arXiv subjects

Peter Zvengrowski

Publications and source records attributed to Peter Zvengrowski.

5 recordsLinked to original sources

arXiv2021

On generalized projective product spaces and Dold manifolds

D. Davis introduced projective product spaces in 2010 as a generalization of real projective spaces and discussed some of their topological properties. On the other hand, Dold manifolds were introduced by A. Dold in 1956 to study the generators of the non-oriented cobordism ring. Recently, in 2019, A. Nath and P. Sankaran made a modest generalization of Dold manifolds. In this paper we simultaneously generalize both the notions of projective product spaces and Dold manifolds, leading to infinitely many different classes of new smooth manifolds. Our main goal will be to study the integral homology groups. cohomology rings, stable tangent bundles, and vector field problems, on certain generalized projective product spaces and Dold manifolds.

math.AT↗

arXiv2020

Vector fields on projective Stiefel manifolds and the Browder-Dupont invariant

We develop strong lower bounds for the span of the projective Stiefel manifolds $X_{n,r}=O(n)/(O(n-r)\times \mathbb Z/2)$, which enable very accurate (in many cases exact) estimates of the span. The technique, for the most part, involves elementary stability properties of vector bundles. However, the case $X_{n,2}$ with $n$ odd presents extra difficulties, which are partially resolved using the Browder-Dupont invariant. In the process, we observe that the symmetric lift due to Sutherland does not necessarily exist for all odd dimensional closed manifolds, and therefore the Browder-Dupont invariant, as he formulated it, is not defined in general. We will characterize those $n$'s for which the Browder-Dupont invariant is well-defined on $X_{n,2}$. Then the invariant will be used in this case to obtain the lower bounds for the span as a corollary of a stronger result.

math.GT↗

arXiv2015

A Generalized Goursat Lemma

In this note the usual Goursat lemma, which describes subgroups of the direct product of two groups, is generalized to describing subgroups of a direct product $A_1\times A_2 \times...\times A_n$ of a finite number of groups. Other possible generalizations are discussed and applications characterizing several types of subgroups are given. Most of these applications are straightforward, while somewhat deeper applications occur in the case of profinite groups, cyclic groups, and the Sylow $p$-subgroups (including infinite groups that are virtual $p$-groups).

math.GR↗

arXiv2012

Horizontal Monotonicity of the Modulus of the Riemann Zeta Function and Related Functions

It is shown that the absolute values of Riemann's zeta function and two related functions strictly decrease when the imaginary part of the argument is fixed to any number with absolute value at least 8 and the real part of the argument is negative and increases up to 0; extending this monotonicity to the increase of the real part up to 1/2 is shown to be equivalent to the Riemann Hypothesis. Another result is a double inequality relating the real parts of the logarithmic derivatives of the three functions under consideration.

math.NT↗

arXiv2009

Remarks on the cohomology of finite fundamental groups of 3-manifolds

Computations based on explicit 4-periodic resolutions are given for the cohomology of the finite groups G known to act freely on S^3, as well as the cohomology rings of the associated 3-manifolds (spherical space forms) M = S^3/G. Chain approximations to the diagonal are constructed, and explicit contracting homotopies also constructed for the cases G is a generalized quaternion group, the binary tetrahedral group, or the binary octahedral group. Some applications are briefly discussed.

math.AT↗