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Register a new userComputing $ u$-invariants of Joyces compact $G_2$-manifolds
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Christopher Scaduto
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Crowley and Nordstr{o}m introduced an invariant of $G_2$-structures on the tangent bundle of a closed 7-manifold, taking values in the integers modulo 48. Using the spectral description of this invariant due to Crowley, Goette and Nordstr{o}m, we compute it for many of the closed torsion-free $G_2$-manifolds defined by Joyces generalized Kummer construction.
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Froyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in three and four-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal $L$-spaces. In particular, we study relations between Froyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive effective upper bounds for the Froyshov invariants of minimal hyperbolic $L$-spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Froyshov invariants of minimal $L$-spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Froyshov invariants for all spin$^c$ structures on the Seifert-Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.
Let $text{G}(n)$ be equal either to $text{PO}(n,1),text{PU}(n,1)$ or $text{PSp}(n,1)$ and let $Gamma leq text{G}(n)$ be a uniform lattice. Denote by $mathbb{H}^n_K$ the hyperbolic space associated to $text{G}(n)$, where $K$ is a division algebra over the reals of dimension $d=dim_{mathbb{R}} K$. Assume $d(n-1) geq 2$. In this paper we generalize natural maps to measurable cocycles. Given a standard Borel probability $Gamma$-space $(X,mu_X)$, we assume that a measurable cocycle $sigma:Gamma times X rightarrow text{G}(m)$ admits an essentially unique boundary map $phi:partial_infty mathbb{H}^n_K times X rightarrow partial_infty mathbb{H}^m_K$ whose slices $phi_x:mathbb{H}^n_K rightarrow mathbb{H}^m_K$ are atomless for almost every $x in X$. Then, there exists a $sigma$-equivariant measurable map $F: mathbb{H}^n_K times X rightarrow mathbb{H}^m_K$ whose slices $F_x:mathbb{H}^n_K rightarrow mathbb{H}^m_K$ are differentiable for almost every $x in X$ and such that $text{Jac}_a F_x leq 1$ for every $a in mathbb{H}^n_K$ and almost every $x in X$. The previous properties allow us to define the natural volume $text{NV}(sigma)$ of the cocycle $sigma$. This number satisfies the inequality $text{NV}(sigma) leq text{Vol}(Gamma backslash mathbb{H}^n_K)$. Additionally, the equality holds if and only if $sigma$ is cohomologous to the cocycle induced by the standard lattice embedding $i:Gamma rightarrow text{G}(n) leq text{G}(m)$, modulo possibly a compact subgroup of $text{G}(m)$ when $m>n$. Given a continuous map $f:M rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.
It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their rational cohomology algebra already -- an as prominent as particular property in rational homotopy theory. Special interest now lies on exceptional holonomy $G_2$ and $Spin(7)$. In this article we provide a method of how to confirm that the famous Joyce examples of holonomy $G_2$ indeed are formal spaces; we concretely exert this computation for one example which may serve as a blueprint for the remaining Joyce examples (potentially also of holonomy $Spin(7)$). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly Kahler manifolds. A connection between these two results can be found in the fact that both special holonomy and nearly Kahler naturally generalize compact Kahler manifolds, whose formality is a classical and celebrated theorem by Deligne-Griffiths-Morgan-Sullivan.
C. Giller proposed an invariant of ribbon 2-knots in S^4 based on a type of skein relation for a projection to R^3. In certain cases, this invariant is equal to the Alexander polynomial for the 2-knot. Gillers invariant is, however, a symmetric polynomial -- which the Alexander polynomial of a 2-knot need not be. After modifying a 2-knot into a Montesinos twin in a natural way, we show that Gillers invariant is related to the Seiberg-Witten invariant of the exterior of the twin, glued to the complement of a fiber in E(2).
M-theory compactified on $G_2$-holonomy manifolds results in 4d $mathcal{N}=1$ supersymmetric gauge theories coupled to gravity. In this paper we focus on the gauge sector of such compactifications by studying the Higgs bundle obtained from a partially twisted 7d super Yang-Mills theory on a supersymmetric three-cycle $M_3$. We derive the BPS equations and find the massless spectrum for both abelian and non-abelian gauge groups in 4d. The mathematical tool that allows us to determine the spectrum is Morse theory, and more generally Morse-Bott theory. The latter generalization allows us to make contact with twisted connected sum (TCS) $G_2$-manifolds, which form the largest class of examples of compact $G_2$-manifolds. M-theory on TCS $G_2$-manifolds is known to result in a non-chiral 4d spectrum. We determine the Higgs bundle for this class of $G_2$-manifolds and provide a prescription for how to engineer singular transitions to models that have chiral matter in 4d.
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