Several identities similar to the Schlaefli formula are established for tetrahedra in a space of constant curvature.

Given an involution on a rational homology 3-sphere Y with quotient the 3-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg--Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants associated to spin structures on Y. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology.

We study the rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature-like quantities for polyhedral surfaces are introduced and are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law. They can be considered as 2-dimensional counterparts of the Schlaefli formula.

It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite regular cover which has a geodesic partially truncated triangulation. The proofs use an extension of a result due to Long and Niblo concerning the separability of peripheral subgroups.

A family of coordinates h for the Teichmller space of a compact surface with boundary was introduced by the second author. Mondello showed that the coordinate 0 can be used to produce a natural cell decomposition of the Teichmller space, invariant under the action of the mapping class group. In this paper, we show that, for any h 0, the coordinate h produces a natural cell decomposition of the Teichmller space.