Volume 29, Issue 2

In this short note we observe that the Hilali conjecture holds for 2-stage spaces, i.e. we argue that the dimension of the rational cohomology is at least as large as the dimension of the rational homotopy groups for these spaces. We also prove the Hilali conjecture for a class of spaces which puts it into the context of fibrations.

We prove that for every ϵ>0${\epsilon>0}$ there exists a δ>0${\delta>0}$ such that every group of order n≥3${n\geq 3}$ has at least δ⁢log2⁡n/(log2⁡log2⁡n)3+ϵ${\delta\log_{2}n/{(\log_{2}\log_{2}n)}^{3+\epsilon}}$ conjugacy classes. This sharpens earlier results of Pyber and Keller. Bertram speculates whether it is true that every finite group of order n has more than log3⁡n${\log_{3}n}$ conjugacy classes. We answer Bertram’s question in the affirmative for groups with a trivial solvable radical.

In this paper we prove a family of identities for Bernoulli numbers parameterized by triples of integers (a,b,c)${(a,b,c)}$ with a+b+c=n-1${a+b+c=n-1}$, n≥4${n\geq 4}$. These identities are deduced by translating into homotopical terms the gauge action on the Maurer–Cartan set of a differential graded Lie algebra. We show that Euler and Miki’s identities, well-known and apparently non-related formulas, are linear combinations of our family and they satisfy a particular symmetry relation.

In this paper we construct faithful representations of saturated fusion systems over discrete p -toral groups and use them to find conditions that guarantee the existence of unitary embeddings of p -local compact groups. These conditions hold for the Clark–Ewing and Aguadé–Zabrodsky p -compact groups. We also show the existence of unitary embeddings of finite loop spaces.

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.

Let ℕ${{\mathbb{N}}}$ be the set of nonnegative integers. A problem about how to transport profitably an organized group of persons leads us to study the set T formed by the integers n such that the system of inequalities, with nonnegative integer coefficients, a1⁢x1+⋯+ap⁢xp<n<b1⁢x1+⋯+bp⁢xp$a_{1}x_{1}+\cdots+a_{p}x_{p}<n<b_{1}x_{1}+\cdots+b_{p}x_{p}$ has at least one solution in ℕp${{\mathbb{N}}^{p}}$. We will see that T∪{0}${T\cup\{0\}}$ is a numerical semigroup. Moreover, we will show that a numerical semigroup S can be obtained in this way if and only if {a+b-1,a+b+1}⊆S${\{a+b-1,a+b+1\}\subseteq S}$, for all a,b∈S∖{0}${a,b\in S\setminus\{0\}}$. In addition, we will demonstrate that such numerical semigroups form a Frobenius variety and we will study this variety. Finally, we show an algorithmic process in order to compute T .

Let p be a prime and let p1,…,pr${p_{1},\ldots,p_{r}}$ be distinct prime divisors of p-1${p-1}$. We prove that the smallest positive integer n which is a simultaneous p1,…,pr${p_{1},\ldots,p_{r}}$-power nonresidue modulo p satisfies n<p1/4-cr+o⁢(1) (p→∞)$n<p^{1/4-c_{r}+o(1)}\quad(p\to\infty)$ for some positive cr${c_{r}}$ satisfying cr=e-(1+o⁢(1))⁢r${c_{r}=e^{-(1+o(1))r}}$ as r→∞${r\to\infty}$.

In [1] Casas-Alvero found decompositions of higher order polars of an irreducible plane complex analytic curve generalizing the results of Merle. We improve his result obtaining a finer decomposition where we find out a kind of branches that we call threshold semi-roots . The existence of threshold semi-roots is a new phenomenon observed for the higher order polars. The topological type and the number of these branches is determined by the topological type of the original curve.

We consider the Wiener sausage for a Brownian motion with a constant drift up to time t associated with a closed ball. In the two or more dimensional cases, we obtain the explicit form of the expected volume of the Wiener sausage. The result says that it can be represented by the sum of the mean volumes of the multi-dimensional Wiener sausages without a drift. In addition, we show that the leading term of the expected volume of the Wiener sausage is written as κ⁢t⁢(1+o⁢[1])${\kappa t(1+o[1])}$ for large t by a constant κ. The expression for κ is of a complicated form, but it converges to the known constant as the drift tends to 0.

We describe the slice spectral sequence of a 32-periodic C4$C_{4}$-spectrum K[2]$K_{[2]}$ related to the C4$C_{4}$ norm MU((C4))=NC2C4⁢MUℝ${\mathrm{MU}^{((C_{4}))}=N_{C_{2}}^{C_{4}}\mathrm{MU}_{\mathbb{R}}}$ of the real cobordism spectrum MUℝ$\mathrm{MU}_{\mathbb{R}}$. We will give it as a spectral sequence of Mackey functors converging to the graded Mackey functor π¯*⁢K[2]$\underline{\pi}_{*}K_{[2]}$, complete with differentials and exotic extensions in the Mackey functor structure. The slice spectral sequence for the 8-periodic real K -theory spectrum Kℝ$K_{\mathbb{R}}$ was first analyzed by Dugger. The C8$C_{8}$ analog of K[2]$K_{[2]}$ is 256-periodic and detects the Kervaire invariant classes θj$\theta_{j}$. A partial analysis of its slice spectral sequence led to the solution to the Kervaire invariant problem, namely the theorem that θj$\theta_{j}$ does not exist for j≥7${j\geq 7}$.

Let 𝔽q${\mathbb{F}_{q}}$ be a finite field of q elements, where q is a large odd prime power and Q=a1⁢x1c1+⋯+ad⁢xdcd∈𝔽q⁢[x1,…,xd],${Q=a_{1}x_{1}^{c_{1}}+\cdots+a_{d}x_{d}^{c_{d}}\in\mathbb{F}_{q}[x_{1},\ldots,% x_{d}]},$ where 2≤ci≤N${2\leq c_{i}\leq N}$, gcd⁡(ci,q)=1${\gcd(c_{i},q)=1}$, and ai∈𝔽q${a_{i}\in\mathbb{F}_{q}}$ for all 1≤i≤d${1\leq i\leq d}$. A Q -sphere is a set of the form {𝒙∈𝔽qd∣Q⁢(𝒙-𝒃)=r},${\bigl{\{}\boldsymbol{x}\in\mathbb{F}_{q}^{d}\mid Q(\boldsymbol{x}-\boldsymbol% {b})=r\bigr{\}}},$ where 𝒃∈𝔽qd,r∈𝔽q${\boldsymbol{b}\in\mathbb{F}_{q}^{d},r\in\mathbb{F}_{q}}$. We prove bounds on the number of incidences between a point set 𝒫${{{\mathcal{P}}}}$ and a Q -sphere set 𝒮${{{\mathcal{S}}}}$, denoted by I⁢(𝒫,𝒮)${I({{\mathcal{P}}},{{\mathcal{S}}})}$, as the following: |I⁢(𝒫,𝒮)-|𝒫|⁢|𝒮|q|≤qd/2⁢|𝒫|⁢|𝒮|.$\Biggl{|}I({{\mathcal{P}}},{{\mathcal{S}}})-\frac{|{{\mathcal{P}}}||{{\mathcal% {S}}}|}{q}\Biggr{|}\leq q^{d/2}\sqrt{|{{\mathcal{P}}}||{{\mathcal{S}}}|}.$ We also give a version of this estimate over finite cyclic rings ℤ/q⁢ℤ${\mathbb{Z}/q\mathbb{Z}}$, where q is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem and a bound on the number of incidences between a random point set and a random Q -sphere set in 𝔽qd${\mathbb{F}_{q}^{d}}$. We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.

In this paper, we discuss mean values of some Witten L -functions with respect to the “character aspect”. In particular, we focus on the SU⁢(2)${\mathrm{SU}(2)}$ case. We prove that such mean values turn out to be 0, 1 or to diverge.

We give a self-contained and purely combinatorial proof of the well-known fact that the cohomology of the braces operad is the operad 𝖦𝖾𝗋${{\mathsf{Ger}}}$ governing Gerstenhaber algebras.

In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic p>0${p>0}$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.