We show that Norbury’s McShane identity for nonorientable cusped hyperbolic surfaces N generalizes to quasifuchsian representations of π1(N) as well as pseudo-Anosov mapping Klein bottles with singular fibers given by N.

In studying the "11/8-Conjecture" on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a "10/8+4"-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from BPin(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the j-based Atiyah-Hirzebruch spectral sequence.

The purpose of this paper is to study the monodromy groups associated to the quasi-bounded holomorphic quadratic forms on punctured surfaces. As a consequence, we obtain a natural family of symplectic structures on the Teichm/iller space Tg,, for n> 0. As another consequence, we show that the projective monodromy map from a class of Fuchsian equations to the representation variety is generically a local diffeomorphism.

In this paper the centroaffine Minkowski problem, a critical case of the L p-Minkowski problem in the n+ 1 dimensional Euclidean space, is studied. By its variational structure and the method of blow-up analyses, we obtain two sufficient conditions for the existence of solutions, for a generalized rotationally symmetric case of the problem.

The face numbers of simplicial complexes without missing faces of dimension larger than$i$are studied. It is shown that among all such ($d$−1)-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal$f$-vector; and moreover, among all such 2-Cohen–Macaulay (2-CM) complexes, the same sphere has the componentwise minimal$h$-vector. It is also verified that the$l$-skeleton of a flag ($d$−1)-dimensional 2-CM complex is 2($d$−$l$)-CM, while the$l$-skeleton of a flag piecewise linear ($d$−1)-sphere is 2($d$−$l$)-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension and the number of vertices are established.