We present some new findings concerning branched covers in topos theory. Our discussion involves a particular subtopos of a given topos that can be described as the smallest subtopos closed under small coproducts in the including topos. Our main result is a description of the covers of this subtopos as a category of fractions of branched covers, in the sense of Fox [10], of the including topos....

Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we give a formula for the the Casson-Walker invariants of these 3-manifolds in terms of residues of a rational function (which measures the 2-loop part of the Kontsevich integral of a knot) and the s...

Starting with the classical result of Alexander [1], asserting that every closed, orientable PL–manifold of dimension n admits a branched cover onto the n–dimensional sphere S , a series of authors such as Berstein and Edmonds [3], Edmonds [11], Hilden [15], Hirsch [16], Iori and Piergallini [18], Montesinos [19] and Piergallini [20] have proved the existence of branched covers satisfying certa...

Introduction. The problem of devising optimal methods for numerically approximating the roots of a polynomial has been of interest for several centuries, and is far from solved. There are numerous recent works on root-finding algorithms and their cost, for example, the work of Jenkins and Traub [JT70], Renegar [Ren87], Schönhage [Sch82], and Shub and Smale [SS85, SS86, Sma85]. This list is far ...

Let Σ̃ and Σ be closed, connected, and orientable surfaces, and let f : Σ̃ → Σ be a branched cover. For each branching point x ∈ Σ the set of local degrees of f at f−1(x) is a partition of the total degree d. The total length of the various partitions is determined by χ(Σ̃), χ(Σ), d and the number of branching points via the Riemann-Hurwitz formula. A very old problem asks whether a collection of ...

This work identifies a class of moves on knots which translate to m-equivalences of the associated p-fold branched cyclic covers, for a fixed m and any p (with respect to the Goussarov-Habiro filtration). These moves are applied to give a flexible (if specialised) construction of knots for which the Casson-Walker-Lescop invariant (for example) of their p-fold branched cyclic covers may be readi...

A (real) manifold of dimension n is a (Hausdorff, second countable) space which is locally homeomorphic to an open subset of Rn. If we wish to make a definition of a complex manifold, we could replace Rn by Cn, but then we see that we have simply given the definition of an even-dimensional real manifold. As with real differentiable manifolds, the key is to add some additional structure via a we...

Let p : Σ → Σ be a finite Galois cover, possibly branched, with Galois group G. We are interested in the structure of the cohomology of Σ as a module over G. We treat the cases of branched and unbranched covers separately. In the case of branched covers, we give a complete classification of possible module structures of H1(Σ , C). In the unbranched case, we algebro-geometrically realize the rep...