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Register a new userOn the torsion in the center conjecture
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We present a condition for towers of fiber bundles which implies that the fundamental group of the total space has a nilpotent subgroup of finite index whose torsion is contained in its center. Moreover, the index of the subgroup can be bounded in terms of the fibers of the tower. Our result is motivated by the conjecture that every almost nonnegatively curved closed m-dimensional manifold M admits a finite cover M for which the number of leafs is bounded in terms of m such that the torsion of the fundamental group of M lies in its center.
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We show that the analogue of the Peterson conjecture on the action of Steenrod squares does not hold in motivic cohomology.
We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M, omega, tau)$ with nonempty and compact real part $L={rm Fix}(tau)$. For given $Lambdain (0, +infty]$ and $minNcup{0}$ we show the equivalence of the following two claims: (i) $sharp(Lcapphi^H_1(L))ge m$ for any Hamiltonian function $Hin C_0^infty([0, 1]times M)$ with Hofers norm $|H|<Lambda$; (ii) $sharp {cal P}(H,tau)ge m$ for every $Hin C^infty_0(R/Ztimes M)$ satisfying $H(t,x)=H(-t,tau(x));forall (t,x)inmathbb{R}times M$ and with Hofers norm $|H|<2Lambda$, where ${cal P}(H, tau)$ is the set of all $1$-periodic solutions of $dot{x}(t)=X_{H}(t,x(t))$ satisfying $x(-t)=tau(x(t));forall tinR$ (which are also called brake orbits sometimes). Suppose that $(M, omega)$ is geometrical bounded for some $Jin{cal J}(M,omega)$ with $tau^ast J=-J$ and has a rationality index $r_omega>0$ or $r_omega=+infty$. Using Hofers method we prove that if the Hamiltonian $H$ in (ii) above has Hofers norm $|H|<r_omega$ then $sharp(Lcapphi^H_1(L))gesharp {cal P}_0(H,tau)ge {rm Cuplength}_{F}(L)$ for $F=Z_2$, and further for $F=Z$ if $L$ is orientable, where ${cal P}_0(H,tau)$ consists of all contractible solutions in ${cal P}(H,tau)$.
We study the analytic torsion of odd-dimensional hyperbolic orbifolds $Gamma backslash mathbb{H}^{2n+1}$, depending on a representation of $Gamma$. Our main goal is to understand the asymptotic behavior of the analytic torsion with respect to sequences of representations associated to rays of highest weights.
The evenness conjecture for the equivariant unitary bordism groups states that these bordism groups are free modules over the unitary bordism ring on even-dimensional generators. In this paper we review the cases in which the conjecture is known to hold and we highlight the properties that permit one to prove the conjecture in these cases.
Let an n-algebra mean an algebra over the chain complex of the little n-cubes operad. We give a proof of Kontsevichs conjecture, which states that for a suitable notion of Hochschild cohomology in the category of n-algebras, the Hochschild cohomology complex of an n-algebra is an (n+1)-algebra. This generalizes a conjecture by Deligne for n=1, now proven by several authors.
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