Algebraic topology in differential geometry

Please take this page in conjunction with the Part III Guide to Courses Geometry and Topology section.

Prerequisite areas

Differential Geometry

The course generally starts from scratch, and since it is taken by people with a variety of interests (including topology, analysis and physics) it is usually fairly accessible. It is an important stepping stone for many other geometry courses.

You will find this helpful for the following Part III courses:

• Complex Manifolds
• (Algebraic Topology)
• Other geometry and geometric analysis courses which change from year to year (eg Riemannian Geometry)
• Theoretical Physics courses (eg General Relativity, Symmetries, Fields and Particles, Applications of Differential Geometry to Physics)

Relevant undergraduate courses are:

• Differential Geometry
• Riemann Surfaces
• Algebraic Topology
• Geometry 1B

First level prerequisites

Linear algebra: abstract vector spaces and linear maps, bilinear forms. See e.g. Ib Linear Algebra. 

Multi-variable calculus: derivatives of functions as linear maps, the chain rule, partial derivatives, Taylor's theorem in several variables. See e.g.Ib Analysis II. You can check if you are at the required level by doing the following exercises: Analysis II 2015-16 Sheet 4 (Questions 4, 5, 11).

Solutions to first-order differential equations (Picard's theorem)

Elementary point-set topology: topological spaces, continuity, compactness etc. See e.g. Ib Metric and Topological Spaces. You can check if you are at the required level by doing the following exercises: Met & Top 2015-16 Example Sheet 1

Second level prerequisites

Some exposure to ideas of classical differential geometry, e.g. Riemannian metrics on surfaces, curvature, geodesics.

Useful books and resources

Notes from the Part II Course.

Milnor's classic book "Topology from the Differentiable Viewpoint" is a terrific introduction to differential topology as covered in Chapter 1 of the Part II course. It is quite different in feel from the Part III course but would be great to look at in preparation.

Nakahara "Geometry, Topology and Physics". This is not a pure maths book, so comes with a warning that it is not always completely precise and rigorous. It also covers lots of material outside the Part III course. However, it is excellent for giving an intuitive picture of the concepts, and may be particularly helpful to physicists taking the course.

Algebraic Topology

Relevant undergraduate courses are:

• Part II Algebraic Topology

First level prerequisites

Second level prerequisites

Some experience of some version of homology in algebraic topology. For example you should know about:

• Homotopic maps and homotopy equivalence of spaces.
• Chain complexes and exact sequences.
• Simplicial homology. (Or another type of homology.)

Useful books and resources

Chapter 1, Algebraic Topology, Allen Hatcher, CUP, 2009

Part II notes for Algebraic Topology on Oscar Randal-Williams’ teaching page 

This is an article in our series on Faculty Researchers. This series of articles will highlight mathematics faculty research contributions within the various curricular areas in the mathematics department.

Professor Christine Escher’s research falls into two major areas of mathematics: algebraic topology and differential geometry. Her current research emphasizes algebraic topology to explore an important link with differential geometry. In joint work with Catherine Searle (Wichita State University), they ask whether geometric properties of a manifold, such as the existence of a metric with positive or non-negative curvature, imply specific restrictions on the topology of the manifold. One of the major challenges in this area is to understand how local invariants such as curvature, telling how much a space “bends,” relate to global topological invariants, such as the fundamental group, which tells how “connected” a manifold is. Manifolds with curvature bounds have been of interest since the beginning of global Riemannian geometry (early 1900's). By now, the mathematical community has a good understanding of the topology of the class of complete Riemannian manifolds with an upper curvature bound of zero. However, the class of Riemannian manifolds with strictly positive or zero lower curvature bound remains unclassified, and to date, only a small number of examples of such manifolds are known. Some of the oldest conjectures in the field, for example the Hopf conjecture (which posits that the four dimensional manifold that is the product of two 2-spheres does not admit a metric of positive sectional curvature), also fit into this subject. The work of Escher and Searle explores ways to better understand the topology of spaces with lower curvature bounds under the additional assumption of symmetries. One part of this project now also includes graduate student Zheting Dong, whom Escher and Searle are co-advising.

In the past two years Escher received several grants including funding from the Mathematical Sciences Research Institute (MSRI), the Association for Women in Mathematics (AWM) and the Max Planck Institut for Mathematics, in Bonn, Germany to pursue this research. Starting this Fall, she has also received funding from the Simons Foundation’s Mathematics and Physical Sciences (MPS) division for a Simons Collaboration grant in MPS.

During the past year Escher has advised a Masters student, Suresh Ramasamy, who is also completing his PhD in mechanical engineering. They have been working on an expository paper on the use of Lie algebra theory in the analysis of the movement of a snakeboard, which is a modified version of a skateboard in which the front and back pairs of wheels are independently actuated.

“It has been a pleasure to work with Suresh and learn a different application of differential geometry.”

During 2017-18, Escher ran a seminar in a new field, called topological data analysis. The goal is to use tools from algebraic topology, more recently also differential geometry, to study the structure of large data sets. During the year, she invited several colloquium speakers in topological data analysis, including Vin de Silva of Pomona College, Bryn Keller from Intel, Bala Krishnamoorthy from Washington State University, and Dev Sinha from University of Oregon. In May of 2018, her Masters student Branwen Purdy finished an expository paper in topological data analysis, in particular on sensor coverage.

This article was created by the Newsletter/Media Committee. Please email with any comments.

First of all, the concept of a "manifold" is certainly not exclusive to differential geometry. Manifolds are one of the basic objects of study in topology, and are also used extensively in complex analysis (in the form of Riemann surfaces and complex manifolds) and algebraic geometry (in the form of varieties).

Within topology, manifolds can be studied purely as topological spaces, but it is also common to consider manifolds with either a piecewise-linear or differentiable structure. The topological study of piecewise-linear manifolds is sometimes called piecewise-linear topology, and the topological study of differentiable manifolds is sometimes called differential topology.

I'm not sure I would necessarily describe these as distinct subfields of topology -- they are more like points of view towards geometric topology, and for the most part one can study the same geometric questions from each of the three main points of view. However, there are questions that only make sense from one of these points of view, e.g. the classification of exotic spheres, and there are certainly topology researchers who specialize in either piecewise-linear or differentiable methods. Differential topology can be found in position 57Rxx on the 2010 Math Subject Classification.

Differential geometry, on the other hand, is a major field of mathematics with many subfields. It is concerned primarily with additional structures that one can put on a smooth manifold, and the properties of such structures, as well as notions such as curvature, metric properties, and differential equations on manifolds. It corresponds to the heading 53-XX on the MSC 2010, and the MSC divides differential geometry into four large subfields:

• Classical differential geometry, i.e. the study of the geometry of curves and surfaces in $\mathbb{R}^2$ and $\mathbb{R}^3$, and more generally submanifolds of $\mathbb{R}^n$.

• Local differential geometry, which studies Riemannian manifolds (and manifolds with similar structures) from a local point of view.

• Global differential geometry, which studies Riemannian manifolds (and manifolds with similar structures) from a global point of view.

• Symplectic and contact geometry, which studies manifolds that have certain rich structures that are significantly different from a Riemannian structure.

As a general rule, anything that requires a Riemannian metric is part of differential geometry, while anything that can be done with just a differentiable structure is part of differential topology. For example, the classification of smooth manifolds up to diffeomorphism is part of differential topology, while anything that involves curvature would be part of differential geometry.