In [7], Serre proves that if G is a p-group which is not elementary abelian, then a product of Bocksteins of one dimensional classes is zero in the mod p cohomology algebra of G, provided that the product includes at least one nontrivial class from each line in H(G,Fp). For p = 2, this gives that (σG) = 0 where σG is the product of all nontrivial one dimensional classes in H(G,F2). In this note...

Recall that the rank of a finitely generated group is the minimal number of elements needed to generate it. In [Whi02], M. White proved that the injectivity radius of a closed hyperbolic 3-manifold M is bounded above by some function of rank(π1(M)). Building on a technique that he introduced, we determine the ranks of the fundamental groups of a large class of hyperbolic 3-manifolds fibering ov...

Using the u-plane integral of Moore and Witten, we derive a simple expression for the Donaldson invariants of product ruled surfaces Σg × S, where Σg is a Riemann surface of genus g. This expression generalizes a theorem of Morgan and Szabó for g = 1 to any genus g. We give two applications of our results: (1) We derive Thaddeus’ formulae for the intersection pairings on the moduli space of ran...

We combine ionized gas ([N II] λ6583) and stellar central velocity dispersions for a sample of 345 galaxies, with and without active galactic nuclei (AGNs), to study the dynamical state of the nuclear gas and its physical origin. The gas dispersions strongly correlate with the stellar dispersions over the velocity range of σ ≈ 30 − 350 km s−1, such that σg/σ∗ ≈ 0.6 − 1.4, with an average value ...

Thirteen bands for the B(1)Δg-A(1)Πu system and eleven bands for the B'(1)Σg(+)-A(1)Πu system of C2 were identified in the Fourier transform infrared emission spectra of hydrocarbon discharges. The B'(1)Σg(+)v = 4 and the B(1)Δg v = 6, 7, and 8 vibrational levels involved in nine bands were studied for the first time. A direct global analysis with Dunham parameters was carried out satisfactoril...

Let Σg,r be a compact oriented surface of genus g with r boundary components. We determine the abelianization of the symmetric mapping class group M̂(g,r)(p2) of a double unbranched cover p2 : Σ2g−1,2r → Σg,r using the Riemann constant, Schottky theta constant, and the theta multiplier. We also give lower bounds of the abelianizations of some finite index subgroups of the mapping class group.

Let Σg,n be a compact oriented genus g surface with n boundary components. The mapping class group of Σg,n, denoted Modg,n, is the group of orientation-preserving diffeomorphisms of Σg,n that restrict to the identity on ∂Σg,n, modulo isotopies that fix ∂Σg,n. The group Modg,n plays a fundamental role in many areas of mathematics, ranging from low-dimensional topology to algebraic geometry. At l...

We use Bott’s equivariant Morse theory to show that compact Riemannian three-manifolds with negative sectional curvature possess closed minimal surfaces of arbitrarily high Morse index. 1 Sources of noncompactness for parametrized minimal surfaces In this article, we apply the theory of parametrized two-dimensional minimal surfaces in a compact n-dimensional Riemannian manifold (M, 〈·, ·〉) to t...

Let Diff(D, zn) denote the group of smooth diffeomorphisms of the 2-dimensional disc that fix a neighborhood of ∂D and preserve a set zn consisting of n points. Let Diff0(D, zn) denote the identity component of this group. Then the mapping class group Diff(D, zn)/Diff0(D, zn) is isomorphic to Brn, the braid group on n strands. There is a natural “geometric” map ψ : Br2g+2 → Modg,2 induced by li...

To investigate the relationship between black holes and their host galaxies, many groups have used the width of the [O III] λ5007 line as a substitute for the stellar velocity dispersion (σ∗) of galaxy bulges. We directly test this assumption with a large and homogeneous sample of narrow-line active galactic nuclei from the Sloan Digital Sky Survey. We consider multiple transitions ([O II] λ372...