arXiv subjects
Publications and source records attributed to Donald M. Davis.
We use the spectrum tmf to obtain new nonimmersion results for many real projective spaces RP^n for n as small as 113. The only new ingredient is some new calculations of tmf-cohomology groups. We present an expanded table of nonimmersion results. Our new theorem is new for 17% of the values of n between 2^i and 2^i + 2^14 for i > 14.
The span of a manifold is its maximum number of linearly independent vector fields. We discuss the question, still unresolved, of whether span(P^m x P^n) always equals span(P^m) + span(P^n). Here P^n denotes real projective space. We use BP-cohomology to obtain new upper bounds for span(P^m x P^n), much stronger than previously known bounds.
We study three connective versions of the spectrum for topological modular forms of level 3. All three were described briefly by Mahowald and Rezk in [Pure Appl Math Quar (2009)], but we add much detail to their discussion. Letting tmf(3) denote our connective model which is a ring spectrum, we compute the tmf(3)-homology of RP^infty.
We determine the p-exponent in many of the coefficients in the power series (log(1+x)/x)^t, where t is any integer. In our proof, we introduce a variant of multinomial coefficients. We also characterize the power series x/log(1+x) by certain zero coefficients in its powers.
We determine, within 1, the value of N for which sum (s1 choose i)(s2 choose N)(s1 choose N-i)(N choose i) achieves its maximum value. Here s1 and s2 are fixed integers. This problem arises in studying the most likely value for the size of the union of A, B, and C if A and C are disjoint sets of size s1, and B is a set of size s2. Attempting to remove the 1 unit of indeterminacy leads to interesting conjectures about a family of rational functions.
Let nbar=(n_1,...,n_r). The quotient space P_nbar:=(S^{n_1} x...x S^{n_r})/(x ~ -x)is what we call a projective product space. We determine the integral cohomology ring and the action of the Steenrod algebra. We give a splitting of Sigma P_nbar in terms of stunted real projective spaces, and determine when S^{n_i} is a product factor. We relate the immersion dimension and span of P_nbar to the much-studied sectioning question for multiples of the Hopf bundle over real projective spaces. We show that the immersion dimension of P_nbar depends only on min(n_i), sum n_i, and r, and determine its precise value unless all n_i exceed 9. We also determine exactly when P_nbar is parallelizable.
We prove that RP^{2^e-1} can be immersed in R^{2^{e+1}-e-7} provided e>6. If e>13, this is 1 better than previously known immersions. Our method is primarily an induction on geometric dimension, with compatibility of liftings being a central issue.
We determine the v1-periodic homotopy groups of all irreducible p-compact groups (BX,X). In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomials to these homotopy groups. We show that, if p is odd, every irreducible p-compact group has X of the homotopy type of a product of explicit spaces related to p-completed Lie groups.
Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence $$p^{°f}\sum_{k=r(mod p^a)}\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\sum_{i=a}^{\infty}[n/p^i]})$$ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If a is greater than one, and $l,s,t$ are nonnegative integers with $s,t<p$, then $$\frac{1}{[n/p^{a-1}]!} \sum_{k=r(mod p^a)} \binom{pn+s}{pk+t}(-1)^{pk}((k-r)/p^{a-1})^l =\frac {1}{[n/p^{a-1}]!} \sum_{k=r(mod p^a)}\binom{n}{k}\binom{s}{t}(-1)^k((k-r)/p^{a-1})^l (mod p).$$ We also present an application of the first congruence to Bernoulli polynomials, and apply the second congruence to show that a p-adic order bound given by the authors in a previous paper can be attained when p=2.
The Dwyer-Wilkerson space DI(4) is the only exotic 2-compact group. We compute its v1-periodic homotopy groups.
In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for RP^n with n as small as 48, and yet it was stated there that the first new result occurred when n=1536. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map f must satisfy f^*(X)=X_1+X_2. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in the original paper. Finally, we present a complete determination of tmf^{8*}(RP^\infty\times RP^\infty) and tmf^*(CP^\infty\times CP^\infty) in positive dimensions.
We use methods of combinatorial number theory to prove that, for each $n>1$ and any prime $p$, some homotopy group $π_i(SU(n))$ contains an element of order $p^{n-1+ord_p([n/p]!)}$, where $ord_p(m)$ denotes the largest integer $α$ such that $p^α$ divides $m$.
We prove the following two new optimal immersion results for complex projective space. First, if n equiv 3 mod 8 but n not equiv 3 mod 64, and alpha(n)=7, then CP^n can be immersed in R^{4n-14}. Second, if n is even and alpha(n)=3, then CP^n can be immersed in R^{4n-4}. Here alpha(n) denotes the number of 1's in the binary expansion of n. The first contradicts a result of Crabb, who said that such an immersion does not exist, apparently due to an arithmetic mistake.
For X a simply-connected finite H-space, there is a Bousfield-Kan spectral sequence which converges to the homotopy of its K-completion. When X=Spin(2n+1), we expect that these homotopy groups equal the v1-periodic homotopy groups in dimension greater than n^2. In this paper, we accomplish two things. (1) We prove that, for any X, the 1-line of this spectral sequence is determined in an explicit way from K-theory and Adams operations. (2) For X=Spin(2n+1), we make an explicit computation of this 1-line.
Bousfield recently gave a formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations. We apply Bousfield's theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of compact simple Lie groups, a project suggested by Mimura in 1989. The method involves no homotopy theoretic input and no spectral sequences. The input is the second eexterior power operation in the representation ring of E8, which we determine using specialized software. This can be interpreted as giving the Adams operation psi^2 in K(E8). Eigenvectors of psi^2 must also be eigenvectors of psi^k for any k. The matrix of these eigenvectors is the key to the analysis. Its determinant is closely related to the homotopy decomposition of E8 localized at each prime. By taking careful combinations of eigenvectors, we obtain a set of generators of K(E8) on which we have a nice formula for all Adams operations. Bousfield's theorem (and much Maple computation) allows us to obtain the v1-periodic homotopy groups.
We compute the 3-primary v1-periodic homotopy groups of the exceptional Lie group E7. Now E8 at the primes 3 and 5 is the only compact simple Lie group whose odd-primary v1-periodic homotopy groups remian to be computed. The main work is computing the unstable Novikov spectral sequence of ΩE7/Sp(2). Showing that this converges to v1-periodic homotopy groups requires recent work of Bousfield and Bendersky-Thompson.
We use obstruction theory to prove that if alpha(n)=2, then RP^{16n+8} cannot be immersed in R^{32n+3} and RP^{16n+10} cannot be immersed in R^{32n+11}, and that if alpha(n)>2, then RP^{8n+4} can be embedded in R^{16n+1}. These are new results.