Homology of the double loop space of the homogeneous space SU(n)/SO(n)

2001 ◽  
Vol 77 (3) ◽  
pp. 222-232
Author(s):  
Y. Choi
Keyword(s):  
Homogeneous Space ◽  
Loop Space ◽  
Double Loop

scholarly journals The firstk-invariant of a double loop space is trivial

1990 ◽  
Vol 54 (1) ◽  
pp. 84-92 ◽  
Author(s):  
Dominique Arlettaz
Keyword(s):  
Loop Space ◽  
Double Loop

Delooping the total Stiefel–Whitney class

1988 ◽  
Vol 103 (1) ◽  
pp. 83-87
Author(s):  
A. Kozlowski
Keyword(s):  
Loop Space ◽  
Cohomology Ring ◽  
Group Homomorphism ◽  
Double Loop ◽  
Bott Periodicity ◽  
Group Of Units ◽  
Periodicity Theorem ◽  
Whitney Class ◽  
Infinite Loop ◽  
Infinite Loop Space

Let FH(X) denote the group of units of the classical cohomology ring H(X) = Πn≥0Hn(X; Z/2) of a CW-complex X. The total Stiefel–Whitney class can be viewed as a group homomorphism where is the reduced real K-theory of X. Both and FH( ) are representable functors, with representing spaces BO and FH, and thus w can be represented by a map w: BO → FH. By the Bott periodicity theorem, BO is an infinite loop space, and by a theorem of G. Segal[9] so is FH. However, it is well known that w is not an infinite loop map; this was first shown in [10]. The purpose of this paper is to prove the following:Theorem 0·1. w: BO → FHis a loop map but not a double loop map.

scholarly journals ON THE OTHER SIDE OF THE BIALGEBRA OF CHORD DIAGRAMS

2007 ◽  
Vol 16 (05) ◽  
pp. 575-629 ◽  
Author(s):  
V. TOURTCHINE
Keyword(s):  
Spectral Sequence ◽  
Lie Algebras ◽  
Loop Space ◽  
The Other ◽  
Dimensional Sphere ◽  
Double Loop ◽  
Homology Groups ◽  
Chord Diagrams ◽  
Future Work ◽  
Hochschild Complex

In this paper we describe complexes whose homologies are naturally isomorphic to the first term of the Vassiliev spectral sequence computing (co)homology of the spaces of long knots in ℝd, d ≥ 3. The first term of the Vassiliev spectral sequence is concentrated in some angle of the second quadrant. In homological case the lower line of this term is the bialgebra of chord diagrams (or its superanalog if d is even). We prove in this paper that the groups of the upper line are all trivial. In the same bigradings we compute the homology groups of the complex spanned only by strata of immersions in the discriminant (maps having only self-intersections). We interpret the obtained groups as subgroups of the (co)homology groups of the double loop space of a (d - 1)-dimensional sphere. In homological case the last complex is the normalized Hochschild complex of the Poisson or Gerstenhaber (depending on parity of d) algebras operad. The upper line bigradings are spanned by the operad of Lie algebras. To describe the cycles in these bigradings, we introduce new homological operations on Hochschild complexes. We show in future work that these operations are in fact the Dyer–Lashof–Cohen operations induced by the action of the singular chains operad of little squares on Hochschild complexes.

scholarly journals A POLYNOMIAL MODEL FOR THE DOUBLE-LOOP SPACE OF AN EVEN SPHERE

2004 ◽  
Vol 47 (1) ◽  
pp. 155-162
Author(s):  
Yasuhiko Kamiyama
Keyword(s):  
Loop Space ◽  
Polynomial Model ◽  
Subject Classification ◽  
Double Loop ◽  
Holomorphic Maps ◽  
Primary 55P35

AbstractIt is well known that $\varOmega^2S^{2n+1}$ is approximated by $\textrm{Rat}_{k}(\mathbb{C}P^{n})$, the space of based holomorphic maps of degree $k$ from $S^2$ to $\mathbb{C}P^{n}$. In this paper we construct a space $G_{k}^{n}$ which is an analogue of $\textrm{Rat}_{k}(\mathbb{C}P^{n})$, and prove that under the natural map $j_k:G_{k}^{n}\to\varOmega^2S^{2n}$, $G_{k}^{n}$ approximates $\varOmega^2S^{2n}$.AMS 2000 Mathematics subject classification: Primary 55P35

scholarly journals The fibre of the degree 3 map, Anick spaces and the double suspension

2020 ◽  
Vol 63 (3) ◽  
pp. 830-843
Author(s):  
Steven Amelotte
Keyword(s):  
Loop Space ◽  
Homotopy Equivalent ◽  
Double Loop ◽  
Decomposition Problem ◽  
Homotopy Fibre ◽  
Space Decomposition ◽  
Homotopy Decomposition ◽  
Invariant Element

AbstractLet S2n+1{p} denote the homotopy fibre of the degree p self map of S2n+1. For primes p ≥ 5, work by Selick shows that S2n+1{p} admits a non-trivial loop space decomposition if and only if n = 1 or p. Indecomposability in all but these dimensions was obtained by showing that a non-trivial decomposition of ΩS2n+1{p} implies the existence of a p-primary Kervaire invariant one element of order p in $\pi _{2n(p-1)-2}^S$. We prove the converse of this last implication and observe that the homotopy decomposition problem for ΩS2n+1{p} is equivalent to the strong p-primary Kervaire invariant problem for all odd primes. For p = 3, we use the 3-primary Kervaire invariant element θ3 to give a new decomposition of ΩS55{3} analogous to Selick's decomposition of ΩS2p+1{p} and as an application prove two new cases of a long-standing conjecture stating that the fibre of the double suspension $S^{2n-1} \longrightarrow \Omega ^2S^{2n+1}$ is homotopy equivalent to the double loop space of Anick's space.

Order of the canonical vector bundle over configuration spaces of spheres

2018 ◽  
Vol 30 (5) ◽  
pp. 1265-1277
Author(s):  
Shiquan Ren
Keyword(s):  
Vector Bundle ◽  
Riemann Surfaces ◽  
Loop Space ◽  
American Mathematical Society ◽  
Configuration Spaces ◽  
Double Loop ◽  
Stable Order ◽  
Pure Math ◽  
Ann Arbor ◽  
Canonical Vector

AbstractGiven a vector bundle, its (stable) order is the smallest positive integer t such that the t-fold self-Whitney sum is (stably) trivial. So far, the order and the stable order of the canonical vector bundle over configuration spaces of Euclidean spaces have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54], [F. R. Cohen, M. E. Mahowald and R. J. Milgram, The stable decomposition for the double loop space of a sphere, Algebraic and Geometric Topology (Stanford 1976), Proc. Sympos. Pure Math. 32 Part 2, American Mathematical Society, Providence 1978, 225–228], and [S.-W. Yang, Order of the Canonical Vector Bundle on {C_{n}(k)/\Sigma_{k}}, ProQuest LLC, Ann Arbor, 1978]. Moreover, the order and the stable order of the canonical vector bundle over configuration spaces of closed orientable Riemann surfaces with genus greater than or equal to one have been studied in [F. R. Cohen, R. L. Cohen, N. J. Kuhn and J. A. Neisendorfer, Bundles over configuration spaces, Pacific J. Math. 104 1983, 1, 47–54]. In this paper, we mainly study the order and the stable order of the canonical vector bundle over configuration spaces of spheres and disjoint unions of spheres.

scholarly journals The Borel cohomology of the loop space of a homogeneous space

2013 ◽  
Vol 160 (12) ◽  
pp. 1313-1332
Author(s):  
Kentaro Matsuo
Keyword(s):  
Homogeneous Space ◽  
Loop Space
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