Mathematical Field Notes

Archive for the ‘Geometry’ Category

Some simple spectral sequences

leave a comment »

I keep finding myself trying to explain how the very simplest spectral sequences arise (spectral sequence to compute the cohomology of a cone or an iterated cone), so I have taken the time to TeX the explanation into a sequence of guided exercises:

Some simple spectral sequences

This is all very formal and diagram-chasy. One of the off-putting things about spectral sequences is all the indices; in these exercises I have suppressed gradings and concentrated on the very simplest cases to avoid overcomplicating the notation. Once you’ve seen how the proof goes, you should go and look in Bott-Tu or McCleary for some actual examples and computations.

Please let me know of any errors in the exercises!

Written by Jonny Evans

April 28, 2015 at 3:10 pm

Gromoll filtration

leave a comment »

In my latest preprint with Georgios Dimitroglou Rizell, we use the topology of diffeomorphism groups of high-dimensional spheres to produce interesting examples of nontrivial topology in symplectomorphism groups of cotangent bundles. Until we started thinking about this, I didn’t know much about the higher homotopy groups of Diff(S^n) so here is some interesting stuff I learned while we were writing this paper.

Read the rest of this entry »

Written by Jonny Evans

July 14, 2014 at 7:16 am

Geometry and undecidability

leave a comment »

These are the notes from a talk I gave to the UCL Undergraduate Mathematics Colloquium in early October and I would like to thank them for being such an attentive audience with so many good questions. The talk is a gentle introduction to the work of Nabutovsky and Weinberger, on how logical complexity gives rise to complexity for sublevel sets of functionals in geometry.

Read the rest of this entry »

Written by Jonny Evans

November 21, 2013 at 11:13 pm

Posted in Geometry, Teaching

The geometric definition of the Johnson homomorphism

leave a comment »

I have recently been thinking about Torelli groups.

The Torelli group of a surface is the subgroup of mapping classes which act trivially on cohomology. Consider the case of an orientable surface with g handles and one boundary component (diffeomorphisms are required to fix the boundary). There is a famous homomorphism from this group to the free abelian group of rank \binom{2g}{3} called the Johnson homomorphism. The usual definition is pretty algebraic-looking (involving the mapping class group action on the fundamental group and its commutator subgroup). This week I read an alternative (extremely beautiful, geometric) definition of this homomorphism in Johnson’s survey paper on the Torelli group (D. Johnson, A survey of the Torelli group, Contemp. Math. (1983) vol. 20, 165-179). This definition is probably very well-known, but I didn’t formerly know it and I thought it was too nice not to blog about.

Fix a point p on a genus g complex curve C. Consider the Abel-Jacobi map A\colon C\to\mathrm{Jac}(C) from C into its Jacobian torus \mathrm{Jac}(C)\cong T^{2g} which sends q to \mathcal{O}(q-p). Precompose this embedding with a Torelli diffeomorphism fixing p (almost equivalent to fixing the boundary of the complement of a neighbourhood of p, except that boundary-parallel twists are now trivial…but the Johnson homomorphism would vanish on these anyway). This gives another embedding A'; A and A' are now homotopic because based homotopy classes of maps C=K(\pi,1)\to T^{2g}=K(Z^{2g},1) are determined by the induced map on cohomology (which is the same because the diffeomorphism is Torelli). This homotopy traces out a 3-cycle in T^{2g}, i.e. an element of \Lambda^3 H_1(C;Z)\cong\mathbf{Z}^{\binom{2g}{3}}. This is the Johnson homomorphism.

Alternatively, you take the universal curve over the quotient of Teichmueller space by Torelli (possible because automorphisms of a Riemann surface act nontrivially on cohomology so the universal curve exists as a bundle, not a stack). The corresponding universal Jacobian bundle is trivial (because the Jacobian is cohomological and the monodromies are Torelli). There is a universal Abel-Jacobi embedding of the universal curve into the universal Jacobian and you project that embedding into a single T^{2g} fibre using a trivialisation of the universal Jacobian. Now homology classes in the quotient of Teichmueller space by the Torelli group (equivalently the classifying space of the Torelli group) pullback to classes in the universal curve (taking preimages) and then pushforward to classes in T^{2g}. The induced map on H_1 is the Johnson homomorphism (and there are higher maps on higher group homology of the Torelli group). You need Torelli diffeomorphisms fixing a point in order to talk about the relative Abel-Jacobi map.

Written by Jonny Evans

April 26, 2013 at 6:56 pm

Posted in Geometry

Convex Integration (talk notes)

leave a comment »

In case you were unable to take notes from my talk (either because you were unable to attend or because you were present) I’ve written them up below in some detail. They’re basically just a summary of Borrelli’s notes.

Also relevant is the preliminary cartoon which contains many of the essential ideaas.

Warning: Images are large may take some time to load.

Read the rest of this entry »

Written by Jonny Evans

February 15, 2013 at 1:32 pm

Posted in Geometry, h-principle

Convex integration

leave a comment »

On Thursday I’ll be giving a talk on convex integration at the London h-principle learning seminar. This will be a dry and technical subject, so I thought I’d create some light-hearted preliminary reading. Here then, in cartoon-form, is the simplest version of 1-dimensional convex integration, used to construct an immersed loop in the plane whose tangent vector has winding number zero.

The technical details of the talk will be heavily based on these notes by Vincent Borrelli, which is an excellent place to learn all this stuff from.

Warning: The cartoon is big (about 1MB) and may take time to load.

Read the rest of this entry »

Written by Jonny Evans

February 10, 2013 at 9:06 pm

Kronheimer’s argument: Small resolutions and Dehn twists

leave a comment »

I want to amplify an expository argument I gave in a recent lecture which shows that the squared Dehn twist on a symplectic 4-manifold is smoothly isotopic to the identity map. This is an old argument of Kronheimer and I only managed to sketch it hurriedly in the lecture. A few people have asked me to explain this to them in the past, so here’s an explanation to which I can point people in future.

Read the rest of this entry »

Written by Jonny Evans

January 27, 2013 at 5:53 pm