In this article, we compute the mapping class group of the total space S(xi) of the sphere bundle of a 3-dimensional real vector bundle xi over the complex projective plane P2 with < p1(xi),[P2]>=8n+5 . Examples of these manifolds include the Milnor hypersurface M1 and its generalizations Mk={(z,z ')is an element of P2 & times;P2|& sum;zikzi '=0} with k odd.
In this paper, we compute some Gromoll filtration groups Gamma i+1n+1 for certain i when 8 <= n <= 17 and n=4k+2 >= 18 . Using these results, we obtain some new information on pi 1Diff partial derivative(Dn) for 6 <= n <= 15 and on pi 2Diff partial derivative(D4k+3) for 4k+3 >= 15 .
In [], a cohomological criterion is given for a Lie g -foliation on a compact manifold to be rigid among nearby Lie foliations. Our aim is to look for examples of this rigidity statement in case the Lie foliation is modeled on the two-dimensional non-abelian Lie algebra g=aff(1) . We study the relevant cohomology group in detail, showing that it can be expressed in terms of Morse-Novikov cohomology. We find the precise conditions under which it vanishes, which yields many examples of rigid Lie affine foliations. In particular, we show that any Lie affine foliation on a compact, connected, orientable manifold of dimension 3 or 4 is rigid when deformed as a Lie foliation. Our results rely on a computation of the Morse-Novikov cohomology groups associated with a nowhere-vanishing closed one-form with discrete period group, which may be of independent interest.
We consider simple supercuspidal representations of GSp4 over a p-adic field and show that they have conductor exponent 5. We study (paramodular) newvectors and minimal vectors in these representations, obtain formulas for their matrix coefficients, and compute key local integrals involving these as test vectors. Our local computations lead to several explicit global period formulas involving automorphic representations pi of GSp4(A) whose local components (at ramified primes) are simple supercuspidal representations, and where the global test vectors are chosen to be (diagonal shifts of) newforms or automorphic forms of minimal type. As an analytic application of our work to the sup-norm problem, we show the existence of paramodular newforms on GSp4(A) of conductor p5 that take "large values" on a fixed compact set as p ->infinity .
