then all continuous maps S m → S n are in the same homotopy class, so [ S m , S n ] = {∗} . ◮ If n > 0, the homotopy classes of continuous maps S n → S n are in canonical bijection with the integers, so [ S n , S n ] = Z . ◮ For m � 1 the set [ S m , X ] has the structure of a group, called the fundamental group for m = 1. Abelian for m > 1. ◮ π m ( S n ) is finitely generated, but not well understood in the cases m > n , especially when m > n + 100.
9 Homotopy type ◮ 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 id Y and g ◦ f is homotopic to id X . ◮ Example R n +1 � { 0 } has the same homotopy type as S n . ◮ Example R 2 minus two distinct points has the same homotopy type as a “figure eight” in the plane. ◮ Example The M¨ obius band is homotopy equivalent to a circle.
10 Vector calculus and homotopy types of manifolds ◮ In low dimensions, standard vector calculus gives some information about homotopy types. ◮ Let U be a nonempty open set in R 3 . 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 . ◮ Vector calculus provides linear maps grad curl div A − − − − → B − − − − → B − − − − → A such that any two consecutive ones compose to zero. ◮ Therefore im(grad) ⊂ ker(curl) and im(curl) ⊂ ker(div). If U = R 3 , 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 cohomology ◮ In our course, vector calculus will be generalised to be applicable to arbitrary smooth n -manifolds M . The above sequence of grad, div and curl generalises to a sequence of vector spaces and linear maps d n − 1 d 0 d 1 d 2 Ω 0 ( M ) → Ω 1 ( M ) → Ω 2 ( M ) → Ω n ( M ) − − − − − − − − − − − − → · · · − − − − where d i ◦ d i − 1 = 0, so that im( d i − 1 ) ⊆ ker( d i ). ◮ The dimensions of the vector spaces H i ( M ) = ker( d i ) / im( d i − 1 ) are invariants of the homotopy type of M . ◮ If M is compact, the dimensions are finite.
12 Riemannian manifolds ◮ 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 curve segment in M . Following Gauss, Riemann and others, we will isolate the intrinsic aspects of curvature in terms of length measurements. ◮ Curvature properties of a Riemannian manifold are often related to the homotopy type of the manifold. Examples in 2 dimensions: ◮ For any Riemannian metric on S 2 , there will be points where the curvature is positive. ◮ For any Riemannian metric on the surface of a smooth pretzel, there will be points where the curvature is negative. ◮ These statements follow from the Gauss-Bonnet theorem. We will see some generalisations to higher dimensions.
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