An Explicit View of the Hitchin Fibration on the Betti Side for ℙ1 minus Five Points

The dual complex of the divisor at infinity of the character variety of local systems on P1−{t1,…,t5} with monodromies in prescribed conjugacy classes Ci⊂SL2(C), was shown by Komyo to be the sphere S3. This chapter compares in some detail the projection from a tubular neighbourhood to this dual complex, with the corresponding Hitchin fibration at infinity.

REAL ANALYTIC GERMS $f \bar{g}$ AND OPEN-BOOK DECOMPOSITIONS OF THE 3-SPHERE

Let f,g: (C2,0) → (C,0) be two holomorphic germs with isolated singularities and no common branches and let Lf, [Formula: see text] be their links. We prove that the real analytic germ [Formula: see text] has an isolated singularity at 0 if and only if the link Lf ∪ -Lg is fibred. This was conjectured by Rudolph in [14]. If this condition holds, then the underlying Milnor fibration is an open-book fibration of the link Lf ∪ -Lg which coincides with [Formula: see text] in a tubular neighbourhood of this link. This enables one to realize a large family of fibrations of plumbing links in S3 as the Milnor fibrations of some real analytic germs [Formula: see text].

Spherical Functions on SO0(p, q)/ SO(p) × SO(q)

An integral formula is derived for the spherical functions on the symmetric space G/K = SO0(p, q)/ SO(p) × SO(q). This formula allows us to state some results about the analytic continuation of the spherical functions to a tubular neighbourhood of the subalgebra a of the abelian part in the decomposition G = KAK. The corresponding result is then obtained for the heat kernel of the symmetric space SO0(p, q)/ SO(p) × SO(q) using the Plancherel formula.In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel.

On braid monodromies of non-simple braided surfaces

A braided surface of degree m is a compact oriented surface S embedded in a bidisk such that is a branched covering map of degree m and , where is the projection. It was defined L. Rudolph [14, 16] with some applications to knot theory, cf. [13, 14, 15, 16, 17, 18]. A similar notion was defined O. Ya. Viro: A (closed) 2-dimensional braid in R4 is a closed oriented surface F embedded in R4 such that and pr2 │F: F → S2 is a branched covering map, where is the tubular neighbourhood of a standard 2-sphere in R4. It is related to 2-knot theory, cf. [8, 9, 10]. Braided surfaces and 2-dimensional braids are called simple if their associated branched covering maps are simple. Simple braided surfaces and simple 2-dimensional braids are investigated in some articles, [5, 8, 9, 14, 16], etc. This paper treats of non-simple braided surfaces in the piecewise linear category. For braided surfaces a natural weak equivalence relation, called braid ambient isotopy, appears essentially although it is not important for classical dimensionai braids Artin's argument [1].

Orientable surfaces in the 4-sphere associated with non-orientable knotted surfaces

Let F be a closed connected and non-orientable surface smoothly embedded in the 4-sphere S4 with normal Euler number e(F) = 0. We note that if e(F) = 0, then the non-orientable genus n is even (ef. [7]) and the tubular neighbourhood N(F) of F in S4 which is a D2-bundle over F has a trivial I-subbundle. Let τ be a trivial I-subbundle of N(F) and let τ* = F × I ⊂ N(F) be its orthogonal I-subbundle which is twisted. Then is a closed connected genus n – 1 orientable surface smoothly embedded in S4 and doubly covers F. We call this surface a doubled surface of F in S4 (associated with τ). If a trivial I-subbundle τ is given, then we see that the knot type of F* ⊂ S4 is uniquely determined.

On the h-enclosability of spheres

Let Bk, Mn, Np be manifolds in the category C = Top, Duff or PL. Define Mn and Np to be h-enclosable in Bk if (1) k = n + p + 1, (2) there are C-imbeddings i:Mn ⊆ Bk and j: Np ⊆ Bk with disjoint images and (3) there are de formation retractions of Bk − i(Mn) onto j(Np) and of Bk − j(NP) onto i(Mn). This is expressed as Bk = [Mn, NP/i, j] (mod C). The manifolds are trivially h-enclosable in Bk if, in addition, each manifold has a product tubular neighbourhood in the category C.

On tubular neighbourhoods of manifolds. I

Introduction. LetXbe a submanifold ofY, in either the topological, smooth, or piecewise linear ( =PL) categories. Anormal cell bundleonXinYis a bundle ξ = (p, E, X) in the category whose fibre is a closed cell, and such thatEis a neighbourhood ofXinYandp: E → Xis a retraction. The triple (Y, X, ξ) is atubular neighbourhood, or briefly, atube. For convenience we may refer to a tube by its cell bundle.