MOD p HOMOLOGY OF THE DOUBLE LOOP SPACE OF THE HOMOGENEOUS SPACE SO(2n)/U(n)

2003 ◽  
Vol 40 (2) ◽  
pp. 281-288
Author(s):  
Younggi Park
Keyword(s):  
Homogeneous Space ◽  
Loop Space ◽  
Double Loop ◽  
Mod P

scholarly journals A Spectral Sequence for the Hochschild Cohomology of a Coconnective DGA

2013 ◽  
Vol 112 (2) ◽  
pp. 182 ◽  
Author(s):  
Shoham Shamir
Keyword(s):  
Spectral Sequence ◽  
Loop Space ◽  
Hochschild Cohomology ◽  
Derived Category ◽  
Orientable Manifold ◽  
Support Varieties ◽  
Mod P

A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence presented in [7] for the computation of the loop homology of a closed orientable manifold. Using this spectral sequence we identify a class of spaces for which the Hochschild cohomology of their mod-$p$ cochain algebra is Noetherian. This implies, among other things, that for such a space the derived category of mod-$p$ chains on its loop-space carries a theory of support varieties.

scholarly journals On vector bundles for a Morse decomposition of $L\mathbb{C}\mathrm{P}^n$

2017 ◽  
Vol 121 (2) ◽  
pp. 186
Author(s):  
Iver Ottosen
Keyword(s):  
Loop Space ◽  
Vector Bundles ◽  
Morse Decomposition ◽  
Energy Integral ◽  
Chern Classes ◽  
Free Loop Space ◽  
Stiefel Manifolds ◽  
Mod P

We give a description of the negative bundles for the energy integral on the free loop space $L\mathbb{C}\mathrm{P}^n$ in terms of circle vector bundles over projective Stiefel manifolds. We compute the mod $p$ Chern classes of the associated homotopy orbit bundles.

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.

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