Hasil untuk "math.AT"

R. Bruner

A colleague asked about the Adams filtrations of the homotopy classes in the homotopy of the fiber of a particular map between GEMs. The theorem proved in arXiv:2105.02601v3 [math.AT] proves to be effective in answering this (Theorem 4.4). We show that this and some related Adams spectral sequences all collapse at $E_3$ and we determine the value of $E_3 = E_\infty$. Notably, we do not need to determine the cohomology of the fiber or the $E_2$ term of the Adams spectral sequence to do this.

R. Bruner

We describe the variety of `symmetric' left actions of the mod 2 Steenrod algebra $\mathcal{A}$ on its subalgebra $\mathcal{A}(2)$. These arise as the cohomology of $\text{v}_2$ self maps $\Sigma^7 Z \longrightarrow Z$, as in arXiv:1608.06250 [math.AT]. There are $256$ $\mathbb{F}_2$ points in this variety, arising from $16$ such actions of $Sq^8$ and, for each such, $16$ actions of $Sq^{16}$. We describe in similar fashion the 1600 $\mathcal{A}$ actions on $\mathcal{A}(2)$ found by Roth(1977) and the inclusion of the variety of symmetric actions into the variety of all actions. We also describe two related varieties of $\mathcal{A}$ actions, the maps between these and the behavior of Spanier-Whitehead duality on these varieties. Finally, we note that the actions which have been used in the literature correspond to the simplest choices, in which all the coordinates equal zero.

John Jones, D. Rumynin, Adam R. Thomas

Following on from arXiv:2310.14365 [math.AT], we make a detailed study of the $32$-dimensional Rosenfeld projective plane which is the symmetric space EIII in Cartan's list of compact symmetric spaces.

Y. Manin, M. Marcolli

Arguably, the first bridge between vast, ancient, but disjoint domains of mathematical knowledge, – topology and number theory, – was built only during the last fifty years. This bridge is the theory of spectra in the stable homotopy theory. In particular, it connects Z, the initial object in the theory of commutative rings, with the sphere spectrum S: see [Sc01] for one of versions of it. This connection poses the challenge: discover a new information in number theory using the developed independently machinery of homotopy theory. (Notice that a passage in reverse direction has already generated results about computability in the homotopy theory: see [FMa20] and references therein.) In this combined research/survey paper we suggest to apply homotopy spectra to the problem of distribution of rational points upon algebraic manifolds. Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT); Topology (math.AT) Comments: 72 pages. MSC-classes: 16E35, 11G50, 14G40, 55P43, 16E20, 18F30. CONTENTS 0. Introduction and summary 1. Homotopy spectra: a brief presentation 2. Diophantine equations: distribution of rational points on algebraic varieties 3. Rational points, sieves, and assemblers 4. Anticanonical heights and points count 5. Sieves “beyond heights” ? 6. Obstructions and sieves 7. Assemblers and spectra for Grothendieck rings with exponentials References 1

F. Muro

We correct a mistake in the construction of push-outs along free morphisms of algebras over a nonsymmetric operad in arXiv:1101.1634 [math.AT], and we fix the affected results in arXiv:1101.1634 [math.AT] and arXiv:1304.6641 [math.AT].

S. Vrecica, R. Živaljević

Sinivsa T. Vre'cica, Rade T. vZivaljevi'c

We give a proof of the result of Edgar Ramos which claims that two finite, continuous Borel measures $\mu_1$ and $\mu_2$ defined on $\mathbb{R}^5$ admit an equipartition by a collection of three hyperplanes. Our proof illuminates one of the central methods developed and used in our earlier papers and may serve as a good `test case' for addressing (and resolving) the `issues' raised in the paper "Topology of the Gr\"unbaum-Hadwiger-Ramos hyperplane mass partition problem", arXiv:1502.02975 [math.AT]. We also offer a degree-theoretic interpretation of the `parity calculation method' developed by Ramos and demonstrate that, up to minor corrections or modifications, it remains a rigorous and powerful tool for proving results about mass equipartitions.

M. Crabb, A. S. Mishchenko, Q. Morales Meléndez et al.

Alexander Berglund, K. Hess

In this article we revisit the theory of homotopic Hopf-Galois extensions introduced in arXiv:0902.3393v2 [math.AT], in light of the homotopical Morita theory of comodules established in arXiv:1411.6517 [math.AT]. We generalize the theory to a relative framework, which we believe is new even in the classical context and which is essential for treating the Hopf-Galois correspondence in forthcoming work of the second author and Karpova. We study in detail homotopic Hopf-Galois extensions of differential graded algebras over a commutative ring, for which we establish a descent-type characterization analogous to the one Rognes provided in the context of ring spectra. An interesting feature in the differential graded setting is the close relationship between homotopic Hopf-Galois theory and Koszul duality theory. We show that nice enough principal fibrations of simplicial sets give rise to homotopic Hopf-Galois extensions in the differential graded setting, for which this Koszul duality has a familiar form.

Kathryn Hess, Brooke Shipley

Let $X$ be a simplicial set. We construct a novel adjunction between the categories of retractive spaces over $X$ and of $X_{+}$-comodules, then apply recent work on left-induced model category structures (arXiv:1401.3651v2 [math.AT],arXiv:1509.08154 [math.AT]) to establish the existence of a left proper, simplicial model category structure on the category of $X_+$-comodules, with respect to which the adjunction is a Quillen equivalence after localization with respect to some generalized homology theory. We show moreover that this model category structure stabilizes, giving rise to a model category structure on the category of $Σ^\infty X_{+}$-comodule spectra. The Waldhausen $K$-theory of $X$, $A(X)$, is thus naturally weakly equivalent to the Waldhausen $K$-theory of the category of homotopically finite $Σ^\infty X_{+}$-comodule spectra, with weak equivalences given by twisted homology. For $X$ simply connected, we exhibit explicit, natural weak equivalences between the $K$-theory of this category and that of the category of homotopically finite $Σ^{\infty}(ΩX)_+$-modules, a more familiar model for $A(X)$. For $X$ not necessarily simply connected, we have localized versions of these results. For $H$ a simplicial monoid, the category of $Σ^{\infty}H_{+}$-comodule algebras admits an induced model structure, providing a setting for defining homotopy coinvariants of the coaction of $Σ^{\infty}H_{+}$ on a $Σ^{\infty}H_{+}$-comodule algebra, which is essential for homotopic Hopf-Galois extensions of ring spectra as originally defined by Rognes in arXiv:math/0502183v2} and generalized in arXiv:0902.3393v2 [math.AT]. An algebraic analogue of this was only recently developed, and then only over a field (arXiv:1401.3651v2 [math.AT]).

Alexander Berglund, Kathryn Hess

In this article we revisit the theory of homotopic Hopf-Galois extensions introduced in arXiv:0902.3393v2 [math.AT], in light of the homotopical Morita theory of comodules established in arXiv:1411.6517 [math.AT]. We generalize the theory to a relative framework, which we believe is new even in the classical context and which is essential for treating the Hopf-Galois correspondence in forthcoming work of the second author and Karpova. We study in detail homotopic Hopf-Galois extensions of differential graded algebras over a commutative ring, for which we establish a descent-type characterization analogous to the one Rognes provided in the context of ring spectra. An interesting feature in the differential graded setting is the close relationship between homotopic Hopf-Galois theory and Koszul duality theory. We show that nice enough principal fibrations of simplicial sets give rise to homotopic Hopf-Galois extensions in the differential graded setting, for which this Koszul duality has a familiar form.

K. Ko, H. Park

We give formulae for the first homology of the n-braid group and the pure 2-braid group over a finite graph in terms of graph-theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the n-braid group over the graph is torsion-free and the conjectures about the first homology of the pure 2-braid groups over graphs in Farber and Hanbury (arXiv:1005.2300 [math.AT]) can be verified. We discover more characteristics of graph braid groups: the n-braid group over a planar graph and the pure 2-braid group over any graph have a presentation whose relators are words of commutators, and the 2-braid group and the pure 2-braid group over a planar graph have a presentation whose relators are commutators. The latter was a conjecture in Farley and Sabalka (J. Pure Appl. Algebra, 2012) and so we propose a similar conjecture for higher braid indices.

Wang Ru-fa

Boris Botvinnik, Ib Madsen

We study the moduli and determine a homotopy type of the space of all generalized Morse functions on d-manifolds for given d. This moduli space is closely connected to the moduli space of all Morse functions studied in the paper math.AT/0212321, and the classifying space of the corresponding cobordism category.

A. Dimca, S. Papadima

We describe a new relation between the topology of hypersurface complements, Milnor fibers and degree of gradient mappings. In particular we show that any projective hypersurface has affine parts which are bouquets of spheres. The main tools are the polar curves and the affine Lefschetz theory developped by H. Hamm, D.T. L\^e and A. N\'emethi. In the special case of the hyperplane arrangements, we strengthen some results due to Orlik and Terao (see Math. Ann. 301(1995)) and obtain the minimality of hyperplane arrangements (see Randell math.AT/0011101 for another proof of this result). This is then used to compute some higher homotopy groups of hyperplane arrangements using the ideas from Papadima-Suciu, see math.AT/0002251. The second version contains applications of the above ideas to the polar Cremona transformations and gives a positive answer to Dolgachev's Conjecture (see Michigan Math. J. 48 (2000), volume dedicated to W. Fulton). The third version corrects some errors and provides new applications.

LU Hong-peng

Wang Gang