Composition methods in the homotopy groups of ring spectra

2001 ◽  
pp. 131-148
Author(s):  
Brayton Gray
Keyword(s):  
Homotopy Groups ◽  
Composition Methods ◽  
Ring Spectra

scholarly journals Uniqueness of MoravaK-theory

2010 ◽  
Vol 147 (2) ◽  
pp. 633-648 ◽  
Author(s):  
Vigleik Angeltveit
Keyword(s):  
Spectral Sequence ◽  
Moduli Space ◽  
Homotopy Groups ◽  
Algebra Structure ◽  
Ring Spectra ◽  
Set Up

AbstractWe show that there is an essentially uniqueS-algebra structure on the MoravaK-theory spectrumK(n), whileK(n) has uncountably manyMUor$\widehat {E(n)}$-algebra structures. Here$\widehat {E(n)}$is theK(n)-localized Johnson–Wilson spectrum. To prove this we set up a spectral sequence computing the homotopy groups of the moduli space ofA∞structures on a spectrum, and use the theory ofS-algebrak-invariants for connectiveS-algebras found in the work of Dugger and Shipley [Postnikov extensions of ring spectra, Algebr. Geom. Topol.6(2006), 1785–1829 (electronic)] to show that all the uniqueness obstructions are hit by differentials.

A nilpotent Whitehead theorem for $\mathsf{TQ}$-homology of structured ring spectra

2018 ◽  
Vol 11 (3) ◽  
pp. 69-79
Author(s):  
Michael Ching ◽  
John E. Harper
Keyword(s):  
Ring Spectra

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.

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