arXiv subjects
Publications and source records attributed to Donald M. Davis.
An n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.
Recently, Cohen and Vandembroucq proved that the reduced topological complexity of the Klein bottle is 4. Simultaneously and independently, we announced a proof of the same result. Mistakes were found in our argument, which was quite different than theirs. After correcting these, we found that our description of the obstruction class agreed with theirs. Our approach to showing that this obstruction is nonzero failed to do so, while theirs did not fail. Here we discuss our approach, which deals more directly with the simplicial structure of the Klein bottle.
Hausmann and Rodriguez classified spaces of isometry classes of planar n-gons according to their genetic code, which is a collection of sets (called genes) containing n. Omitting the n yields what we call gees. We prove that, for a set of gees with largest gee of size k>0, the topological complexity (TC) of the associated space of n-gons is either 2n-5 or 2n-6 if n>2k+2. We present evidence that suggests that it is very rare that the TC is not equal to 2n-5 or 2n-6.
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 potential application to topological complexity.
Using information about the rational cohomology ring of the space of oriented isometry classes of planar n-gons with specified side lengths, we obtain bounds for the zero-divisor-cup-length (zcl) of these spaces, which provide lower bounds for their topological complexity (TC). In many cases our result about the cohomology ring is complete and we determine the precise zcl. We find that there will usually be a significant gap between the bounds for TC implied by zcl and dimensional considerations.
We determine lower bounds for the topological complexity of many planar polygon spaces mod isometry. With very few exceptions, the upper and lower bounds given by dimension and cohomology considerations differ by 1. This is true for 130 of the 134 generic 7-gon spaces. Our results apply to spaces of n-gons for all n, but primarily for those whose genetic codes, in the sense of Hausmann and Rodriguez, are moderately small.
Using known results about their mod-2 cohomology ring, we prove that the topological complexity of the space of isometry classes of n-gons in the plane with one side of length r and all others of length 1 equals either 2n-5 or 2n-6, provided that n-r is not an odd integer.
Using known results about their integral cohomology ring, we prove that the topological complexity of the space of n-gons in R^3 with prescribed side lengths equals 2n-5, assuming that the space is nonempty and does not contain any straight-line polygons.
We use ku-cohomology to determine lower bounds for the topological complexity of 2-torsion lens spaces. In the process, we give an almost-complete description of the tensor product of two copies of the ku-homology of infinite mod 2^e lens space, proving a conjecture of Gonzalez about the annihilator ideal of the bottom class. Our proof involves an elaborate row reduction of presentation matrices of arbitrary size.
We prove that real projective space RP^{n-3} is homeomorphic to the space of all isometry classes of n-gons in the plane with one side of length n-2 and all other sides of length 1. This makes the topological complexity of real projective space more relevant to robotics.
Let $P_n(x)=\frac1{n!}\sum\binom n{2i+1}(2i+1)^x$. This extends to a continuous function on the 2-adic integers, the $n$th 2-adic partial Stirling function. We show that $(-1)^{n+1}P_n$ is the only 2-adically continuous approximation to $S(x,n)$, the Stirling number of the second kind. We present extensive information about the zeros of $P_n$, for which there are many interesting patterns. We prove that if $e\ge2$ and $2^e+1\le n\le 2^e+4$, then $P_n$ has exactly $2^{e-1}$ zeros, one in each mod $2^{e-1}$ congruence. We study the relationship between the zeros of $P_{2^e+Δ}$ and $P_Δ$, for $1\leΔ\le 2^e$, and the convergence of $P_{2^e+Δ}(x)$ as $e\to\infty$.
Given k sets such that no one is contained in another, there is an associated lattice on the power set P([k]) corresponding to inclusion relations among unions of the sets. Two lattices on P([k]) are equivalent if there is a permutation of [k] under which they correspond. We show that for k=1, 2, 3, and 4, there are 1, 1, 4, and 50 equivalence classes of lattices on P([k]) obtained from sets in this way. We cannot find a reference to previous work on this enumeration problem in the literature, and so wish to introduce it for subsequent investigation. We explain how the problem arose from algebraic topology.
Let S(n,k) denote the Stirling numbers of the second kind. We prove that the p-adic limit of S(p^e a + c, p^e b + d) as e goes to infinity exists for all integers a, b, c, and d. We call the limiting p-adic integer S(p^\infty a + c, p^\infty b + d). When a equiv b mod (p-1) or d \le 0, we express them in terms of p-adic binomial coefficients introduced in a recent paper.
If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/p^{v(n)} the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)^{pke} u((kp^e)!) as e goes to infinity is a well-defined p-adic integer, which we call z_k. In terms of these, we give a formula for the p-adic limit of binom{a p^e +c, b p^e +d) as e goes to infinity, which we call binom(a p^\infty +c, b p^\infty +d). Here a \ge b are positive integers, and c and d are integers.
Let $f(n)=\sum_k \binom nk^{-1}$. In a previous paper, we defined for a p-adic integer x that f(x) is p-definable if lim $f(x_j)$ exists in $Q_p$, where $x_j$ denotes the mod $p^j$ reduction of $x$. We proved that if p is odd, then -1 is the only element of $Z_p-N$ for which f(x) is p-definable. For p=2, we proved that if the 1's in the binary expansion of x are eventually extraordinarily sparse, then f(x) is 2-definable. Here we present some conjectures that f(x) is 2-definable for many more 2-adic integers. We discuss the extent to which we can prove these conjectures.
Let f(n)= Sum binomial(n,k)^(-1). First, we show that f:N to Q_p is nowhere continuous in the p-adic topology. If x is a p-adic integer, we say that f(x) is p-definable if lim f(x_j) exists in Q_p, where x_j denotes the jth partial sum for x. We prove that f(-1) is p-definable for all primes p, and if p is odd, then -1 is the only element of Z_p - N for which f(x) is p-definable. For p=2, we show that if k is a positive integer, then f(-k-1) is not 2-definable, but that if the 1's in the binary expansion of x are eventually very sparse, then f(x) is 2-definable. Some of our proofs require that p satisfy one of two conditions. There are three small primes that do not satisfy the relevant condition, but our theorems can be proved directly for these primes. No other prime less than 100,000,000 fails to satisfy the conditions.
The partial Stirling numbers T_n(k) used here are defined as the sum over odd values of i of (n choose i) i^k. Their 2-exponents nu(T_n(k)) are important in algebraic topology. We provide many specific results, applying to all values of n, stating that, for all k in a certain congruence class mod 2^t, nu(T_n(k)) = nu(k - k0) + c0, where k0 is a 2-adic integer and c0 a positive integer. Our analysis involves several new general results for nu(sum (n choose 2i+1) i^j), the proofs of which involve a new family of polynomials. Following Clarke, we interpret T_n as a function on the 2-adic integers, and the 2-adic integers k0 described above as the zeros of these functions.
We determine precisely the largest v1-periodic homotopy groups of SU(2^e) and SU(2^e + 1). This gives new results about the largest actual homotopy groups of these spaces. Our proof relies on results about 2-divisibility of restricted sums of binomial coefficients times powers proved by the author in a companion paper.