On bifurcation braid monodromy of elliptic fibrations

We propose to study a new kind of monodromy homomorphism for families of regular elliptic fibrations of a given differentiable fibration type to get a hold on topological properties of moduli stacks of elliptic surfaces.In specific cases, including the most significant one, when all singular fibres are nodal irreducible rational curves, we compute the corresponding monodromy group, a subgroup of the mapping class group of the fibration base punctured at the singular values of the fibration.We study a tentative algebraic characterisation and give implications for the group of diffeomorphisms compatible with the fibration.     

Dihedral manifold approximate fibrations over the circle

Consider the cyclic group C ₂ of order two acting by complex-conjugation on the unit circle S ¹. The main result is that a finitely dominated manifold W of dimension > 4 admits a cocompact, free, discontinuous action by the infinite dihedral group D _∞ if and only if W is the infinite cyclic cover of a free C ₂-manifold M such that M admits a C ₂-equivariant manifold approximate fibration to S ¹. The novelty in this setting is the existence of codimension-one, invariant submanifolds of M and W. Along the way, we develop an equivariant sucking principle for orthogonal actions of finite groups on Euclidean space.     

Symplectic homology of Lefschetz fibrations and Floer homology of the monodromy map

We construct a spectral sequence converging to symplectic homology of a Lefschetz fibration whose E ¹ page is related to Floer homology of the monodromy symplectomorphism and its iterates. We use this to show the existence of fixed points of certain symplectomorphisms.     

PL fibrations are fibrations in the PL category

Proper PL maps which are Hurewicz fibrations have the covering homotopy property in the PL category.     

Group-groupoids and monodromy groupoids

This paper gives an introduction to some results on monodromy groupoids and the monodromy principle, and then develops the notion of monodromy groupoid for group groupoids.     

Mapping fibrations

We present a fibration theorem for mappings from C ⁿ to C ^p , withn <p that resembles the Milnor fibration theorem for isolated complete intersection singularities which is due to H. Hamm.     

Borel fibrations and G-spaces

Considering discrete groups G only, we present an elementary proof of the familiar equivalence of the category of G-spaces (with maps equivariant up to homotopy) and the category of Borel fibrations over BG.     

Three-dimensional conical fibrations

The investigation of conical fibrations initiated by M. Artin and D. Mumford, and by A. N. Tyurin (for the case of a nonsingular degeneracy divisor) is extended to the general case. The main results are: a canonical birational model is found for varieties of a given type; the cohomology groups are computed for the canonical models.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 745–758, June, 1977.The author thanks V. A. Iskovskikh, without whose substantial participation, this paper would not have appeared.     

Submersions, fibrations and bundles

When does a submersion have the homotopy lifting property? When is it a locally trivial fibre bundle? We establish characterizations in terms of consistency in the topology of the neighbouring fibres.

     

Skew Hopf fibrations

We introduce class F_R(S²⁺¹) of analytic fibrations of sphere S²ⁿ⁺¹,n1, by great circles for which there exists a tensor R, with the algebraic properties of a curvature tensor, such that 1) for almost everyx (R ²ⁿ +2 there exists a unique plane )x, Ofith the condition R (x, u, x)=x ² u, (u x, u (); 2) for planes spanned by fibers condition R(x, u x)=x ² u, (u x, u, x () is fulfiled. We show that F_R(S²ⁿ⁺¹) consists of skew Hopf fibrations (for n=1 see Rzh. Mat. 1987, 11A822). This implies a negative answer to the conjecture expressed in Rzh. Mat. 1972, 11A559 that this class consists of Hopf fibrations. The proof is based on the following result: skew Hopf fibrations are characterized, in the class of all analytic fibrations of a sphere by great circles, by the property that for any pair of orthogonal fibers the great three-dimensional sphere containing them inherits a skew Hopf fibration.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 101–104, 1990.     

Versal braid monodromy

We extend the method of Zariski to determine the braid monodromy group of the discriminant of a versal unfolding of a hypersurface singularity from low-dimensional generic subunfoldings to highly non-generic ones. At the expense of an induction over adjacent singularities, it is thus possible to neglect genericity issues and perturb by very simple polynomials only. To cite this article: M. Lönne, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Arithmetic monodromy groups

Mathematische Annalen -