scholarly journals Classification of knotted tori

2019 ◽  
Vol 150 (2) ◽  
pp. 549-567
Author(s):  
A. Skopenkov
Keyword(s):  
Abelian Group ◽  
Exact Sequence ◽  
Smooth Manifold ◽  
Group Structure ◽  
Extension Problem ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stiefel Manifolds

AbstractFor a smooth manifold N denote by Em(N) the set of smooth isotopy classes of smooth embeddings N → ℝm. A description of the set Em(Sp × Sq) was known only for p = q = 0 or for p = 0, m ≠ q + 2 or for 2m ⩾ 2(p + q) + max{p, q} + 4. (The description was given in terms of homotopy groups of spheres and of Stiefel manifolds.) For m ⩾ 2p + q + 3 we introduce an abelian group structure on Em(Sp × Sq) and describe this group ‘up to an extension problem’. This result has corollaries which, under stronger dimension restrictions, more explicitly describe Em(Sp × Sq). The proof is based on relations between sets Em(N) for different N and m, in particular, on a recent exact sequence of M. Skopenkov.

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.

scholarly journals When is the set of embeddings finite up to isotopy?

2015 ◽  
Vol 26 (07) ◽  
pp. 1550051 ◽  
Author(s):  
Mikhail Skopenkov
Keyword(s):  
Exact Sequence ◽  
Disjoint Union ◽  
Group Structure ◽  
Classification Problems ◽  
Natural Case

Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddingsN → Smfinite? In case when the manifold N is a sphere the answer was given by A. Haefliger in 1966. In case when the manifold N is a disjoint union of spheres the answer was given by D. Crowley, S. Ferry and the author in 2011. We consider the next natural case when N is a product of two spheres. In the following theorem, FCS (i, j) ⊂ ℤ2 is a specific set depending only on the parity of i and j which is defined in the paper. Theorem.Assume thatm > 2p + q + 2andm < p + 3q/2 + 2. Then the set of isotopy classes ofC1-smooth embeddingsSp × Sq → Smis infinite if and only if eitherq + 1orp + q + 1is divisible by 4, or there exists a point(x, y)in the set FCS (m - p - q, m - q)such that(m - p - q - 2)x + (m - q - 2)y = m - 3. Our approach is based on a group structure on the set of embeddings and a new exact sequence, which in some sense reduces the classification of embeddings Sp × Sq → Sm to the classification of embeddings Sp+q ⊔ Sq → Sm and Dp × Sq → Sm. The latter classification problems are reduced to homotopy ones, which are solved rationally.

On the classification of some two-dimensional Markov shifts with group structure

1992 ◽  
Vol 12 (4) ◽  
pp. 823-833 ◽  
Author(s):  
Mark A. Shereshevsky
Keyword(s):  
Abelian Group ◽  
Measure Space ◽  
Group Structure ◽  
Finite Abelian Group ◽  
Two Dimensional ◽  
Finite Abelian Groups ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Markov Shifts

AbstractFor a finite Abelian group G define the two-dimensional Markov shift for all . Let μG be the Haar measure on the subgroup . The group ℤ2 acts on the measure space (XG, MG) by shifts. We prove that if G1, and G2 are p-groups and E(G1,) ≠ E(G2), where E(G) is the least common multiple of the orders of the elements of G, then the shift actions on and are not measure-theoretically isomorphic. For any finite Abelian groups G1 and G2 the shift actions on and are topologically conjugate if and only if G1 and G2 are isomorphic.

scholarly journals Whitehead products in the complex Stiefel manifolds

1983 ◽  
Vol 26 (2) ◽  
pp. 241-251 ◽  
Author(s):  
Yasukuni Furukawa
Keyword(s):  
Stiefel Manifold ◽  
Isotropy Group ◽  
Homotopy Groups ◽  
Stiefel Manifolds

The complex Stiefel manifoldWn,k, wheren≦k≦1, is a space whose points arek-frames inCn. By using the formula of McCarty [4], we will make the calculations of the Whitehead products in the groups π*(Wn,k). The case of real and quaternionic will be treated by Nomura and Furukawa [7]. The product [[η],j1l] appears as generator of the isotropy group of the identity map of Stiefel manifolds. In this note we use freely the results of the 2-components of the homotopy groups of real and complex Stiefel manifolds such as Paechter [8], Hoo-Mahowald [1], Nomura [5], Sigrist [9] and Nomura-Furukawa [6].

scholarly journals ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

scholarly journals A Conjecture on Homotopy Groups of Spheres, Details on the Algebra of Higher Cohomology Operations

2009 ◽  
Vol 16 (1) ◽  
pp. 1-12
Author(s):  
Hans-Joachim Baues
Keyword(s):  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Cohomology Operations

Abstract The computation of the algebra of secondary cohomology operations in [Baues, The algebra of secondary cohomology operations, Birkhäuser Verlag, 2006] leads to a conjecture concerning the algebra of higher cohomology operations in general and an Ext-formula for the homotopy groups of spheres. This conjecture is discussed in detail in this paper.

Some infinite families in the stable homotopy of spheres

1987 ◽  
Vol 101 (3) ◽  
pp. 477-485 ◽  
Author(s):  
Wen-Hsiung Lin
Keyword(s):  
Spectral Sequence ◽  
Adams Spectral Sequence ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Homotopy Groups Of Spheres ◽  
Stable Homotopy Groups ◽  
Stable Homotopy Of Spheres

The classical Adams spectral sequence [1] has been an important tool in the computation of the stable homotopy groups of spheres . In this paper we make another contribution to this computation.

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