Donald M. Davis
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View article: The connective KO-theory of the Eilenberg–MacLane space $$K({\mathbb Z}_2,2)$$, I: the $$E_2$$ page
The connective KO-theory of the Eilenberg–MacLane space $$K({\mathbb Z}_2,2)$$, I: the $$E_2$$ page Open
We compute the $$E_2$$ page of the Adams spectral sequence converging to the connective KO -theory of the second mod 2 Eilenberg–MacLane space, $$ko_*(K({\mathbb Z}_2,2))$$ , where $${\mathbb Z}_2$$ is the cyclic…
View article: Orientable manifolds with nonzero dual Stiefel-Whitney classes of largest possible grading
Orientable manifolds with nonzero dual Stiefel-Whitney classes of largest possible grading Open
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…
View article: The connective KO-theory of the Eilenberg-MacLane space K(Z_2,2), I: the E_2 page
The connective KO-theory of the Eilenberg-MacLane space K(Z_2,2), I: the E_2 page Open
We compute the $E_2$ page of the Adams spectral sequence converging to the connective KO-theory of the second mod 2 Eilenberg-MacLane space, $ko_*(K(Z/2,2))$. This required a careful analysis of the structure of $H^*(K(Z/2,2);Z_2)$ as a mo…
View article: The cohomology of the connective spectra for {K}-theory revisited
The cohomology of the connective spectra for {K}-theory revisited Open
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…
View article: Geodesic complexity of a cube
Geodesic complexity of a cube Open
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 geodesi…
View article: Geodesic complexity of a tetrahedron
Geodesic complexity of a tetrahedron Open
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.
View article: Isomorphism classes of cut loci for a cube
Isomorphism classes of cut loci for a cube Open
We prove that a face of a cube can be optimally partitioned into connected 193 sets on which the cut locus, or ridge tree, is constant up to isomorphism as a labeled graph. These are 60 connected open sets, curves bounding them, and inters…
View article: The connective K-theory of the Eilenberg-MacLane space K(Z/p,2)
The connective K-theory of the Eilenberg-MacLane space K(Z/p,2) Open
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-ext…
View article: Two robots moving geodesically on a tree
Two robots moving geodesically on a tree Open
We study the geodesic complexity of the ordered and unordered configuration\nspaces of graphs in both the $\\ell_1$ and $\\ell_2$ metrics. We determine the\ngeodesic complexity of the ordered two-point $\\varepsilon$-configuration space\no…
View article: The connective Morava K-theory of the second mod p Eilenberg-MacLane space
The connective Morava K-theory of the second mod p Eilenberg-MacLane space Open
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…
View article: Gorenstein Duality and Universal Coefficient Theorems
Gorenstein Duality and Universal Coefficient Theorems Open
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 …
View article: Duality in BP (co)homology
Duality in BP (co)homology Open
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 Pon…
View article: The mod-2 connected KU-homology of the Eilenberg-MacLane space K(Z/2,2)
The mod-2 connected KU-homology of the Eilenberg-MacLane space K(Z/2,2) Open
We compute the mod-2 connected KU-homology of the Eilenberg-MacLane space K(Z/2,2), using a novel Adams spectral sequence analysis.
View article: A note about immersions of orientable manifolds
A note about immersions of orientable manifolds Open
We give a new interpretation to decades-old results about immersions of all compact orientable n-manifolds in Euclidean space.
View article: Geodesics in the configuration spaces of two points in R^n
Geodesics in the configuration spaces of two points in R^n Open
We determine explicit formulas for geodesics (in the Euclidean metric) in the configuration space of ordered pairs (x,x') of points in R^n which satisfy d(x,x')>=epsilon. We interpret this as two or three (depending on the parity of n) geo…
View article: The geodesic complexity of n-dimensional Klein bottles
The geodesic complexity of n-dimensional Klein bottles Open
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…
View article: K-theory and immersions of spatial polygon spaces
K-theory and immersions of spatial polygon spaces Open
For ell a generic n-tuple of positive numbers, N(ell) denotes the space of isometry classes of oriented n-gons in R^3 with side lengths specified by ell. We determine the algebra K(N(ell)) and use this to obtain nonimmersions of the 2(n-3)…
View article: $BP$-homology of elementary abelian 2-groups: $BP$-module structure
$BP$-homology of elementary abelian 2-groups: $BP$-module structure Open
We determine the $BP_*$-module structure, mod higher filtration, of the main part of the $BP$-homology of elementary abelian 2-groups. The action is related to symmetric polynomials and to Dickson invariants.
View article: Manifold properties of planar polygon spaces
Manifold properties of planar polygon spaces Open
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 …
View article: The tangent bundle of planar polygon spaces
The tangent bundle of planar polygon spaces Open
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 …
View article: Bounds for higher topological complexity of real projective space\n implied by BP
Bounds for higher topological complexity of real projective space\n implied by BP Open
We use Brown-Peterson cohomology to obtain lower bounds for the higher\ntopological complexity, TC_k(RP^n), of real projective spaces, which are often\nmuch stronger than those implied by ordinary mod-2 cohomology.\n
View article: On the cohomology classes of planar polygon spaces
On the cohomology classes of planar polygon spaces Open
We obtain an explicit formula for the Poincare duality isomorphism H^{n-3}(Mbar(ell)) to Z/2 for the space of isometry classes of n-gons with specified side lengths, if ell is monogenic in the sense of Hausmann-Rodriguez. This has potentia…
View article: BP-homology of elementary abelian 2-groups: BP-module structure
BP-homology of elementary abelian 2-groups: BP-module structure Open
We determine the BP-module structure, mod higher filtration, of the main part of the BP-homology of elementary abelian 2-groups. The action is related to symmetric polynomials and to Dickson invariants.