Borsuk–Ulam theorem for filtered spaces

Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T : X → X {T:X\to X} and S : Y → Y {S:Y\to Y} , respectively. Suppose that there exists a sequence ( X i , T i ) ⁢ ⟶ h i ⁢ ( X i + 1 , T i + 1 ) for ⁢ 1 ≤ i ≤ k , (X_{i},T_{i})\overset{h_{i}}{\longrightarrow}(X_{i+1},T_{i+1})\quad\text{for }% 1\leq i\leq k, where, for each i, X i {X_{i}} is a pathwise connected and paracompact Hausdorff space equipped with a free involution T i {T_{i}} , such that X k + 1 = X {X_{k+1}=X} , and h i : X i → X i + 1 {h_{i}:X_{i}\to X_{i+1}} is an equivariant map, for all 1 ≤ i ≤ k {1\leq i\leq k} . To achieve Borsuk–Ulam-type theorems, in several results that appear in the literature, the involved spaces X in the statements are assumed to be cohomological n-acyclic spaces. In this paper, by considering a more wide class of topological spaces X (which are not necessarily cohomological n-acyclic spaces), we prove that there is no equivariant map f : ( X , T ) → ( Y , S ) {f:(X,T)\to(Y,S)} and we present some interesting examples to illustrate our results.

CONFORMAL COURANT ALGEBROIDS AND ORIENTIFOLD T-DUALITY

We introduce conformal Courant algebroids, a mild generalization of Courant algebroids in which only a conformal structure rather than a bilinear form is assumed. We introduce exact conformal Courant algebroids and show they are classified by pairs (L, H) with L a flat line bundle and H ∈ H3(M, L) a degree 3 class with coefficients in L. As a special case gerbes for the crossed module (U(1) → ℤ2) can be used to twist TM ⊕ T*M into a conformal Courant algebroid. In the exact case there is a twisted cohomology which is 4-periodic if L2 = 1. The structure of Conformal Courant algebroids on circle bundles leads us to construct a T-duality for orientifolds with free involution. This incarnation of T-duality yields an isomorphism of 4-periodic twisted cohomology. We conjecture that the isomorphism extends to an isomorphism in twisted KR-theory and give some calculations to support this claim.

The arithmetic of Prym varieties in genus 3

Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to perform Chabauty- and Brauer–Manin-type calculations for curves of genus 5 with an fixed-point-free involution. As an application, we determine the rational points on a smooth plane quartic and give examples of curves of genus 3 and 5 violating the Hasse principle. The methods are, in principle, applicable to any genus 3 curve with a double cover. We also show how these constructions can be used to design smooth plane quartics with specific arithmetic properties. As an example, we give a smooth plane quartic with all 28 bitangents defined over $\mathbb {Q}(t)$. By specialization, this also gives examples over $\mathbb {Q}$.

STANDARD MODEL BUNDLES OF THE HETEROTIC STRING

We show how to construct supersymmetric three-generation models with gauge group and matter content of the Standard Model in the framework of non-simply-connected elliptically fibered Calabi–Yau manifolds Z. The elliptic fibration on a cover Calabi–Yau, where the model has six generations of SU(5) and the bundle is given via the spectral cover description, has a second section leading to the needed free involution. The relevant involution on the defining spectral data of the bundle is identified for a general Calabi–Yau of this type and invariant bundles are generally constructible.

TOWARDS THE STANDARD MODEL SPECTRUM FROM ELLIPTIC CALABI–YAU MANIFOLDS

We show that it is possible to construct supersymmetric three-generation models with the Standard Model gauge group in the framework of non-simply-connected elliptically fibered Calabi–Yau threefolds, without section but with a bi-section. The fibrations on a cover Calabi–Yau threefold, where the model has six generations of SU(5) and the bundle is given via the spectral cover description, use a different description of the elliptic fiber which leads to more than one global section. We present two examples of a possible cover Calabi–Yau threefold with a free involution: one is a fiber product of rational elliptic surfaces dP9; another example is an elliptic fibration over a Hirzebruch surface. We compute the necessary amount of chiral matter by "turning on" a further parameter which is related to singularities of the fibration and the branching of the spectral cover.

Flows on hypermaps

The combinatorial investigation of graphs embedded on surfaces leads one to consider a pair of permutations (σ, α) that generate a transitive group [7]. The permutation α is a fixed-point-free involution and the pair is called a map. When this condition on α is dropped the combinatorial object that arises is called a hypermap. Both maps and hypermaps have a topological description: for maps a classical reference is [13] and for hypermaps such a description can be found in [4] and [6]; a brief account of it will be given below. However, the relationship between maps and hypermaps is not simply that the latter generalize the former. Actually, with every hypermap there is associated a map, its bipartite map, and conversely every bipartite map arises in this way. We do not enter into the details of this question; we refer the reader to the work of Walsh [16]. In this sense hypermaps are, at the same time, a generalization and a special case of maps.

Limits of stable homotopy and cohomotopy groups

In this paper we formulate and prove generalizations of a theorem of Lin [7]. Let X be a CW complex with base point x0. Define a free involution T on S∞×(X Λ X) by T (w, xΛy) = (−w, yΛx). The quadratic construction on X is the complexThis construction can be applied to spectra. A complete and thorough account will appear in the work on equivariant stable homotopy theory in preparation by L. G. Lewis, J. P. May, J. McLure and M. Steinberger. Some of the results are announced in [8].