ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Mon, 26 Sep 2022 00:49:06 GMT2022-09-26T00:49:06Z50311- Combinatorial descriptions of homotopy groups of certain spaceshttps://scholarbank.nus.edu.sg/handle/10635/102999Title: Combinatorial descriptions of homotopy groups of certain spaces
Authors: Wu, J.
Abstract: We give a combinatorial description of the homotopy groups of the suspension of a K(π, 1) and of wedges of 2-spheres. In particular, all of the homotopy groups of the 2-sphere are given as the centres of certain combinatorially described groups. © 2001 Cambridge Philosophical Society.
Tue, 01 May 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1029992001-05-01T00:00:00Z
- Decompositions of looped co-H-spaceshttps://scholarbank.nus.edu.sg/handle/10635/103106Title: Decompositions of looped co-H-spaces
Authors: Grbić, J.; Theriault, S.; Wu, J.
Abstract: We prove two homotopy decomposition theorems for the loops on simply-connected co-H-spaces, including a generalization of the Hilton-Milnor Theorem. Several examples are given. © 2012 American Mathematical Society.
Tue, 01 Jan 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031062013-01-01T00:00:00Z
- Delta-structures on mapping class groups and braid groupshttps://scholarbank.nus.edu.sg/handle/10635/103120Title: Delta-structures on mapping class groups and braid groups
Authors: Berrick, A.J.; Hanbury, E.; Wu, J.
Abstract: We describe a Delta-group structure on the mapping class groups of surfaces, and show that it is compatible with the Delta-group structures of the braid groups of surfaces given by Berrick-Cohen-Wong-Wu. We then prove an isomorphism theorem relating these two Delta-groups. This is the first of a pair of papers on this topic. © 2013 American Mathematical Society Reverts to public domain 28 years from publication.
Wed, 01 Jan 2014 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031202014-01-01T00:00:00Z
- Functorial decompositions of looped coassociative co-H spaceshttps://scholarbank.nus.edu.sg/handle/10635/103310Title: Functorial decompositions of looped coassociative co-H spaces
Authors: Selick, P.; Theriault, S.; Wu, J.
Abstract: Selick and Wu gave a functorial decomposition of ΩΣX for path-connected, p-local CW-complexes X which obtained the smallest nontrivial functorial retract A min(X) of ΩΣX, This paper uses methods developed by the second author in order to extend such functorial decompositions to the loops on coassociative co-H spaces. © Canadian Mathematical Society 2006.
Tue, 01 Aug 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033102006-08-01T00:00:00Z
- On homotopy rigidity of the functor ?? on co-H-spaceshttps://scholarbank.nus.edu.sg/handle/10635/123558Title: On homotopy rigidity of the functor ?? on co-H-spaces
Authors: Grbi?, Jelena; Wu, Jie
Thu, 01 Jan 2015 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1235582015-01-01T00:00:00Z
- Relative homotopy abelian H-spaceshttps://scholarbank.nus.edu.sg/handle/10635/179004Title: Relative homotopy abelian H-spaces
Authors: Theriault, S; Wu, J
Abstract: We introduce the notion of a relatively homotopy associative and homotopy commutative H-space, construct one for any path-connected space X, and describe several useful properties, including exponent properties. © 2018, The Author(s).
Tue, 01 Jan 2019 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1790042019-01-01T00:00:00Z
- The functor Amin on p-local spaceshttps://scholarbank.nus.edu.sg/handle/10635/104302Title: The functor Amin on p-local spaces
Authors: Selick, P.; Wu, J.
Abstract: In a previous paper, the authors gave the finest functorial decomposition of the loop suspension of a p-torsion suspension. The purpose of this paper is to generalize this theorem to the loop suspension of arbitrary p-local path connected spaces.
Sat, 01 Jul 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043022006-07-01T00:00:00Z
- Rank p - 1 mod-p H-spaceshttps://scholarbank.nus.edu.sg/handle/10635/104028Title: Rank p - 1 mod-p H-spaces
Authors: Grbić, J.; Harper, J.; Mimura, M.; Theriault, S.; Wu, J.
Abstract: Different constructions by Cooke, Harper and Zabrodsky and by Cohen and Neisendorfer produce torsion free finite p-local H-spaces of rank l < p - 1. The first construction goes through when l = p - 1 and we show the second does as well. However, the space produced need not be an H-space. We give a criterion for when an H-space is obtained. In the special case of rank 2 mod-3 H-spaces, we also give a practical test for when the criterion holds, and use this to give many new examples of finite H-spaces. © 2012 Hebrew University Magnes Press.
Tue, 01 Jan 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1040282013-01-01T00:00:00Z
- Artin's braid groups, free groups, and the loop space of the 2-spherehttps://scholarbank.nus.edu.sg/handle/10635/102885Title: Artin's braid groups, free groups, and the loop space of the 2-sphere
Authors: Cohen, F.R.; Wu, J.
Abstract: The purpose of this article is to describe connections between the loop space of the 2-sphere and Artin's braid groups. The current article exploits Lie algebras associated with Vassiliev invariants in the work of Kohno (Linear representations of braid groups and classical Yang-Baxter equations, Cont. Math. 78 (1988), 339-369 and Vassiliev invariants and de Rham complex on the space of knots, Symplectic Geometry and Quantization, Contemp. Math. 179 (1994), Am. Math. Soc. Providence, RI, 123-138), and provides connections between these various topics.Two consequences are as follows: the homotopy groups of spheres are identified as 'natural' sub-quotients of free products of pure braid groups, andan axiomatization of certain simplicial groups arising from braid groups is shown to characterize the homotopy types of connected CW-complexes. © 2010 Published by Oxford University Press. All rights reserved.
Thu, 01 Dec 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028852011-12-01T00:00:00Z
- Decomposition of loop spaces and periodic problem on π*https://scholarbank.nus.edu.sg/handle/10635/103104Title: Decomposition of loop spaces and periodic problem on π*
Authors: Weidong, C.; Wu, J.
Abstract: We provide a family of spaces localized at 2, whose stable homotopy groups are summands of their unstable homotopy groups. Applications to mod 2 Moore spaces are given.
Mon, 23 Sep 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031042013-09-23T00:00:00Z
- Homology decompositions of the loops on 1-stunted Borel constructions of C2 -actionshttps://scholarbank.nus.edu.sg/handle/10635/103381Title: Homology decompositions of the loops on 1-stunted Borel constructions of C2 -actions
Authors: Gao, M.; Wu, J.
Abstract: The Carlsson construction is a simplicial group whose geometric realization is the loop space of the 1-stunted reduced Borel construction. Our main results are: (i) given a pointed simplicial set acted upon by the discrete cyclic group C2 of order 2, if the orbit projection has a section, then the loop space on the geometric realization of the Carlsson construction has a mod 2 homology decomposition; (ii) in addition, if the reduced diagonal map of the C2 -invariant set is homologous to zero, then the pinched sets in the above homology decomposition themselves have homology decompositions in terms of the C2 -invariant set and the orbit space. Result (i) generalizes a previous homology decomposition of the second author for trivial actions. To illustrate these two results, we compute the mod 2 Betti numbers of an example.
Wed, 11 Sep 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033812013-09-11T00:00:00Z
- Module structure on Lie powers and natural coalgebra-split sub-Hopf algebras of tensor algebrashttps://scholarbank.nus.edu.sg/handle/10635/103557Title: Module structure on Lie powers and natural coalgebra-split sub-Hopf algebras of tensor algebras
Authors: Li, J.Y.; Lei, F.C.; Wu, J.
Abstract: We investigate the functors from modules to modules that occur as the summands of tensor powers and the functors from modules to Hopf algebras that occur as natural coalgebra summands of tensor algebras. The main results provide some explicit natural coalgebra summands of tensor algebras. As a consequence, we obtain some decompositions of Lie powers over the general linear groups. © 2011 Edinburgh Mathematical Society.
Tue, 01 Mar 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1035572011-03-01T00:00:00Z
- On symmetric commutator subgroups, braids, links and homotopy groupshttps://scholarbank.nus.edu.sg/handle/10635/103763Title: On symmetric commutator subgroups, braids, links and homotopy groups
Authors: Li, J.Y.; Wu, J.
Abstract: In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This gives a connection between links and homotopy groups. Similar results hold for braid and surface groups. © 2011 American Mathematical Society.
Fri, 01 Jul 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037632011-07-01T00:00:00Z
- On the Hodge decomposition of differential graded bi-algebrashttps://scholarbank.nus.edu.sg/handle/10635/103806Title: On the Hodge decomposition of differential graded bi-algebras
Authors: Wu, J.; Gerstenhaber, M.; Stasheff, J.
Abstract: We give a natural decomposition of a connected commutative differential graded bi-algebra over a commutative algebra in the case of characteristic zero. This gives the ordinary Hodge decomposition of the Hochschild homology when we apply this natural decomposition to the cyclic bar complex of a commutative algebra. In the case of characteristic p>0, we show that, in the spectral sequence induced by the augmentation ideal filtration of the cyclic bar complex of a commutative algebra, the only possible non-trivial differentials are dk(p-1) for k≥1. Also we show that the spectral sequence which converges to the Hochschild cohomology is multiplicative with respect to the Gerstenhaber brackets and the cup products. © 2001 Elsevier Science B.V.
Wed, 08 Aug 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1038062001-08-08T00:00:00Z
- Some calculations of Lie (n)max for low nhttps://scholarbank.nus.edu.sg/handle/10635/104155Title: Some calculations of Lie (n)max for low n
Authors: Selick, P.; Wu, J.
Abstract: In a previous paper, the authors gave the finest functorial decomposition of the loop suspension of a p-torsion suspension. In order to determine the decomposition, one must know the maximum projective submodule of Lie (n), which we label Liemax (n). The purpose of this paper is to give some sample calculations of Liemax (n) for low n when p = 2. © 2008 Elsevier B.V. All rights reserved.
Sat, 01 Nov 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041552008-11-01T00:00:00Z
- On co-H-maps to the suspension of the projective planehttps://scholarbank.nus.edu.sg/handle/10635/103690Title: On co-H-maps to the suspension of the projective plane
Authors: Wu, J.
Abstract: We study co-H-maps from a suspension to the suspension of the projective plane and provide examples of non-suspension 3-cell co-H-spaces. © 2001 Elsevier Science B.V. All rights reserved.
Mon, 30 Sep 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036902002-09-30T00:00:00Z
- Configurations, braids, and homotopy groupshttps://scholarbank.nus.edu.sg/handle/10635/103037Title: Configurations, braids, and homotopy groups
Authors: Berrick, A.J.; Cohen, F.R.; Wong, Y.L.; Wu, J.
Sat, 01 Apr 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030372006-04-01T00:00:00Z
- Functorial homotopy decompositions of looped co-H spaceshttps://scholarbank.nus.edu.sg/handle/10635/103311Title: Functorial homotopy decompositions of looped co-H spaces
Authors: Selick, P.; Theriault, S.; Wu, J.
Abstract: In recent work of the first and third authors, functorial coalgebra decompositions of tensor algebras were geometrically realized to give functorial homotopy decompositions of loop suspensions. Later work by all three authors generalized this to functorial decompositions of looped coassociative co-H spaces. In this paper we use different methods which allow for the coassociative hypothesis to be removed. © 2009 Springer-Verlag.
Tue, 01 Feb 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033112011-02-01T00:00:00Z
- Artin braid groups and homotopy groupshttps://scholarbank.nus.edu.sg/handle/10635/102884Title: Artin braid groups and homotopy groups
Authors: Li, J.; Wu, J.
Abstract: We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as summands of the centres of the quotient groups of Artin pure braid groups modulo boundary Brunnian braids. The results give new connections between the braid groups and the general higher homotopy groups of spheres. © 2009 London Mathematical Society.
Sun, 01 Nov 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028842009-11-01T00:00:00Z
- Homotopy groups as centres of finitely presented groupshttps://scholarbank.nus.edu.sg/handle/10635/103383Title: Homotopy groups as centres of finitely presented groups
Authors: Mikhailov, R.V.; Wu, J.
Abstract: For every finite Abelian group A and integer n > 3 we construct a finitely presented group defined by explicit generators and relations such that its centre is isomorphic to n(K(A,1)). © 2013 RAS(DoM) and LMS.
Tue, 01 Jan 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1033832013-01-01T00:00:00Z
- The functor A min for (p - 1)-cell complexes and EHP sequenceshttps://scholarbank.nus.edu.sg/handle/10635/104301Title: The functor A min for (p - 1)-cell complexes and EHP sequences
Authors: Wu, J.
Abstract: Let X be a co-H-space of (p - 1)-cell complex with all cells in even dimensions. Then the loop space ΩX admits a retract Āmin(X) that is the evaluation of the functor Āmin on X. In this paper, we determine the homology H*(Āmin(X)) and give the EHP sequence for the spaces Āmin(X). © 2010 Hebrew University Magnes Press.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043012010-01-01T00:00:00Z
- Modular representations and the homotopy of low rank p-local CW-complexeshttps://scholarbank.nus.edu.sg/handle/10635/103556Title: Modular representations and the homotopy of low rank p-local CW-complexes
Authors: Beben, P.; Wu, J.
Abstract: Fix an odd prime p and let X be the p-localization of a finite suspended CW-complex. Given certain conditions on the reduced mod-p homology H̃*(X; ℤp) of X, we use a decomposition of ΩΣX due to the second author and computations in modular representation theory to show there are arbitrarily large integers i such that ΩΣiX is a homotopy retract of ΩΣX. This implies the stable homotopy groups of ΣX are in a certain sense retracts of the unstable homotopy groups, and by a result of Stanley, one can confirm the Moore conjecture for ΣX. Under additional assumptions on H̃*(X; ℤp), we generalize a result of Cohen and Neisendorfer to produce a homotopy decomposition of ΩΣX that has infinitely many finite H-spaces as factors. © 2012 Springer-Verlag.
Mon, 01 Apr 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1035562013-04-01T00:00:00Z
- Symmetric ideals in group rings and simplicial homotopyhttps://scholarbank.nus.edu.sg/handle/10635/104235Title: Symmetric ideals in group rings and simplicial homotopy
Authors: Mikhailov, R.; Passi, I.B.S.; Wu, J.
Abstract: In this paper homotopical methods for the description of subgroups determined by ideals in group rings are introduced. It is shown that in certain cases the subgroups determined by symmetric product of ideals in group rings can be described with the help of homotopy groups of spheres. © 2010 Elsevier B.V.
Sun, 01 May 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042352011-05-01T00:00:00Z
- A braided simplicial grouphttps://scholarbank.nus.edu.sg/handle/10635/104510Title: A braided simplicial group
Authors: Wu, J.
Wed, 01 May 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1045102002-05-01T00:00:00Z
- Combinatorial group theory and the homotopy groups of finite complexeshttps://scholarbank.nus.edu.sg/handle/10635/103000Title: Combinatorial group theory and the homotopy groups of finite complexes
Authors: Mikhailov, R.; Wu, J.
Abstract: For n > k≥3, we construct a finitely generated group with explicit generators and relations obtained from braid groups, whose center is exactly πn.(S k). Our methods can be extended to obtain combinatorial descriptions of homotopy groups of finite complexes. As an example, we also give a combinatorial description of the homotopy groups of Moore spaces.
Thu, 07 Mar 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030002013-03-07T00:00:00Z
- Brunnian subgroups of mapping class groups and braid groupshttps://scholarbank.nus.edu.sg/handle/10635/102957Title: Brunnian subgroups of mapping class groups and braid groups
Authors: Berrick, A.J.; Hanbury, E.; Wu, J.
Abstract: This is the second of a pair of papers on the Delta-group structure on the braid and mapping class groups of a surface. We obtain a description of the homotopy groups of these Delta-groups and generalize to an arbitrary surface the Berrick-Cohen-Wong-Wu exact sequence relating the Brunnian braid groups of the 2-sphere to its homotopy groups. We prove a similar result for Brunnian mapping class groups. © 2013 London Mathematical Society.
Tue, 01 Oct 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1029572013-10-01T00:00:00Z
- On homotopy groups of the suspended classifying spaceshttps://scholarbank.nus.edu.sg/handle/10635/103715Title: On homotopy groups of the suspended classifying spaces
Authors: Mikhailov, R.; Wu, J.
Abstract: In this paper, we determine the homotopy groups π4(∑K(A, 1)) and π5(∑K(A, 1))for abelian groups A by using the following methods from group theory and homotopy theory: derived functors, the Carlsson simplicial construction, the Baues-Goerss spectral sequence, homotopy decompositions and the methods of algebraic K-theory. As the applications, we also determine πi(∑K(G, 1)) with i =4; 5 for some nonabelian groups G =∑3 and SL(Z), and π4(∑K(A4, 1)) for the 4-th alternating group A4.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037152010-01-01T00:00:00Z
- The intersecting kernels of heegaard splittingshttps://scholarbank.nus.edu.sg/handle/10635/104308Title: The intersecting kernels of heegaard splittings
Authors: Lei, F.; Wu, J.
Abstract: Let V ∪sW be a Heegaard splitting for a closed orientable 3-manifold M. The inclusion-induced homomorphisms π1 (S) → π 1 (S)→ π 1 (W) are both surjective. The paper is principally concerned with the kernels K = Ker (π1(S) → π1 (V)), L = Ker(π1(S) → π1 (W)), their intersection K ∩ L and the quotient (K∩L)/[K,L]. The module (K ∩ L)/=[K,L] is of special interest because it is isomorphic to the second homotopy module π2(M). There are two main results. (1) We present an exact sequence of Z (π1(M)-modules of the form (K∩L)/=[K,L] {right arrow, hooked} R{xl,...,xg}/J →Tπ R{y1,...yg}→theta; R↠∈Z where R=Z(φ1(M)), J is a cyclic R-submodule of R{X1...xg}, Tθ and are explicitly described morphisms of R-modules and T involves Fox derivatives related to the gluing data of the Heegaard splitting M = V Us W. (2) Let K be the intersection kernel for a Heegaard splitting of a connected sum, and k1, K2 the intersection kernels of the two summands. We show that there is a surjection K → K1 * K2 onto the free product with kernel being normally generated by a single geometrically described element.
Sat, 01 Jan 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043082011-01-01T00:00:00Z
- Suspension splittings and James-Hopf invariantshttps://scholarbank.nus.edu.sg/handle/10635/104233Title: Suspension splittings and James-Hopf invariants
Authors: Grbić, J.; Theriault, S.; Wu, J.
Abstract: James constructed a functorial homotopy decomposition for path-connected, p ointed CW-complexes X. We generalize this to a p-local functorial decomposition of ΣA, where A is any functorial retract of a looped co-H-space. This is used to construct Hopf invariants in a more general context. In addition, when A = ΩY is the loops space of a co-H-space, we show that the wedge summands of ΣΩY further functorially decompose by using an action of an appropriate symmetric group. As a valuable example, we give an application to the theory of quasi-symmetric functions. © 2014 The Royal Society of Edinburgh.
Sat, 01 Feb 2014 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042332014-02-01T00:00:00Z
- On functorial decompositions of self-smash productshttps://scholarbank.nus.edu.sg/handle/10635/103709Title: On functorial decompositions of self-smash products
Authors: Selick, P.; Wu, J.
Abstract: We give a decomposition formula for n -fold self smash of a two-cell suspension X localized at 2. The mod 2 homology of each factor in the decomposition is explicitly given as a module over the Steenrod algebra and in the case where X is formed by suspending one of ℝP2, ℂP2, ℍP2 or double-struck K sign P2, this is a complete decomposition into indecomposable pieces. The method has consequences in the modular representation theory of the symmetric group where it leads to a computation of the submatrix for the decomposition matrix of the group algebra ℤ/2[Sn] which correspond to partitions of length 2. In particular this yields a derivation of the explicit formula due to Erdmann which gives the multiplicities in the decomposition of ℤ/2[S n] of the indecomposable projective modules which correspond to those partitions.
Fri, 01 Aug 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037092003-08-01T00:00:00Z
- The decomposition of the loop space of the mod 2 More spacehttps://scholarbank.nus.edu.sg/handle/10635/104278Title: The decomposition of the loop space of the mod 2 More space
Authors: Grbić, J.; Selick, P.; Wu, J.
Abstract: In 1979 Cohen, Moore and Neisendorfer determined the decomposition into indecomposable pieces, up to homotopy, of the loop space on the mod p Moore space for primes p>2 and used the results to find the best possible exponent for the homotopy groups of spheres and for Moore spaces at such primes. The corresponding problems for p=2 are still open. In this paper we reduce to algera the determination of the base in decomposable factor in the decomposition of the mod 2 Moore space. The algebraic problems involved in determining detailed information about this factor are formidable, related to deep unsolved problems in the modular representation theory of the symmetric groups. Our decomposition has not led (thus far) to a proof of the conjectured existence of an exponent for the homotopy groups of the mod 2 Moore space or to an improvement in the known bounds for the exponent of the 2 -torsion in the homotopy groups of spheres. © 2008 Algebraic & Geometric Topology.
Tue, 01 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042782008-01-01T00:00:00Z