An Exotic Free Involution on S 4

1981 ◽  
Vol 113 (2) ◽  
pp. 357 ◽  
Author(s):  
Ronald Fintushel ◽  
Ronald J. Stern
Keyword(s):  
Free Involution

scholarly journals STANDARD MODEL BUNDLES OF THE HETEROTIC STRING

2006 ◽  
Vol 21 (06) ◽  
pp. 1261-1281 ◽  
Author(s):  
GOTTFRIED CURIO
Keyword(s):  
Standard Model ◽  
Spectral Data ◽  
Gauge Group ◽  
Matter Content ◽  
Heterotic String ◽  
Spectral Cover ◽  
The Standard Model ◽  
Free Involution ◽  
Simply Connected ◽  
Calabi Yau Manifolds

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.

scholarly journals CONFORMAL COURANT ALGEBROIDS AND ORIENTIFOLD T-DUALITY

2012 ◽  
Vol 10 (02) ◽  
pp. 1250084 ◽  
Author(s):  
DAVID BARAGLIA
Keyword(s):  
Line Bundle ◽  
Bilinear Form ◽  
Conformal Structure ◽  
Crossed Module ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Circle Bundles ◽  
Twisted Cohomology ◽  
Free Involution ◽  
Special Case

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.

scholarly journals TOWARDS THE STANDARD MODEL SPECTRUM FROM ELLIPTIC CALABI–YAU MANIFOLDS

2004 ◽  
Vol 19 (12) ◽  
pp. 1987-2014 ◽  
Author(s):  
BJÖRN ANDREAS ◽  
GOTTFRIED CURIO ◽  
ALBRECHT KLEMM
Keyword(s):  
Standard Model ◽  
Global Section ◽  
Elliptic Surfaces ◽  
Spectral Cover ◽  
Elliptic Fiber ◽  
The Standard Model ◽  
Free Involution ◽  
Simply Connected ◽  
Rational Elliptic Surfaces ◽  
Standard Model Gauge Group

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.

scholarly journals Framed manifolds with a fixed point free involution.

1976 ◽  
Vol 23 (3) ◽  
pp. 257-260 ◽  
Author(s):  
Edgar H. Brown
Keyword(s):  
Fixed Point ◽  
Free Involution

Limits of stable homotopy and cohomotopy groups

1983 ◽  
Vol 94 (3) ◽  
pp. 473-482 ◽  
Author(s):  
J. D. S. Jones ◽  
S. A. Wegmann
Keyword(s):  
Homotopy Theory ◽  
Base Point ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
Free Involution ◽  
Image Position ◽  
Equivariant Stable Homotopy

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].

On the ARF Invariant of an Involution

1970 ◽  
Vol 22 (3) ◽  
pp. 519-524 ◽  
Author(s):  
Peter Orlik
Keyword(s):  
Abelian Group ◽  
Smooth Manifold ◽  
Finite Abelian Group ◽  
Homotopy Equivalent ◽  
Smooth Manifolds ◽  
Connected Sum ◽  
Group Operation ◽  
Free Involution ◽  
Arf Invariant

Let Σ4k+1 denote a smooth manifold homeomorphic to the (4k + 1)-sphere, S4k+1k ≧ 1, and T: Σ4k+1 → Σ4k+1 a differentiate free involution. Our aim in this note is to derive a connection between the differentiate structure on Σ4k+1 and the properties of the free involution T.To be more specific, recall [5] that the h-cobordism classes of smooth manifolds homeomorphic (or, what is the same, homotopy equivalent) to S4k+1, k ≧ 1, form a finite abelian group θ4k+1 with group operation connected sum. The elements are called homotopy spheres. Those homotopy spheres that bound parallelizable manifolds form a subg roup bP4k+2 ⊂ θ4k+1. It is proved in [5, Theorem 8.5] that bP4k+2 is either zero or cyclic of order 2. In the latter case the two distinct homotopy spheres are distinguished by the Arf invariant of the parallelizable manifolds they bound.

Continuous Families of Curves

1966 ◽  
Vol 18 ◽  
pp. 529-537 ◽  
Author(s):  
Branko Grünbaum
Keyword(s):  
Fixed Point ◽  
Jordan Curve ◽  
Convex Sets ◽  
General Setting ◽  
Jordan Curve Theorem ◽  
Planar Line ◽  
Families Of Curves ◽  
Planar Convex ◽  
Free Involution

The present paper is an attempt to find the unifying principle of results obtained by different authors and dealing—in the original papers—with areabisectors, chords, or diameters of planar convex sets, with outwardly simple planar line families, and with chords determined by a fixed-point free involution on a circle. The proofs in the general setting seem to be simpler and are certainly more perspicuous than many of the original ones. The tools required do not transcend simple continuity arguments and the Jordan curve theorem. The author is indebted to the referee for several helpful remarks.

Free S1-actions and involutions on homotopy seven spheres

1973 ◽  
Vol 73 (3) ◽  
pp. 455-458 ◽  
Author(s):  
K. H. Mayer
Keyword(s):  
Vector Bundle ◽  
Rational Function ◽  
Equivalence Classes ◽  
Free Involution

In this note the concept of modest vector bundle, defined in (5), is used to define an invariant for certain free S1-actions. This invariant is a rational function and classifies the equivalence classes of free S1-actions on homotopy seven spheres. In a similar way an invariant for free involutions is defined, which is a generalization of the Spin invariant defined in (4). Using this invariant and Hirzebruch's α-invariant it is possible to state conditions for a free involution of a homotopy seven sphere to be embedded in a free S1-action.

scholarly journals The arithmetic of Prym varieties in genus 3

2008 ◽  
Vol 144 (2) ◽  
pp. 317-338 ◽  
Author(s):  
Nils Bruin
Keyword(s):  
Double Cover ◽  
Hasse Principle ◽  
Prym Variety ◽  
Prym Varieties ◽  
Arithmetic Properties ◽  
Field Of Definition ◽  
Smooth Plane ◽  
Free Involution ◽  
Genus 2 ◽  
Unramified Double Cover

AbstractGiven 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 $\mathbb {Q}$.

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