Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal to 1. Consequently, we obtain an elementary proof that, in general, the integral simplicial volume of (triangulated) manifolds is not computable.

We propose a finite-dimensional variational principle on triangulated 3-manifolds so that its critical points are related to solutions to Thurstons gluing equation and Hakens normal surface equation. The action functional is the volume.

For smooth compact oriented Riemannian manifolds$M$of dimension 4$k$+2,$k$≥0, with or without boundary, and a vector bundle$F$on$M$with an inner product and a flat connection, we construct a modification of the Hodge star operator on the middle-dimensional (parabolic) cohomology of$M$twisted by$F$. This operator induces a canonical complex structure on the middle-dimensional cohomology space that is compatible with the natural symplectic form given by integrating the wedge product. In particular, when$k$=0 we get a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface, with monodromies along boundary components belonging to fixed conjugacy classes when the surface has boundary, that is compatible with the standard symplectic form on the moduli space.

We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescribed cone angles at all edges is (if non-empty) a smooth complex manifold of dimension the sum of the genera of the vertex links. Moreover, we show that the complex lengths of a collection of peripheral elements give a local holomorphic parameterisation of this manifold.

Here we classify$J$-embeddable surfaces, i.e. surfaces whose secant varieties have dimension at most 4, when the surfaces have two components at most.