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Donald M. Davis

Publications and source records attributed to Donald M. Davis.

At least 19 recordsLinked to original sources
arXiv2026

Spin manifolds with nonzero dual Stiefel-Whitney classes of large grading

The dual Stiefel-Whitney classes wbar_j(M) of a manifold M are elements of H^j(M;Z/2) which give information about embedding and immersing M in Euclidean space. We consider the question of finding, for each n, the largest j such that there is an n-dimensional Spin manifold with wbar_j nonzero. We obtain upper and lower bounds. Since one theorem proves existence of manifolds without giving explicit examples, we consider a parallel question of finding explicit manifolds with wbar_j nonzero for j as large as possible.

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arXiv2025

Some 2-adic integers related to the odd part of 2^e!

The odd part of 2^e! as e approaches infinity leads to a 2-adic integer z. The bits of z were publicized in OEIS-A359349, where two conjectures were made, relevant to computing z. We prove both of those conjectures. A second 2-adic integer, the limit of ((2^e-1)!!-1)/2^e, plays a key role in one proof.

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arXiv2025

Orientable manifolds with nonzero dual Stiefel-Whitney classes of largest possible grading

It is known that, for all n, there exist compact differentiable orientable n-manifolds with dual Stiefel-Whitney class wbar_{n-ahat(n)} nonzero, and this is best possible, but the proof is nonconstructive. Here ahat(n) equals the number of 1's in the binary expansion of n if n equiv 1 mod 4 and exceeds this by 1 otherwise. We find, for all n nonzero mod 4, examples of real Bott manifolds with this property.

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arXiv2024

The cohomology of the connective spectra for {K}-theory revisited

The stable mod 2 cohomologies of the spectra for connective real and complex K-theories are well known and easy to work with. However, the known bases are in terms of the anti-automorphism of Milnor basis elements. We offer simple bases in terms of admissible sequences of Steenrod operations that come from the Adem relations. In particular, the basis for the complex case is that you don't use any Steenrod operations in degree one or $2^n+1$, $n > 0$.

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arXiv2023

Geodesic complexity of a cube

The topological (resp. geodesic) complexity of a topological (resp. metric) space is roughly the smallest number of continuous rules required to choose paths (resp. shortest paths) between any points of the space. We prove that the geodesic complexity of a cube exceeds its topological complexity by exactly 2. The proof involves a careful analysis of cut loci of the cube.

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arXiv2023

Geodesic complexity of a tetrahedron

We prove that the geodesic complexity of a regular tetrahedron exceeds its topological complexity by 1 or 2. The proof involves a careful analysis of minimal geodesics on the tetrahedron.

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arXiv2023

The connective K-theory of the Eilenberg-MacLane space K(Z/p,2)

We compute ku^*(K(Z/p,2)) and ku_*(K(Z/p,2)), the connective KU-cohomology and connective KU-homology groups of the mod-p Eilenberg-MacLane space K(Z/p,2), using the Adams spectral sequence. We obtain a striking interaction between h_0-extensions and exotic extensions. The mod-p connective KU-cohomology groups, computed elsewhere, are needed in order to establish higher differentials and exotic extensions in the integral groups.

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arXiv2022

Gorenstein Duality and Universal Coefficient Theorems

The paper describes a duality phenomenon for cohomology theories with the character of Gorenstein rings. For a connective cohomology theory with the p-local integers in degree 0, and coefficient ring R_* Gorenstein of shift 0, this states that for X with R_*(X) torsion, we have R^*(X)=Σ^a Hom( R_*(X), Z/p^{\infty}). A corresponding statement for modules over a commutative Gorenstein ring spectrum is also proved. [Minor typographical and bibliographic changes to the last version.]

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arXiv2022

The connective Morava K-theory of the second mod p Eilenberg-MacLane space

We develop tools for computing the connective n-th Morava K-theory of spaces. Starting with a Universal Coefficient Theorem that computes the cohomology version from the homology version, we show that every step in the process of computing one is mirrored in the other and that this can be used to make computations. As our example, we compute the connective n-th Morava K-theory of the second mod p Eilenberg-MacLane space.

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arXiv2022

Duality in BP (co)homology

Let E=BP denote the Johnson-Wilson spectrum, localized at p. It is proved that if E_*(X) is locally finite, then there is an isomorphism of right E_*-modules E^*(X) = (E_*(Sigma^{D+n+1}X))^V, where D=Sum |v_i| and M^V=Hom(M,Q/Z) is the Pontryagin dual. This result was motivated by work of the author and W.S.Wilson regarding the 2-local ku-homology and -cohomology of the Eilenberg-MacLane space K(Z/2,2).

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arXiv2021

Geodesic complexity for non-geodesic spaces

We define the notion of near geodesic between points of a metric space when no geodesic exists, and use this to extend Recio-Mitter's notion of geodesic complexity to non-geodesic spaces. This has potential application to topological robotics. We determine explicit near geodesics and geodesic complexity in a variety of cases.

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arXiv2021

Stiefel-Whitney classes and immersions of orientable and Spin manifolds

We determine a nice simple formula for the largest Euclidean space for which there is an orientable n-manifold with a nonimmersion detected by Stiefel-Whitney classes. For Spin manifolds, we prove the analogue of the upper bound and establish the complete answer for n<24 and n=33,34. Results similar to many of these were obtained some 50 years ago, but in a much less tractable form. The sharp results for Spin manifolds require detailed calculations of ko-homology groups of mod-2 Eilenberg-MacLane spaces.

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arXiv2020

Two robots moving geodesically on a tree

We study the geodesic complexity of the ordered and unordered configuration spaces of graphs in both the $\ell_1$ and $\ell_2$ metrics. We determine the geodesic complexity of the ordered two-point $\varepsilon$-configuration space of any star graph in both the $\ell_1$ and $\ell_2$ metrics and of the unordered two-point configuration space of any tree in the $\ell_1$ metric, by finding explicit geodesics from any pair to any other pair, and arranging them into a minimal number of continuously-varying families. In each case the geodesic complexity matches the known value of the topological complexity.

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arXiv2019

The geodesic complexity of n-dimensional Klein bottles

The geodesic complexity of a metric space X is the smallest k for which there is a partition of X x X into ENRs E_0,...,E_k on each of which there is a continuous choice of minimal geodesic sigma(x_0,x_1) from x_0 to x_1. We prove that the geodesic complexity of an n-dimensional Klein bottle equals 2n. Its topological complexity remains unknown for n>2.

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arXiv2019

On the unordered configuration space C(RP^n,2)

We prove that, if n is a 2-power, the unordered configuration space C(RP^n,2) cannot be immersed in R^{4n-2} nor embedded as a closed subspace of R^{4n-1}, optimal results, while if n is not a 2-power, C(RP^n,2) can be immersed in R^{4n-3}. We also obtain cohomological lower bounds for the topological complexity of C(RP^n,2), which are nearly optimal when n is a 2-power. We also give a new description of the mod-2 cohomology algebra of the Grassmann manifold G_{n+1,2}.

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arXiv2018

Manifold properties of planar polygon spaces

We prove that the tangent bundle of a generic space of planar n-gons with specified side lengths, identified under isometry, plus a trivial line bundle is isomorphic to (n-2) times a canonical line bundle. We then discuss consequences for orientability, cobordism class, immersions, and parallelizability.

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