scholarly journals The bigraded homotopy groups πi, jX of a pointed simplicial space X

1995 ◽  
Vol 103 (2) ◽  
pp. 167-188 ◽  
Author(s):  
W.G. Dwyer ◽  
D.M. Kan ◽  
C.R. Stover
Keyword(s):  
Homotopy Groups ◽  
Simplicial Space

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 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.

Toric Geometry ofSL2(ℂ) Free Group Character Varieties from Outer Space

2018 ◽  
Vol 70 (2) ◽  
pp. 354-399 ◽  
Author(s):  
Christopher Manon
Keyword(s):  
Free Group ◽  
Outer Space ◽  
Integrable Hamiltonian System ◽  
Outer Automorphism ◽  
Character Variety ◽  
Toric Geometry ◽  
Character Varieties ◽  
Simplicial Space ◽  
Proper Map ◽  
Divisorial Valuations

AbstractCuller and Vogtmann defined a simplicial spaceO(g), calledouter space, to study the outer automorphism group of the free groupFg. Using representation theoretic methods, we give an embedding ofO(g) into the analytification of X(Fg,SL2(ℂ)), theSL2(ℂ) character variety ofFg, reproving a result of Morgan and Shalen. Then we show that every pointvcontained in a maximal cell ofO(g) defines a flat degeneration of X(Fg,SL2(ℂ)) to a toric varietyX(PΓ). We relate X(Fg,SL2(ℂ)) andX(v) topologically by showing that there is a surjective, continuous, proper map Ξv:X(Fg,SL2(ℂ)) →X(v). We then show that this map is a symplectomorphism on a dense open subset of X(Fg, SL2(ℂ)) with respect to natural symplectic structures on X(Fg, SL2(ℂ)) andX(v). In this way, we construct an integrable Hamiltonian system in X(Fg, SL2(ℂ)) for each point in a maximal cell ofO(g), and we show that eachvdefines a topological decomposition of X(Fg, SL2(ℂ)) derived from the decomposition ofX(PΓ) by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell inO(g) all arise as divisorial valuations built from an associated projective compactification of X(Fg, SL2(ℂ)).

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