An introduction to homotopy groups

2022 ◽  
pp. 251-284
Keyword(s):  
Homotopy Groups

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].

The Symmetric Commutator Homology of Link Towers and Homotopy Groups of 3-Manifolds

2015 ◽  
Vol 3 (4) ◽  
pp. 497-526 ◽  
Author(s):  
Fuquan Fang ◽  
Fengchun Lei ◽  
Jie Wu
Keyword(s):  
Homotopy Groups

scholarly journals Homotopy groups of diagonal complements

2016 ◽  
Vol 16 (5) ◽  
pp. 2949-2980 ◽  
Author(s):  
Sadok Kallel ◽  
Ines Saihi
Keyword(s):  
Homotopy Groups

scholarly journals The structure of motivic homotopy groups

2016 ◽  
Vol 23 (1) ◽  
pp. 389-397 ◽  
Author(s):  
Bogdan Gheorghe ◽  
Daniel C. Isaksen
Keyword(s):  
Homotopy Groups ◽  
Motivic Homotopy

scholarly journals Omega vs. pi, and 6d anomaly cancellation

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Joe Davighi ◽  
Nakarin Lohitsiri
Keyword(s):  
Gauge Theories ◽  
Necessary Conditions ◽  
Homotopy Groups ◽  
Mapping Torus ◽  
Anomaly Cancellation ◽  
Global Anomaly ◽  
Gauge Anomalies ◽  
Local Anomalies ◽  
Dimensional Mapping

Abstract In this note we review the role of homotopy groups in determining non-perturbative (henceforth ‘global’) gauge anomalies, in light of recent progress understanding global anomalies using bordism. We explain why non-vanishing of πd(G) is neither a necessary nor a sufficient condition for there being a possible global anomaly in a d-dimensional chiral gauge theory with gauge group G. To showcase the failure of sufficiency, we revisit ‘global anomalies’ that have been previously studied in 6d gauge theories with G = SU(2), SU(3), or G2. Even though π6(G) ≠ 0, the bordism groups $$ {\Omega}_7^{\mathrm{Spin}}(BG) $$ Ω 7 Spin BG vanish in all three cases, implying there are no global anomalies. In the case of G = SU(2) we carefully scrutinize the role of homotopy, and explain why any 7-dimensional mapping torus must be trivial from the bordism perspective. In all these 6d examples, the conditions previously thought to be necessary for global anomaly cancellation are in fact necessary conditions for the local anomalies to vanish.

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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