Geometric definition of the Johnson homomorphism
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 \({2g\choose 3}\) 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 \(\Lambda^3 H_1(C;Z)\cong\mathbf{Z}^{2g\choose 3}\). 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.