Betti Numbers and Injectivity Radii
نویسندگان
چکیده
The theme of this paper is the connection between topological properties of a closed orientable hyperbolic 3-manifold M and the maximal injectivity radius of M . In [4] we showed that if the first Betti number of M is at least 3 then the maximal injectivity radius of M is at least log 3. By contrast, the best known lower bound for the maximal injectivity radius of M with no topological restriction on M is the lower bound of arcsinh( 4 ) = 0.24746 . . . due to Przeworski [7]. One of the results of this paper, Corollary 4, gives a lower bound of 0.32798 for the case where the first Betti number of M is 2 and M does not contain a “fibroid” (see below). Our main result, Theorem 3, is somewhat stronger than this
منابع مشابه
On a special class of Stanley-Reisner ideals
For an $n$-gon with vertices at points $1,2,cdots,n$, the Betti numbers of its suspension, the simplicial complex that involves two more vertices $n+1$ and $n+2$, is known. In this paper, with a constructive and simple proof, wegeneralize this result to find the minimal free resolution and Betti numbers of the $S$-module $S/I$ where $S=K[x_{1},cdots, x_{n}]$ and $I$ is the associated ideal to ...
متن کاملPositive Pinching , Volume and Second Betti Number
Our main theorem asserts that for all odd n ≥ 3 and 0 < δ ≤ 1, there exists a small constant, i(n, δ) > 0, such that if a simply connected n-manifold, M , with vanishing second Betti number admits a metric of sectional curvature, δ ≤ KM ≤ 1, then the injectivity radius of M is greater than i(n, δ).
متن کاملGauss Equation and Injectivity Radii for Subspaces in Spaces of Curvature Bounded Above
A Gauss Equation is proved for subspaces of Alexandrov spaces of curvature bounded above by K. That is, a subspace of extrinsic curvature ≤ A, defined by a cubic inequality on the difference of arc and chord, has intrinsic curvature ≤ K +A. Sharp bounds on injectivity radii of subspaces, new even in the Riemannian case, are derived.
متن کاملNonuniform Thickness and Weighted Distance
Nonuniform tubular neighborhoods of curves in Rn are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtained for nonuniform thickness. All singularities within almost injectivity radius are classifie...
متن کاملPrecompactness of solutions to the Ricci flow in the absence of injectivity radius estimates
Consider a sequence of pointed n–dimensional complete Riemannian manifolds {(Mi, gi(t), Oi)} such that t ∈ [0, T ] are solutions to the Ricci flow and gi(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n–dimensional solution to the Ricci...
متن کامل