The primary intent of the present paper is to adapt familiar notions of algebraic topology, such as the fundamental group, homology, and cohomology to the context of algebraic maps and their (ramified) covering projections. The notion of a principal derived map is already fairly well understood in terms of the defining voltages; however, once it is recognized that voltages are essentially cohom...

PROOF. We first observe that the pair (M, dM) is (m — k — 1)connected. Indeed, since inclusion induces an isomorphism of fundamental groups, we can use the relative Hurewicz theorem to compute the first nonvanishing relative homotopy group: TTi(M, dM) =TTi(My dM) (where M denotes the universal cover), 7Ti(M, dM) Ç^Hi(M, dM)=HTM~\M), by duality, where c denotes compact cohomology, and H?~\M)^HTM...

The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X , there is a neighborhood Ω of Y such that, for any covering space Υ: X̂ → X in which Ŷ ≡ Υ(Y ) has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a C∞ exhaustion function φ on X̂ which is strongly q-convex on Ω̂ ...

We demonstrate that the question whether or not a given topological ramified covering map of the 2-sphere is Thurston equivalent to a rational map is algorithmically decidable.

Proof: Writing Y (Γ) = Γ\H, it is natural to ask (as we did in the “parabolic” case C/Λ) which elements σ ∈ PSL2(R) = Aut(H) descend to give automorphisms of Y (Γ). The condition we need is clearly that for all z ∈ H and γ ∈ Γ, there exists γ′ ∈ Γ such that σ(γz) = γ′(σz). Of course this holds for all z ⇐⇒ σγ = γ′σ; in other words the condition is that σΓσ−1 = Γ, i.e., that σ ∈ N(Γ). This defin...

We shall give, in an optimal form, a sufficient numerical condition for the finiteness of the fundamental group of the smooth locus of a normal K3 surface. We shall moreover prove that, if the normal K3 surface is elliptic and the above fundamental group is not finite, then there is a finite covering which is a complex torus.

The concept of embedded matching in a weighted graph is introduced, and the maximum cardinality of an embedded matching is computed. On the other hand, consider the following problem induced by a short covering. Given a prime power q, the number c(q) denotes the minimum cardinality of a subset H of Fq which satisfies the following property: every element in this space differs in at most 1 coord...

Can folding a piece of paper flat make it larger? We explore whether a shape S must be scaled to cover a flat-folded copy of itself. We consider both single folds and arbitrary folds (continuous piecewise isometries S → R2). The underlying problem is motivated by computational origami, and is related to other covering and fixturing problems, such as Lebesgue’s universal cover problem and force ...

We show that the universal cover of a compact complex surface X is the bidisk H × H, or X is biholomorphic to P × P, if and only if K X > 0 and there exists an invertible sheaf η such that η 2 ∼= OX and H0(X,S2Ω1X(−KX) ⊗ η) 6= 0. The two cases are distinguished by the second plurigenus, P2(X) ≥ 2 in the former case, P2(X) = 0 in the latter. We also discuss related questions.

Macbeath gave a formula for the number of xed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the xed-point set of each non-identity element of a cyclic group of automorphisms acting on a closed non-orientable sur...