Authors:William Browder Pages: 75 - 78 Abstract: A space is defined to be “n-spheroidal” if it has the homotopy type of an n-dimensional CW-complex X with \(H_{n}(X; \mathbb {Z})\) not zero and finitely generated. A group G is called “n-spheroidal” if its classifying space K(G, 1) is n-spheroidal. Examples include fundamental groups of compact manifold K(G, 1)s. Moreover, the class of groups G, which are n-spheroidal for some n, is closed under products, free products, and group extensions. If Y is a space with \(\pi _{1}(Y)\) n-spheroidal, and if \(H_{k}(Y;\mathbb {F}_{p})\) is non-zero and finitely generated, and if \(H_{i}(Y;\mathbb {F}_{p}) = 0\) for \(i>k\) , then \(H_{n+k}(\overline{Y};\mathbb {F}_{p}) \ne 0\) for \(\overline{Y}\) a finite sheeted covering space of Y. Hence, dim \((Y) \ge n+k\) . Thus, it follows that if dim \((Y) < n\) , and if \(H_{k}(Y;\mathbb {F}_{p}) \ne 0\) and \(H_{i}(Y;\mathbb {F}_{p}) = 0\) for \(i>k>0\) , then \(H_{k}(Y;\mathbb {F}_{p})\) is not finitely generated. Similar results follow for \(Y\subset K(G,1)\) . PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0137-3 Issue No:Vol. 23, No. 1 (2017)

Authors:Kee Yuen Lam Pages: 79 - 86 Abstract: This paper is a summary of my talk given at the event “Sam 80” which was a meeting held at El Colegio Nacional in Mexico (2013) to celebrate Professor Samuel Gitler’s 80th birthday. The main purpose of that talk was to present an affirmative answer to the Yuzvinsky Conjecture in the case of square matrices (explained below). I would like to dedicate this write-up to Sam’s memory, to express my appreciation for his warm friendship throughout our 46 years of acquaintance. PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0130-x Issue No:Vol. 23, No. 1 (2017)

Authors:Noé Bárcenas; Daniel Juan-Pineda; Pablo Suárez-Serrato Pages: 119 - 127 Abstract: In this short note we prove the Borel conjecture for a family of aspherical manifolds that includes higher graph manifolds. PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0099-5 Issue No:Vol. 23, No. 1 (2017)

Authors:Donald M. Davis Pages: 129 - 139 Abstract: Let \({\overline{M}}_{n,r}\) denote the space of isometry classes of n-gons in the plane with one side of length r and all others of length 1, and assume that \(1\le r<n-3\) and \(n-r\) is not an odd integer. Using known results about the mod-2 cohomology ring, we prove that its topological complexity satisfies \({\text {TC}}({\overline{M}}_{n,r})\ge 2n-6\) . Since \({\overline{M}}_{n,r}\) is an \((n-3)\) -manifold, \({\text {TC}}({\overline{M}}_{n,r})\le 2n-5\) . So our result is within 1 of being optimal. PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0093-y Issue No:Vol. 23, No. 1 (2017)

Authors:Suyoung Choi; Shizuo Kaji; Stephen Theriault Pages: 153 - 161 Abstract: We give p-local homotopy decompositions of the suspensions of real toric spaces for odd primes p. Our decomposition is compatible with the one given by Bahri, Bendersky, Cohen, and Gitler for the suspension of the corresponding real moment-angle complex, or more generally, the polyhedral product. As an application, we obtain a stable rigidity property for real toric spaces. PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0090-1 Issue No:Vol. 23, No. 1 (2017)

Authors:Ralph L. Cohen; John D. S Jones Pages: 163 - 172 Abstract: Let R be a ring spectrum and \( \mathcal {E}\rightarrow X\) an R-module bundle of rank n. Our main result is to identify the homotopy type of the group-like monoid of homotopy automorphisms of this bundle, \(hAut^R(\mathcal {E})\) . This will generalize the result regarding R-line bundles proven by Cohen and Jones (Mex Bull Math, 2016). The main application is the calculation of the homotopy type of \(BGL_n(End ((\mathcal {L}))\) where \(\mathcal {L}\rightarrow X\) is any R-line bundle, and \(End^R(\mathcal {L})\) is the ring spectrum of endomorphisms. In the case when such a bundle is the fiberwise suspension spectrum of a principal bundle over a manifold, \(G \rightarrow P \rightarrow M\) , this leads to a description of the K-theory of the string topology spectrum in terms of the mapping space from M to \(BGL \left( \Sigma ^\infty (G_+)\right) \) . PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0136-4 Issue No:Vol. 23, No. 1 (2017)

Authors:Andrew Baker Pages: 195 - 231 Abstract: The 2-primary Hopf invariant 1 elements in the stable homotopy groups of spheres form the most accessible family of elements. In this paper, we explore some properties of the \(\mathcal {E}_\infty \) ring spectra obtained from certain iterated mapping cones by applying the free algebra functor. In fact, these are equivalent to Thom spectra over infinite loop spaces related to the classifying spaces \(B\mathrm {SO},\,B\mathrm {Spin},\,B\mathrm {String}\) . We show that the homology of these Thom spectra are all extended comodule algebras of the form \(\mathcal {A}_*\square _{\mathcal {A}(r)_*}P_*\) over the dual Steenrod algebra \(\mathcal {A}_*\) with \(\mathcal {A}_*\square _{\mathcal {A}(r)_*}\mathbb {F}_2\) as an algebra retract. This suggests that these spectra might be wedges of module spectra over the ring spectra \(H\mathbb {Z}\) , \(k\mathrm {O}\) or \(\mathrm {tmf}\) ; however, apart from the first case, we have no concrete results on this. PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0096-8 Issue No:Vol. 23, No. 1 (2017)

Authors:A. Bahri; M. Bendersky; F. R. Cohen; S. Gitler Pages: 257 - 265 Abstract: In this note, it is shown that the Hilbert–Poincaré series for the rational homology of the free loop space on a moment-angle complex is a rational function if and only if the moment-angle complex is a product of odd spheres and a disk. A partial result is included for the Davis–Januszkiewicz spaces. The opportunity is taken to correct the result (Bahri et al., Proceedings of the Steklov Institute of Mathematics, Russian Academy of Sciences, vol. 286, pp. 219–223. doi:10.1134/S0081543814060121, 2014) which used a theorem from Berglund and Jöllenbeck (J Algebra 315:249–273, 2007). PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0124-8 Issue No:Vol. 23, No. 1 (2017)

Authors:Michael A. Hill; M. J. Hopkins; D. C. Ravenel Pages: 289 - 317 Abstract: We study the slice filtration and associated spectral sequence for a family of \(RO(C_{p^{n}})\) -graded suspensions of the Eilenberg–MacLane spectrum for the constant Mackey functor \(\underline{\mathbb Z}\) . Since \(H\underline{\mathbb Z}\) is the zero slice of the sphere spectrum, this begins an analysis of how one can describe the slices of a suspension in terms of the original slices. PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0129-3 Issue No:Vol. 23, No. 1 (2017)

Authors:Bogdan Gheorghe; Daniel C. Isaksen Pages: 389 - 397 Abstract: We study the stable motivic homotopy groups \(\pi _{s,w}\) of the 2-completion of the motivic sphere spectrum over \(\mathbb {C}\) . When arranged in the (s, w)-plane, these groups break into four different regions: a vanishing region, an \(\eta \) -local region that is entirely known, a \(\tau \) -local region that is identical to classical stable homotopy groups, and a region that is not well-understood. PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0094-x Issue No:Vol. 23, No. 1 (2017)

Authors:Iván D. Castrillón; Enrique Reyes Pages: 399 - 412 Abstract: If a simplicial complex \(\Delta \) is vertex decomposable, then the n-sphere moment angle complex \(\mathcal {Z}_{\Delta ^{\vee }}(D^{n},S^{n-1})\) has the homotopy type of the wedges of spheres, where \(\Delta ^{\vee }\) is the Alexander dual of \(\Delta \) . Furthermore, if \(\Delta \) is pure vertex decomposable, then its Stanley–Reisner ring \(k[\Delta ]\) is Cohen–Macaulay. Consequently, the vertex decomposable property is an interesting property from combinatorial, algebraic and topological point of view. In this paper, we characterize the pure vertex decomposable simplicial complexes associated to graphs whose 5-cycles have at least 4 chords. PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0147-1 Issue No:Vol. 23, No. 1 (2017)

Authors:Anton Ayzenberg Pages: 413 - 421 Abstract: Given a simplicial poset S, whose geometrical realization is a closed orientable homology manifold, Novik and Swartz introduced a Poincare duality algebra \((\mathcal {R}[S]/(l.s.o.p.))/I_{NS}\) , which is a quotient of the face ring of the poset S. The ranks of graded components of this algebra are now called \(h''\) numbers of S and can be computed from face numbers and Betti numbers of S. We introduce a topological model for this Poincare duality algebra. Given an \((n-1)\) -dimensional simplicial homology manifold S, we construct a 2n-dimensional homology manifold with boundary \(\widehat{X}\) carrying the action of a compact n torus. The Poincare–Lefschetz duality on \(\widehat{X}\) is used to reconstruct the algebra \((\mathcal {R}[S]/(l.s.o.p.))/I_{NS}\) . PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0135-5 Issue No:Vol. 23, No. 1 (2017)

Authors:Yadira Barreto; Santiago López de Medrano; Alberto Verjovsky Pages: 423 - 437 Abstract: I. We construct open book structures on all moment-angle manifolds and describe the topology of their leaves and bindings under certain restrictions. II. We also show, using a recent deep result about contact forms due to Borman et al. (Acta Math 215:281–361, 2015), that every odd-dimensional moment-angle manifold admits a contact structure. This contrasts with the fact that, except for a few, well-determined cases, even-dimensional ones do not admit symplectic structures. We obtain the same results for large families of more general intersections of quadrics. PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0113-y Issue No:Vol. 23, No. 1 (2017)

Authors:Michael Wiemeler Pages: 501 - 509 Abstract: In this note we prove that equivariantly homeomorphic quasitoric manifolds are diffeomorphic. As a consequence we show that up to finite ambiguity the diffeomorphism type of certain quasitoric manifolds M is determined by their cohomology rings and first Pontrjagin classes. PubDate: 2017-04-01 DOI: 10.1007/s40590-016-0091-0 Issue No:Vol. 23, No. 1 (2017)

Authors:Ali Shojaei-Fard Abstract: The article applies Connes–Kreimer Hopf algebra of Feynman diagrams and theory of graphons to build an operational calculus machinery on the basis of measure theory for Green’s functions of quantum field theory. PubDate: 2017-05-10 DOI: 10.1007/s40590-017-0166-6

Authors:Iz-iddine EL-Fassi; Abdellatif Chahbi; Samir Kabbaj Abstract: Let \((S,\cdot )\) be a semigroup, \(\mathbb {C}\) be the set of complex numbers, and let \(\sigma ,\tau \in Hom(S,S)\) satisfy \(\tau \circ \tau =\sigma \circ \sigma =id.\) We show that any solution \(f:S \rightarrow \mathbb {C}\) of the functional equation $$\begin{aligned} f(x\sigma (y))+\chi (y)f(\tau (y)x)=2f(x)f(y), \quad x,y \in S, \end{aligned}$$ has the form \(f=(m+\chi \, m\circ \sigma \circ \tau )/2\) , where m is a multiplicative function on S and \(\chi :S\rightarrow (\mathbb {C}\backslash \{0\},\cdot )\) is a character on S (i.e., \(\chi (xy)=\chi (x)\chi (y)\) for all \(x,y\in S\) ) which satisfies \(\chi (x\tau (x))=1\) for all \(x\in S\) . PubDate: 2017-05-09 DOI: 10.1007/s40590-017-0168-4

Authors:Gauhar Rahman; Praveen Agarwal; Shahid Mubeen; Muhammad Arshad Abstract: This paper is devoted to the study of fractional calculus with an integral and differential operators containing the following family of extended Mittag–Leffler function: $$\begin{aligned} E_{\alpha ,\beta }^{\gamma ;c}(z; p)=\sum \limits _{n=0}^{\infty }\frac{B_p(\gamma +n, c-\gamma )(c)_{n}}{B(\gamma , c-\gamma )\Gamma (\alpha n+\beta )}\frac{z^n}{n!}, (z,\beta , \gamma \in \mathbb {C}), \end{aligned}$$ in its kernel. Also, we further introduce a certain number of consequences of fractional integral and differential operators containing the said function in their kernels. PubDate: 2017-04-25 DOI: 10.1007/s40590-017-0167-5

Authors:Ernesto Lupercio; Elias Micha Abstract: In his note we briefly analyze the rôle of the oeuvre of Mexican mathematician Samuel Gitler and his influence in twentieth and twenty-first century mathematics. PubDate: 2017-03-11 DOI: 10.1007/s40590-017-0163-9

Abstract: We determine the 2-primary components of the 32-stem homotopy groups of spheres. The method is based on the classical one including the Toda’s composition methods. PubDate: 2016-11-04 DOI: 10.1007/s40590-016-0154-2