Symmetric Powers of Motives

2019 ◽  
pp. 232-252
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Topological Space ◽  
Symmetric Group ◽  
Homotopy Group ◽  
Orbit Space ◽  
Symmetric Power ◽  
Homotopy Groups ◽  
Integral Homology ◽  
Symmetric Powers ◽  
Mac Lane ◽  
Abelian Monoid

This chapter develops the basic theory of symmetric powers of smooth varieties. The constructions in this chapter are based on an analogy with the corresponding symmetric power constructions in topology. If 𝐾 is a set (or even a topological space) then the symmetric power 𝑆𝑚𝐾 is defined to be the orbit space 𝐾𝑚/Σ‎𝑚, where Σ‎𝑚 is the symmetric group. If 𝐾 is pointed, there is an inclusion 𝑆𝑚𝐾 ⊂ 𝑆𝑚+1𝐾 and 𝑆∞𝐾 = ∪𝑆𝑚𝐾 is the free abelian monoid on 𝐾 − {*}. When 𝐾 is a connected topological space, the Dold–Thom theorem says that ̃𝐻*(𝐾, ℤ) agrees with the homotopy groups π‎ *(𝑆∞𝐾). In particular, the spaces 𝑆∞(𝑆 𝑛) have only one homotopy group (𝑛 ≥ 1) and hence are the Eilenberg–Mac Lane spaces 𝐾(ℤ, 𝑛) which classify integral homology.

scholarly journals Some results in quasitopological homotopy groups

2020 ◽  
Vol 72 (12) ◽  
pp. 1663-1668
Author(s):  
T. Nasri ◽  
H. Mirebrahimi ◽  
H. Torabi
Keyword(s):  
Exact Sequence ◽  
Topological Space ◽  
Homotopy Group ◽  
Loop Space ◽  
Homotopy Groups

UDC 515.4 We show that the th quasitopological homotopy group of a topological space is isomorphic to th quasitopological homotopy group of its loop space and by this fact we obtain some results about quasitopological homotopy groups. Finally, using the long exact sequence of a based pair and a fibration in qTop introduced by Brazas in 2013, we obtain some results in this field.

scholarly journals A Group of Automorphisms of the Homotopy Groups

1951 ◽  
Vol 2 ◽  
pp. 73-82
Author(s):  
Hiroshi Uehara
Keyword(s):  
Normal Subgroup ◽  
Topological Space ◽  
Fundamental Group ◽  
Homotopy Group ◽  
Factor Group ◽  
Complete Analysis ◽  
Homotopy Groups ◽  
Group Of Automorphisms ◽  
Arcwise Connected

It is well known that the fundamental group π1(X) of an arcwise connected topological space X operates on the n-th homotopy group πn(X) of X as a group of automorphisms. In this paper I intend to construct geometrically a group 𝒰(X) of automorphisms of πn(X), for every integer n ≥ 1, which includes a normal subgroup isomorphic to π1(X) so that the factor group of 𝒰(X) by π1(X) is completely determined by some invariant Σ(X) of the space X. The complete analysis of the operation of the group on πn(X) is given in §3, §4, and §5,

scholarly journals Le complémentaire des puissances -ièmes dans un corps de nombres est un ensemble diophantien

2015 ◽  
Vol 151 (10) ◽  
pp. 1965-1980 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Jan Van Geel
Keyword(s):  
Prime Number ◽  
Rational Points ◽  
Parameter Family ◽  
Symmetric Power ◽  
Symmetric Powers

For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

scholarly journals The Exponent of the Homotopy Groups of Moore Spectra and the Stable Hurewicz Homomorphism

1996 ◽  
Vol 48 (3) ◽  
pp. 483-495 ◽  
Author(s):  
Dominique Arlettaz
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Homology Theory ◽  
Homotopy Groups ◽  
Integral Homology ◽  
Ring Spectrum ◽  
Bounded Below ◽  
Torsion Free ◽  
Ordinary Integral

AbstractThis paper shows that for the Moore spectrum MG associated with any abelian group G, and for any positive integer n, the order of the Postnikov k-invariant kn+1(MG) is equal to the exponent of the homotopy group πnMG. In the case of the sphere spectrum S, this implies that the exponents of the homotopy groups of S provide a universal estimate for the exponent of the kernel of the stable Hurewicz homomorphism hn: πnX → En(X) for the homology theory E*(—) corresponding to any connective ring spectrum E such that π0E is torsion-free and for any bounded below spectrum X. Moreover, an upper bound for the exponent of the cokernel of the generalized Hurewicz homomorphism hn: En(X) → Hn(X; π0E), induced by the 0-th Postnikov section of E, is obtained for any connective spectrum E. An application of these results enables us to approximate in a universal way both kernel and cokernel of the unstable Hurewicz homomorphism between the algebraic K-theory of any ring and the ordinary integral homology of its linear group.

scholarly journals SYMMETRIC POWER CONGRUENCE IDEALS AND SELMER GROUPS

2018 ◽  
Vol 19 (5) ◽  
pp. 1521-1572
Author(s):  
Haruzo Hida ◽  
Jacques Tilouine
Keyword(s):  
Power Series ◽  
Galois Representation ◽  
Symmetric Power ◽  
Selmer Groups ◽  
Abelian Surfaces ◽  
Symmetric Powers ◽  
Branching Laws ◽  
Hida Family

We prove, under some assumptions, a Greenberg type equality relating the characteristic power series of the Selmer groups over $\mathbb{Q}$ of higher symmetric powers of the Galois representation associated to a Hida family and congruence ideals associated to (different) higher symmetric powers of that Hida family. We use $R=T$ theorems and a sort of induction based on branching laws for adjoint representations. This method also applies to other Langlands transfers, like the transfer from $\text{GSp}(4)$ to $U(4)$. In that case we obtain a corollary for abelian surfaces.

On Localized Unstable K1-groups and Applications to Self-homotopy Groups

2014 ◽  
Vol 57 (2) ◽  
pp. 344-356
Author(s):  
Daisuke Kishimoto ◽  
Akira Kono ◽  
Mitsunobu Tsutaya
Keyword(s):  
Homotopy Group ◽  
The Self ◽  
Explicit Description ◽  
Homotopy Groups

AbstractThe method for computing the p-localization of the group [X, U(n)], by Hamanaka in 2004, is revised. As an application, an explicit description of the self-homotopy group of Sp(3) localized at p ≥ 5 is given and the homotopy nilpotency of Sp(3) localized at p ≥ 5 is determined.

On Derivatives and Norms of Generalized Matrix Functions and Respective Symmetric Powers

2015 ◽  
Vol 30 ◽  
Author(s):  
Sonia Carvalho ◽  
Pedro Freitas
Keyword(s):  
Symmetric Group ◽  
Symmetric Tensor ◽  
Directional Derivatives ◽  
Matrix Functions ◽  
Tensor Power ◽  
Symmetric Powers ◽  
Generalized Matrix ◽  
Derivatives Of

In recent papers, S. Carvalho and P. J. Freitas obtained formulas for directional derivatives, of all orders, of the immanant and of the m-th $\xi$-symmetric tensor power of an operator and a matrix, when $\xi$ is a character of the full symmetric group. The operator bound norm of these derivatives was also calculated. In this paper similar results are established for generalized matrix functions and for every symmetric tensor power.

The Behaviour of Homology in the Localization of Finite Groups

1991 ◽  
Vol 34 (3) ◽  
pp. 311-320 ◽  
Author(s):  
Carles Casacuberta
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Homotopy Groups ◽  
Integral Homology ◽  
A Finite Group

AbstractWe show that, for a finite group G and a prime p, the following facts are equivalent: (i) the p-localization homomorphism l: G —> Gp induces p-localization on integral homology; (ii) the higher homotopy groups of the Bousfield-Kan Zp-completion of a K(G, 1) vanish; (iii) the group G is p-nilpotent.

On first countable quasitopological homotopy groups

2021 ◽  
Vol 71 (3) ◽  
pp. 773-779
Author(s):  
Hamid Torabi
Keyword(s):  
Homotopy Group ◽  
Loop Space ◽  
Homotopy Groups ◽  
Compact Open Topology ◽  
Quotient Maps

Abstract If q: X → Y is a quotient map, then, in general, q × q: X × X → Y × Y may fail to be a quotient map. In this paper, by reviewing the concept of homotopy groups and quotient maps, we find under which conditions the map q × q is a quotient map, where q: Ω n (X, x 0) → πn (X, x 0), is the natural quotient map from the nth loop space of (X, x 0), Ω n (X, x 0), with compact-open topology to the quasitopological nth homotopy group πn (X, x 0). Ultimately, using these results, we found some properties of first countable homotopy groups.

Brunnian braids over the 2-sphere and Artin combed form

2021 ◽  
pp. 2150012
Author(s):  
Duzhin Fedor ◽  
Loh Sher En Jessica
Keyword(s):  
Exact Sequence ◽  
Word Problem ◽  
Open Problem ◽  
Homotopy Group ◽  
Mapping Class Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Homotopy Groups ◽  
Class Groups ◽  
Homotopy Groups Of Spheres

Finding homotopy group of spheres is an old open problem in topology. Berrick et al. derive in [A. J. Berrick, F. Cohen, Y. L. Wong and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006)] an exact sequence that relates Brunnian braids to homotopy groups of spheres. We give an interpretation of this exact sequence based on the combed form for braids over the sphere developed in [R. Gillette and J. V. Buskirk, The word problem and consequences for the braid groups and mapping class groups of the two-sphere, Trans. Amer. Math. Soc. 131 (1968) 277–296] with the aim of helping one to visualize the sequence and to do calculations based on it.

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