Research of Jie Wu

1  RESEARCH PROGRAMME

Research Interests

Algebraic Topology (homotopy theory) and Group Representation Theory

Description of the Research Areas.

Geometry and topology study shapes and forms and their transformations; as well as the deep interactions between basic geometric properties and geometric objects, such as between symmetry and various least action principles. The visual recognition of physical objects in the universe stimulates their initial development. Through interaction with various fields of science (e.g. astronomy, mechanics, physics) and other areas of mathematics such as algebra and analysis, geometry and topology have become subjects with their own independent interests as well as applications in almost every area of science and technology. Algebraic topology mainly uses algebraic methods to study geometric objects.
As a special area in geometry and topology, homotopy theory is an old and important area of mathematics. In addition to links with wide areas of mathematics, its results and methods are now applied to wide areas of theoretical physics, embracing string theory, quantum groups and mathematical physics.
My major research is in homotopy theory, as well as its connections with low dimensional topology and representation theory.
Accomplishments:
Combinatorial determination of homotopy groups
The determination of homotopy groups is a fundamental and central problem in homotopy theory. Simplicial groups are combinatorial models for loop spaces. Homotopy groups can be described as the derived functors of the Moore complexes of simplicial groups. Combinatorial determinations of homotopy groups have been studied from recently. In particular, it gives a specific combinatorial description of the general homotopy groups of the 3-sphere. Recently it has been discovered that one can apply simplicial group techniques to configuration spaces and braid groups. In particular, it describes the general homotopy groups of the 3-sphere as the derived groups of the classical braid groups. This gives new insight in both homotopy theory and low dimensional topology. The braid groups are used in wide areas of mathematics and physics such as number theory, algebraic geometry, string theory and mathematical physics.
My research on this topic started from my Ph. D. thesis, where one of the main results describes the general homotopy group of the sphere as the center of a group with explicit generators and relations. This is the first time to give a combinatorial description of the general homotopy groups of the sphere. The result has been internationally recognized and presented in many international conferences. These results are given in the paper titled "On combinatorial descriptions of the homotopy groups of certain spaces", Math. Proc. Camb. Phil. Soc. (3) 130 (2001), 489-513.
A continuation of this work is given in the paper titled "A braided simplicial group", Proc. London Math. Soc., (3) 84 (2002), 645-662. The main result in this paper shows that the general homotopy group of the sphere is isomorphic to the fixed set of a pure braid group action on an explicit combinational group. We solved the Taylor conjecture in this paper. This is the first time to establish direct relations between the braid groups and the homotopy groups.
Our further study was given in the 4-author paper titled "Configurations, braids and the homotopy groups" (With Jon Berrick, Fred Cohen and Yan Loi Wong), J. Amer. Math. Soc. 19 (2006), no. 2, 265-326. (One of the top journals in mathematics) This is a lengthy paper establishing various connections between algebraic topology and low dimensional topology. Various surprising results are obtained such as the connections between the Brunnian braids and the homotopy groups. Roughly speaking, the higher homotopy groups of the sphere are the quotient of the Brunnian braids over the sphere by the Brunnian braids over the disk. Another result states that the homotopy groups of the 3-sphere are the derived groups of the sequence of the classical Artin braid groups. These results have been highly recognized in the international conferences.
There are some subsequent developments of this project. Also some new results have been obtained recently by joint work with Fred Cohen in the paper titled "braids, free groups and the loop space of the 2-sphere", submitted.
Recent suggestion from Joan Birman is to explore further on this project by studying the Artin representation of braid groups on the descending central series of the free groups. The Artin representation has been studied in our previous paper titled "A braided simplicial group", Proc. London Math. Soc., (3) 84 (2002), 645-662.
The study of this project on the homotopy groups is different from the traditional methods, and is being involved in more and more methods from other areas such as low dimensional topology, group representations and others.
In this direction, it is also important to study simplicial homotopy theory to discover new methods of determination of homotopy groups. Our parallel work consists of the following:
  1. A simplicial sheaf theoretical description of Carlsson's construction is given in the paper titled "On fibrewise simplicial monoids and the Milnor-Carlsson construction", Topology, 37(1998), No. 5, 1113-1134. An application of this work is to give a combinatorial description of the general homotopy groups of SK(p,1) for an arbitrary group p. This work should also prove useful in combining the operadic approach of Hinich-Schechtman with the configuration space approach of Kontsevich. The latter in turn involves deformation theory, which Murray Gerstenhaber at the University of Pennsylvania is one of the founding fathers. In particular, we overlap in the study of sheaf theory approaches to deformation theory. Along these ideas, I have a co-authored paper with Murray Gerstenhaber and Jim Stasheff titled "On the Hodge decomposition of differential graded bi-algebras", J. of Pure and Appl. Algebra, 162 (2001), 103-25.
  2. The study of products on minimal simplicial sets is given in the paper titled "On products on minimal simplicial sets", J. of Pure and Appl. Algebra, 148 (2000), No. 1, 89-111. In this paper, we answered an old problem on minimal simplicial groups proposed by John Moore in 1950s.
Natural decompositions of loop suspensions
In classifying any mathematical structure, it is helpful to analyze the irreducible or indecomposable components. In the case of the homotopy groups of spheres, the torsion is controlled by the homotopy of mod pr Moore spaces. Cohen, Moore, Neisendorfer and Selick were thus able to obtain the global best possible exponent results for the homotopy groups of spheres and for Moore spaces (at odd primes). More generally, for the homotopy theory of a finite complex, the methods of splitting a space and analyzing the pieces has been fruitful in the past. Once the indecomposable factors are under control, there is often some interesting revelation about the whole space.
Our results joint with Paul Selick state that the mod p homology of a special indecomposable factor of X is a subspace of a specific quotient of the tensor algebra of the mod p homology of X. By using these results, we solved the Cohen conjecture. By one of our results, the problem of natural decompositions of the loop suspensions is reduced to the problem of natural coalgebra decompositions of tensor algebras. Moreover, the solution to the problem of natural coalgebra decompositions of tensor algebras is given by finding the maximal projective submodule of the important symmetric group modules Lie(n). This established new important connections between homotopy theory and the modular representation theory of symmetric groups.
These results have been internationally recognized and are given in the lengthy joint paper with Paul Selick titled "On natural coalgebra decompositions of tensor algebras and loop suspensions", Memoirs Amer. Math. Soc., Vol. 148, No. 701, 2000.
Subsequent developments of this project are given in the articles:
  1. With Paul Selick, Some calculations of Liemax(n) for low n, J. of Pure and Appl. Algebra, to appear. In this paper, the methods and results from the modular representation theory of symmetric groups have been largely used to determine Liemax(n) for low n 8.
  2. With Paul Selick, The functor Amin on p-local spaces, Math. Z, to appear. The results in our Memoirs' paper have been generalized to the loop suspension of any path-connected p-local spaces in this paper. The technical progress in this paper allows us to give the further development given next:
  3. With Paul Selick and Stephen Theriault, Functorial decompositions of looped coassociative co-H spaces, Canad. J. Math., to appear. The results in our Memoirs' paper have been generalized to the looped coassociative co-H spaces in this paper.
  4. With Paul Selick and Stephen Theriault, Decompositions of H-spaces, preprint. By using our new methods, the assumptions on coassociativity in the above paper can be removed. The methods and results in this paper give a possibility to decompose more general loop spaces.
This project has been talked in the ICM satellite conference on algebraic topology with positive response.
In this direction, it is also important to study Hopf invariants and natural decompositions of self-smash products. Our parallel work consists of the following:
  1. The study on the James-Hopf invariants is given in the paper titled "On combinatorial calculations of the James-Hopf maps", Topology, 37(1998), No. 5, 1011-1023. In this paper, we use the recent method introduced by F. R. Cohen for giving combinatorial computations of the James-Hopf invariants. Applications to decompositions of loop suspensions are given.
  2. The study on self-smash products is given in the joint paper with Paul Selick titled "On functorial decompositions of self-smash products", Manuscripta Math. 111 (2003), no. 4, 435-457. In this paper, we give a decomposition formula for general self-smash products of a two-cell suspension X localized at 2, in which the mod 2 homology of each factor in the decomposition is explicitly given and is indecomposable over the Steenrod algebra if X is a suspension of a projective plane. The result answers a classical problem in homotopy theory how to decompose general self-smashes of suspensions of a projective plane.
  3. Further study on self-smash products is given in the joint paper with Fred Cohen and Paul Selick titled "Natural decompositions of self-smashes of 2-cell complexes, preprint.
It should be pointed out that the problem on functorial decompositions of self-smashes is equivalent to the fundamental problem in the modular representation theory of the symmetric groups. The problem on the functorial decompositions of loops on co-H-spaces is equivalent to the representation theory of the free Lie algebras as modules over the general linear groups.
Unstable homotopy theory
Part of my work on classical homotopy theory is on mod 2 Moore spaces (that is, suspensions of the real projective plane). These spaces have important applications in homotopy theory and geometry.
In my joint paper with Fred Cohen titled "A remark on the homotopy groups of SnRP2", Contemporary Mathematics, 181(1995), 65-81, we proved that there are infinitely many Z/8Z-summands in the homotopy groups of the mod 2 Moore spaces.
A specific decomposition of the triple loop space of the suspension of the real projective plane is given in "A product decomposition of W30(SRP2)", Topology, 37(1998), No. 5, 1025-1032. As a corollary, we proved that the higher homotopy groups of the 3-sphere are the summands of the homotopy groups of the suspension of the real projective plane. An application to co-H-spaces is given in the paper titled "On co-H-maps to the suspension of the projective plane", Topology and its Applications, 123(2002), 547-571. The result gives a connection between non-suspension 3-cell co-H-spaces and the elements of order 2 in the homotopy groups of the 3-sphere. In particular, it gives infinitely many examples of non-suspension 3-cell co-H-spaces which answers a problem proposed by John Harper.
Systematic study on mod 2 Moore spaces is given in the lengthy paper titled "on the homotopy theory of the suspensions of the projective plane", Memoirs Amer. Math. Soc., Vol. 162, No.769, 2003. The homotopy theory of these important spaces has been largely investigated. Various new results with some applications are given. Also the homotopy groups are computed up to certain range. Although this paper is on the topic of classical homotopy theory, some new methods such as group representations have been introduced. These new methods successfully solved many classical problems.
The second part of my work in this area is on the exponent problem, which is one of the most challenged problems in unstable homotopy theory. The problem is to study the exponents for the p-torsion components of the homotopy groups of certain type of spaces. My recent progress on this topic is given in the paper titled "On maps from loop suspensions to loop spaces and the shuffle relations on the Cohen groups", Memoirs Amer. Math. Soc., Vol. 180, No. 851, 2006. In this paper, the self-maps of loop suspensions are largely investigated using group representations. The shuffle relations on the Cohen groups are given. By using these relations, a universal ring for functorial self maps of double loop spaces of double suspensions is given. The obstructions to the classical exponent problem in homotopy theory are displayed in the extension groups of the dual of the important symmetric group modules Lie(n), as well as in the top cohomology of the Artin braid groups with coefficients in the top homology of the Artin pure braid groups, which gives some connections with the complexity of algorithms studied by Smale and other people. Moreover new invariants are found by applying the Cohen group functor to simplicial groups.
Further development is given in the following preprints:
  1. Jelena Grbic and Jie Wu, Natural transformations of tensor algebras and representations of combinatorial groups, preprint.
  2. Jelena Grbic and Jie Wu, Applications of combinatorial groups to the Hopf invariants and the exponent problem, preprint.
Further study on the exponent problem will be given by discovering further possible relations to the Cohen groups using configurations and simplicial methods, with connections to the equivariant stable homotopy theory.
Configuration Spaces
The cohomology of configuration spaces and their close relatives, moduli spaces, has in recent years appeared in several areas outside of algebraic topology, most notably in mathematical physics and knot theory. In contrast to the original work on such spaces directly, these recent developments have involved various compactifications.
We gave a generalization of the Hilton-Milnor theorem and determined the homology of certain configuration spaces in the paper titled "On the homology of configuration spaces C((M,M0Rn;X)", Math. Z., 22(1998), 235-248.
Our further study was given in the recent 4-author paper titled "Configurations, braids and the homotopy groups" (With Jon Berrick, Fred Cohen and Yan Loi Wong), J. Amer. Math. Soc. 19 (2006), 265-326. This is the first time to study simplicial structure on configuration spaces. It establishes various connections between algebraic topology and low dimensional topology. Various surprising results are obtained such as the connections between the Brunnian braids and the homotopy groups. Roughly speaking, the higher homotopy groups of the sphere are the quotient of the Brunnian braids over the sphere by the Brunnian braids over the disk. Another result states that the homotopy groups of the 3-sphere are the derived groups of the sequence of the classical Artin braid groups. These results have been highly recognized in the international conferences.

2  Summary of Research (Rough Draft for Experts Only)

2.1  Homotopy Groups and Braids

Let G(n) be the group generated by x1,x2,,xn subject to the relations:

    1) the ordered product of the generators x1x2xn=1 and
    2) the iterated commutators on the generators with the property that each generator occurs at least once in the commutator bracket.
Theorem 1 [17,Theorem 1.4] For each n, the homotopy group pn(S2) is isomorphic to the center of G(n).
This result gives a global combinatorial description of the general homotopy groups of the sphere. Although it does not admit computations of higher homotopy groups yet, the systematic description of the defining relations for the group G(n) suggests us to consider the braid group action on G(n) via the classical Artin representation.
Theorem 2 [18,Theorem 1.2] The Artin representation induces an action of the braid group Bn on G(n). Moreover the homotopy group pn(S2) is isomorphic to the fixed set of the pure braid group action on G(n).
It was observed that the Artin representation is simplicial and so the direction was moved to investigate the simplicial structure on the sequence of the braid groups. The simplicial structure on braids was introduced in [2] in the canonical way by using the method of doubling/deleting strands. According to the terminology in low dimensional topology, a braid is called Brunnian if it becomes a trivial braid after removing any one of its strands. (For instance, the Borromean Rings is a link by closing up a 3-strand Brunnian braid.) By using terminology of simplicial groups, the Brunnian braids are the Moore cycles after establishing simplicial or D-structure on braids. Let Brunn(M) denote the group of n-strand Brunnian braids over the manifold M.
Theorem 3 [2,Theorem 1.2] There is an exact sequence of groups
1 Brunn+1(S2) Brunn(D2) f*

 
Brunn(S2)pn-1(S2)1
for n 5, where f* is induced from the canonical embedding f: D2 S2.
Thus the torsion homotopy groups of S2 (or S3) are the invariants for measuring the difference of the Brunnian braids between S2 and D2. Moreover there is a differential on the sequence of the classical Brunnian braids {Brunn(D2)}, which is essentially induced from complex-conjugation operation on configuration spaces. This makes {Brunn(D2)} is a chain complex of non-commutative groups.
Theorem 4 [2,Theorem 1.3] For all n there is an isomorphism of groups
Hn(Brun(D2)) @ pn(S2).
This result describes the homotopy groups of the sphere as the derived groups of the classical Brunnian braids.
Let AP*={Pn+1} be the sequence of classical Artin pure braid groups with the simplicial structure given by deleting/doubling braids. Then AP* is a reduced simplicial group because AP0=P1=1. Since AP1=P2=Z, there is a unique simplicial homomorphism Q: F[S1] AP* that sends the non-degenerate 1-simplex of S1 to the generator of AP1=Z, where F[S1] is Milnor's F[K]-construction on S1.
Theorem 5 [4,Theorem 1.2] The morphism of simplicial groups
Q: F[S1]  AP*
is an embedding. Hence the homotopy groups of F[S1] are natural sub-quotients of AP*, and the geometric realization of quotient simplicial set AP*/F[S1] is homotopy equivalent to the 2-sphere. Furthermore, the image of Q is the smallest simplicial subgroup of AP* which contains A1,2.
The simplicial group AP* is contractible and so one can use it to construct new simplicial groups using simple operations. Let B denote the smallest full sub-category of the category of reduced simplicial groups which satisfies the following properties:
  1. The simplicial group AP* is in B.
  2. If P, and G are in B, then the coproduct PG is in B.
  3. If P is in B, and G is a simplicial subgroup of P, then G is in B.
  4. If P is in B, and G is a simplicial quotient group of P, then G is in B.
Theorem 6 [4,Theorem 1.5] Let X denote simply-connected CW-complex. Then there exist an object GX in B such that the loop space W(X) is homotopy equivalent to the geometric realization of GX.
This result states that the loop space of any simply-connected CW-complex is braided. There might be a possibility to using this combinatorial model to attack the Moore conjecture by considering the filtrations that admit braided means. Moreover [4] also exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno.
The connections between the homotopy groups and the mapping class groups are being investigated [1]. The observation is that, as a group, F[S1S1]g is of rank 2g with potential connections with Riemann surfaces of genus g.
Moreover the simplicial structures on subgroups of Aut(Fn) are being investigated (with F. Cohen) and also on singular braids (with V. Vershinin).

2.2  Loop Spaces and The Exponent Problem

Let bn: Vn Vn be the map defined by
bn(x1xn)=[[x1,x2],,xn].
By using the classical formula bnbn=nbn and considering the composition of the Hopf invariants and the Whitehead products, a product decomposition of WS2 X has been constructed in [3] with applications to mod 2 Moore spaces. Further investigation on the Hopf invariants was given in [12].
Roughly speaking the above decompositions can be obtained by constructing explicit maps using Hopf invariants and Whitehead products. The explicit constructions became mess in the cases when n is divisible by p although we can explicitly construct maps in the cases n=pq with q\not 0 mod p. The ideas in [7] were to directly do representation theory on loop suspensions rather than constructing explicit maps. So the terminology of functorial decompositions of loop suspensions was introduced.
The study on functorial decompositions of loop suspensions was given by several steps. First, in geometry, we need to handle the Cohen group as functorial self maps of WS2X. Secondly, in algebra, we determined the group of functorial self coalgebra maps of the tensor algebras, where the tensor algebra T(V) is Hopf by saying V primitive. Then we proved that the homology functor induces an isomorphism from the geometric universal group (the Cohen group) to the algebraic universal group (the group of coalgebra natural transformations of the tensor algebra functor). Thus the problem on functorial decompositions of loop suspensions is reduced to the algebraic problem on coalgebra decompositions of the tensor algebra functor. Namely, for any coalgebra functorial decomposition
T(V) @ A(V)B(V),
there is a correspondent functorial decomposition
WS2X @ A(X)×B(X)
such that the mod p homology H*(A(X))=A(V) and H*(B(X))=B(V) by inputting V to be the reduced homology of X.
A connection between the coalgebra decompositions of the tensor algebra functor and the modular representation theory is as follows: Let Liemax(n) be the largest projective Sn-submodule of Lie(n) and let Lmaxn(V)=Liemax(n)k(Sn)Vn. Let Bmax(V) be the sub-Hopf algebra of T(V) generated by Lnmax(V) for n 2. Then the quotient Amin(V)=kBmax(V)T(V) is the smallest functorial coalgebra retract of T(V) that contains the bottom cell. In other words, the problem on functorial decompositions is equivalent to the modular representation theory on Lie(n).
Theorem 7 [Cohen Conjecture][7,Theorem 1.1] Let X be a p-torsion suspension and let V be the reduced mod p homology of X. Then there is a natural homotopy decomposition
WSX @ A(X)×B(X)
such that

    1) V H*(A(X));
    2) B(X) is a loop suspension and the injection B(X)WSX is a loop map;
    3) Ln(V) H*(B(X)) if n is not a power of p.
The generalizations of this result were given in [9] for the case when X is a path-connected p-local space and in [11] for the case of the loop space of co-H-spaces.
It should be pointed out that the Poincaré series of Amin(V) is equivalent to the solution of determining the size Liemax(n) which is then essentially equivalent to the (longstanding unsolved) fundamental problem in the modular representation theory of the symmetric groups. Recently, with Paul Selick, we proved that, for Hopf invariant one complex, Amin(V) is the same as the smallest coalgebra retract of T(V) over the Steenrod algebra. Moreover, with Fred Cohen, we constructed explicit functorial retract of WS2X with primitives of tensor lengths of powers of p.
Motivated from the modular representation theory, it is expected that, for a given module V, Amin(V) is closely related to the smallest coalgebra retract of T(V) over the general linear group GL(V) (if the ground field is algebraically closed). Moreover there is a strong connection between the Steenrod algebra and the hyperalgebra in representation theory. It deserves to have further investigations on this project as this canonically links to the representation theory.
For studying the exponent problem (the Barratt conjecture), the obstructions to the exponent problem are displayed in Lie(n) by considering the Cohen group as the universal group for the loop suspensions. Further study is to investigate the self maps of double loop spaces. Motivated from the fact that the reduced diagonal
-
D
 
:WS2X WS2X WS2 X
is null-homotopic after looping, we investigated the quotient of the group of coalgebra self transformations HK=coalgK(T,T) of the tensor algebra functor by the reduced diagonal, denoted by RK, where K is the ground ring. (Note. By considering cobar construction, the reduced diagonal is the first differential.)
Theorem 8 [21,Theorems 1.3 and 1.4] Let K be a commutative ring with identity. Then there is a quotient group RK of HK=coalgK(T,T) with the following properties.

    1) RK is an abelian group. Moreover RK is a ring with the multiplication induced by the composition operation on coalgK(T,T). Furthermore for any ring homomorphism f: K K, there is an induced ring homomorphism
    R(f): RK RK
    for changing the group rings, that is, R defines a functor from commutative rings with identity to rings with identity.
    2) There is a morphism of rings
    q:RK

    n=1 
    HomK(Sn)(LieK(n),LieK(n)).
    If K is a field of characteristic 0, then q is an isomorphism.
    3) There is a tower of epimorphisms of rings
    RKnRKn-1RK1=K
    such that RK=limnRnK is the inverse limit.
    4) Let IKn denote the kernel of RKn Rn-1K. Then there is an exact sequence
    0 ExtK(Sn)(LieK(n)*,LieK(n)*)IKn q

     
    HomK(Sn)(LieK(n),LieK(n)) H0K(Sn)(LieK(n)*;LieK(n)*)0
    for each n.
    5) Let Bn act on LieK(n) via the canonical quotient Bn Sn. Then there is an isomorphism
    InK @ Hn-1(Bn;LieK(n)[-1]) @ Hn-1(Bn;Hn-1(Pn;K))
    where Pn=Ker(Bn Sn) is the Artin pure braid group with the canonical Bn-action on H*(Pn;K).
    6) If K=Z(p) (or Zp), there is a commutative diagram of semirings
    HK
    RK
    q

     


    n=1 
    HomK(Sn)(LieK(n),LieK(n))
    =


    n=1 
    HomK(Sn)(LieK(n),LieK(n))
    e
    e
    ||
    [WS2,WS2]
    W

     
    [W2S2,W2S2]
    Wk-2

     
    [WkS2,WkS2]


    n=1 
    HomK(Sn)(LieK(n),LieK(n))
    for each k 2, where WkS2 is regarded as a functor from p-local (or p-complete) spaces to pointed spaces. Moreover if
    f: S2X Y
    is of order pr in [S2X,Y], then there is a commutative diagram
    RK
    eX

     
    [W2S2X,W2S2X]
    W2 f*
    RZ/pr
    [W2S2X,W2 Y].
Assertion (5) gives an interesting connection between the exponent problem and complexity of algorithms, namely the obstructions to the exponent problem are then displayed in the cohomology of braid groups with coefficients in Lie(n).

2.3  Others.

Mod 2 Moore spaces have been largely investigated in  [20] including the computation of the homotopy groups. An explicit decomposition of the triple loop space of SRP2 was given in [13]. In [19], non-suspension co-H-spaces of the form SRP2en+1 are classified by the elements of order 2 in pn(S3). Fibrewise simplicial monoids (simplicial sheafs on monoids) were investigated in [14], where homological decompositions of W(FPX) are given for F=R, C or H. (Note. There is a connection between W(RP X) and billiards (cyclic configurations) by considering the word-length filtration on Carlsson's free product construction of Z/2.) Computation of certain configuration spaces was given  [15] and the Hodge decomposition of differential graded bi-algebras was investigated in [6]. An answer to Moore's old problem on minimal simplicial groups was given in [16].

References

[1]
J. A. Berrick, D. Biss and J. Wu, Braid and mapping class groups, in preparing.
[2]
J. A. Berrick, F. R. Cohen, Y. L. Wong and J. Wu, Configurations, Braids and Homotopy Groups, J. Amer. Math. Soc., 19 (2006), 265-326.
[3]
F. R. Cohen and J. Wu, A remark on the homotopy groups of SnRP2, Contem. Math. 181 (1995) 65-81.
[4]
F. R. Cohen, and J. Wu, On braid groups, free groups, and the loop space of the 2-sphere, submitted.
[5]
F. R. Cohen, and J. Wu, On braid groups, free groups, and the loop space of the 2-sphere, Progress in Mathematic Techniques 215 (2003), Algebraic Topology: Categorical Decompositions, 93-105.
[6]
Jie Wu, Murray Gerstenhaber and Jim Stasheff, On the Hodge decomposition of differential graded bi-algebras, J. of Pure and Appl. Algebra 162 (2001), 103-25.
[7]
Paul Selick and Jie Wu, On natural decompositions of loop suspensions and natural coalgebra decompositions of tensor algebras, Memoirs AMS 148 (2000), No. 701.
[8]
Paul Selick and Jie Wu, On functorial decompositions of self-smash products, Manuscripta Math. 111 (2003), 435-457.
[9]
Paul Selick and Jie Wu, The functor Amin on p-local spaces,, Math. Z., 253 (2006), 435-451.
[10]
Paul Selick and Jie Wu, Some calculations of Liemax(n) for low n, J. Pure Appl. Algebra, to appear.
[11]
Paul Selick, Stephen Theriault and Jie Wu, Functorial decompositions of looped coassociative co-H spaces, to appear.
[12]
J. Wu, On combinatorial calculations for the James-Hopf maps, Topology 37 (1998), 1011-1023.
[13]
J. Wu, A product decomposition of W30(SRP2), Topology 37 (1998), 1025-1032.
[14]
J. Wu, On fibrewise simplicial monoids and Milnor-Carlsson's constructions, Topology 37 (1998), 1113-1134.
[15]
J. Wu, On the homology of configuration spaces C((M,M0Rn;X), Math. Z. 22 (1998), 235-248.
[16]
J. Wu, On products on minimal simplicial sets, J. of Pure and Appl. Algebra 148 (2000), 89-111.
[17]
J. Wu, On combinatorial descriptions of the homotopy groups of certain spaces, Math. Proc. Camb. Phil. Soc. 130 (2001), 489-513.
[18]
J. Wu, A braided simplicial group, Proc. London Math. Soc. 84 (2002), 645-662.
[19]
J. Wu, On co-H-maps to the suspension of the projective plane, Topology and its Applications 123 (2002), 547-571.
[20]
J. Wu, Homotopy theory of the suspensions of the projective plane, Memoirs AMS 162 (2003), No. 769.
[21]
J. Wu, On maps from loop suspensions to loop spaces and the shuffle relations on the Cohen groups, Memoirs AMS, Vol. 180, No. 851, 2006.



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