We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZalgebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indica...

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).

We use Klee’s Dehn-Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f -vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k an...

We use Klee’s Dehn-Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f -vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k an...

Topological complexity of semialgebraic sets in R has been studied by many researchers over the past fifty years. An important measure of the topological complexity are the Betti numbers. Quantitative bounds on the Betti numbers of a semialgebraic set in terms of various parameters (such as the number and the degrees of the polynomials defining it, the dimension of the set etc.) have proved use...

The purpose of this note is to obtain a restriction on the fundamental groups of nonelliptic compact complex surfaces of class VII in Kodaira’s classification [9]. We recall that these are the compact complex surfaces with first Betti number one and no nonconstant meromorphic functions. This seems to be the class of compact complex surfaces whose structure is least understood. The first and sim...

We study the rate of growth normalized Hodge numbers along a tower abelian covers smooth projective variety with semismall Albanese map. These bounds are in some cases optimal. Moreover, we compute L 2 -Betti irregular varieties that satisfy weak generic Nakano vanishing theorem (e.g., map). Finally, convergence plurigenera towers any variety. As an application, extend result Kollár concerning ...

We prove an upper bound on the rank of abelianised revised fundamental group (called “revised first Betti number”) a compact $\mathsf{RCD^{}}(K,N)$ space, in same spirit celebrated Gromov–Gallot number for smooth Riemannian manifold with Ricci curvature bounded below. When synthetic lower is close enough to (negative) zero and aforementioned saturated (i.e. equal integer part $N$, denoted by $\...