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 3sphere. 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 3sphere 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), 489513.
A continuation of this work is given in the paper titled "A
braided simplicial group", Proc. London Math. Soc., (3) 84 (2002),
645662. 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 4author 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, 265326. (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
3sphere 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 2sphere", 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), 645662.
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:
A simplicial sheaf theoretical description of Carlsson's
construction is given in the paper titled "On fibrewise
simplicial monoids and the MilnorCarlsson construction",
Topology, 37(1998), No. 5, 11131134. 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
HinichSchechtman 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
coauthored paper with Murray Gerstenhaber and Jim Stasheff titled
"On the Hodge decomposition of differential graded
bialgebras", J. of Pure and Appl. Algebra, 162 (2001),
10325.
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, 89111. 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 p^{r} 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:
With Paul Selick, Some calculations of Lie^{max}(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
Lie^{max}(n) for low n £ 8.
With Paul Selick, The functor Amin on plocal spaces, Math. Z, to
appear. The results in our Memoirs' paper have been generalized to
the loop suspension of any pathconnected plocal spaces in this
paper. The technical progress in this paper allows us to give the
further development given next:
With Paul Selick and Stephen Theriault, Functorial
decompositions of looped coassociative coH spaces, Canad. J.
Math., to appear. The results in our Memoirs' paper have been
generalized to the looped coassociative coH spaces in this paper.
With Paul Selick and Stephen Theriault, Decompositions of
Hspaces, 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 selfsmash products. Our parallel work
consists of the following:
The study on the JamesHopf invariants is given in the paper
titled "On combinatorial calculations of the JamesHopf
maps", Topology, 37(1998), No. 5, 10111023. In this paper, we use
the recent method introduced by F. R. Cohen for giving combinatorial
computations of the JamesHopf invariants. Applications to
decompositions of loop suspensions are given.
The study on selfsmash products is given in the joint paper
with Paul Selick titled "On functorial decompositions of
selfsmash products", Manuscripta Math. 111 (2003), no. 4,
435457. In this paper, we give a decomposition formula for general
selfsmash products of a twocell 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
selfsmashes of suspensions of a projective plane.
Further study on selfsmash products is given in the joint
paper with Fred Cohen and Paul Selick titled "Natural
decompositions of selfsmashes of 2cell complexes, preprint.
It should be pointed out that the problem on functorial
decompositions of selfsmashes is equivalent to the fundamental
problem in the modular representation theory of the symmetric
groups. The problem on the functorial decompositions of loops on
coHspaces 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 S^{n}RP^{2}", Contemporary
Mathematics, 181(1995), 6581, we proved that there are infinitely
many Z/8Zsummands 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
W^{3}_{0}(SRP^{2})", Topology, 37(1998),
No. 5, 10251032. As a corollary, we proved that the higher homotopy
groups of the 3sphere are the summands of the homotopy groups of
the suspension of the real projective plane. An application to
coHspaces is given in the paper titled "On coHmaps to
the suspension of the projective plane", Topology and its
Applications, 123(2002), 547571. The result gives a connection
between nonsuspension 3cell coHspaces and the elements of order
2 in the homotopy groups of the 3sphere. In particular, it gives
infinitely many examples of nonsuspension 3cell coHspaces 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 ptorsion
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
selfmaps 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:
Jelena Grbic and Jie Wu, Natural transformations of tensor
algebras and representations of combinatorial groups, preprint.
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 HiltonMilnor theorem and determined
the homology of certain configuration spaces in the paper titled
"On the homology of configuration spaces
C((M,M_{0})×R^{n};X)", Math. Z., 22(1998), 235248.
Our further study was given in the recent 4author paper titled
"Configurations, braids and the homotopy groups" (With
Jon Berrick, Fred Cohen and Yan Loi Wong), J. Amer. Math.
Soc.19 (2006), 265326. 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
3sphere 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)
Let G(n) be the group generated by x_{1},x_{2},¼,x_{n} subject to
the relations:
1) the ordered product of the generators x_{1}x_{2}¼x_{n}=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 p_{n}(S^{2}) 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 B_{n}
on G(n). Moreover the homotopy group p_{n}(S^{2}) 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 3strand
Brunnian braid.) By using terminology of simplicial groups, the
Brunnian braids are the Moore cycles after establishing
simplicial or Dstructure on braids. Let Brun_{n}(M)
denote the group of nstrand Brunnian braids over the manifold
M.
Theorem 3[2,Theorem 1.2]
There is an exact sequence of groups
1® Brun_{n+1}(S^{2})® Brun_{n}(D^{2})
f_{*} ®
Brun_{n}(S^{2})®p_{n1}(S^{2})®1
for n ³ 5, where f_{*} is induced from the canonical embedding
f: D^{2}® S^{2}.
Thus the torsion homotopy groups of S^{2} (or S^{3}) are the
invariants for measuring the difference of the Brunnian braids
between S^{2} and D^{2}. Moreover there is a differential on the
sequence of the classical Brunnian braids
{Brun_{n}(D^{2})}, which is essentially induced from
complexconjugation operation on configuration spaces. This makes
{Brun_{n}(D^{2})} is a chain complex of noncommutative
groups.
Theorem 4[2,Theorem 1.3]
For all n there is an isomorphism of groups
H_{n}(Brun(D^{2})) @ p_{n}(S^{2}).
This result describes the homotopy groups of the sphere as the
derived groups of the classical Brunnian braids.
Let AP_{*}={P_{n+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 AP_{0}=P_{1}=1. Since
AP_{1}=P_{2}=Z, there is a unique simplicial homomorphism
Q: F[S^{1}]® AP_{*} that sends the nondegenerate
1simplex of S^{1} to the generator of AP_{1}=Z, where
F[S^{1}] is Milnor's F[K]construction on S^{1}.
Theorem 5[4,Theorem 1.2]
The morphism of simplicial groups
Q: F[S^{1}] ® AP_{*}
is an embedding.
Hence the homotopy groups of F[S^{1}] are natural subquotients of
AP_{*}, and the geometric realization of quotient
simplicial set AP_{*}/F[S^{1}] is homotopy equivalent to the
2sphere. Furthermore, the image of Q is the smallest
simplicial subgroup of AP_{*} which contains A_{1,2}.
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 subcategory of the
category of reduced simplicial groups which satisfies the following
properties:
The simplicial group AP_{*} is in B.
If P, and G are in B, then the
coproduct PÚG is in B.
If P is in B, and G is a
simplicial subgroup of P, then G is in B.
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 simplyconnected CWcomplex. Then there exist an
object G_{X} in B such that the loop space
W(X) is homotopy equivalent to the geometric realization of
G_{X}.
This result states that the loop space of any simplyconnected
CWcomplex 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[S^{1}ÚS^{1}]_{g} is of rank 2g with potential
connections with Riemann surfaces of genus g.
Moreover the simplicial structures on subgroups of
Aut(F_{n}) are being investigated (with F. Cohen) and also
on singular braids (with V. Vershinin).
By using the classical formula b_{n}°b_{n}=nb_{n} and
considering the composition of the Hopf invariants and the Whitehead
products, a product decomposition of WS^{2} 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 º 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 WS^{2}X. 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)ÄB(V),
there is a correspondent functorial
decomposition
WS^{2}X @ 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 Lie^{max}(n) be the largest projective S_{n}submodule of
Lie(n) and let
L^{max}_{n}(V)=Lie^{max}(n)Ä_{k(Sn)}V^{Än}. Let B^{max}(V) be the subHopf algebra of T(V) generated
by L_{n}^{max}(V) for n ³ 2. Then the quotient
A^{min}(V)=kÄ_{Bmax(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 ptorsion 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 Í H_{*}(A(X));
2) B(X) is a loop suspension and the
injection B(X)®WSX is a loop map;
3) L_{n}(V) Í 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 pathconnected plocal space and in [11]
for the case of the loop space of coHspaces.
It should be pointed out that the Poincaré series of A^{min}(V)
is equivalent to the solution of determining the size
Lie^{max}(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,
A^{min}(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 WS^{2}X with
primitives of tensor lengths of powers of p.
Motivated from the modular representation theory, it is expected
that, for a given module V, A^{min}(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
:WS^{2}X® WS^{2}X ÙWS^{2} X
is nullhomotopic after looping, we investigated
the quotient of the group of coalgebra self transformations
H^{K}=coalg^{K}(T,T) of the tensor algebra functor by the reduced diagonal, denoted by R^{K}, 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 R^{K} of H^{K}=coalg^{K}(T,T) with the
following properties.
1) R^{K} is an abelian group. Moreover R^{K} is a
ring with the multiplication induced by the composition operation on
coalg^{K}(T,T). Furthermore for any ring homomorphism f: K® K¢, there is an induced ring homomorphism
R(f): R^{K}® R^{K¢}
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:R^{K}®
¥ Õ
n=1
Hom_{K(Sn)}(Lie^{K}(n),Lie^{K}(n)).
If K is a field of characteristic 0, then q is an
isomorphism.
3) There is a tower of epimorphisms of rings
¼®R^{K}_{n}®R^{K}_{n1}®¼®R^{K}_{1}=K
such that R^{K}=lim_{n}R_{n}^{K} is the inverse limit.
4) Let I^{K}_{n} denote the kernel of R^{K}_{n}® R_{n1}^{K}. Then
there is an exact sequence
where P_{n}=Ker(B_{n}® S_{n}) is the Artin pure braid group with the
canonical B_{n}action on H_{*}(P_{n};K).
6) If K=Z_{(p)} (or Z_{p}), there is a commutative
diagram of semirings
H^{K}
®
R^{K}
q ®
¥ Õ
n=1
Hom_{K(Sn)}(Lie^{K}(n),Lie^{K}(n))
=
¥ Õ
n=1
Hom_{K(Sn)}(Lie^{K}(n),Lie^{K}(n))
¯e
¯e

[WS^{2},WS^{2}]
W ®
[W^{2}S^{2},W^{2}S^{2}]
W^{k2} ®
[W^{k}S^{2},W^{k}S^{2}]
®
¥ Õ
n=1
Hom_{K(Sn)}(Lie^{K}(n),Lie^{K}(n))
for each k ³ 2, where W^{k}S^{2} is regarded as a functor
from plocal (or pcomplete) spaces to pointed spaces. Moreover
if
f: S^{2}X® Y
is of order p^{r} in [S^{2}X,Y],
then there is a commutative diagram
R^{K}
e_{X} ®
[W^{2}S^{2}X,W^{2}S^{2}X]
¯
¯W^{2} f_{*}
R^{Z/pr}
®
[W^{2}S^{2}X,W^{2} 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).
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 SRP^{2} was given in [13].
In [19], nonsuspension coHspaces of the form SRP^{2}Èe^{n+1} are classified by the elements
of order 2 in p_{n}(S^{3}). Fibrewise simplicial monoids
(simplicial sheafs on monoids) were investigated in [14],
where homological decompositions of
W(FP^{¥}ÙX) are given for
F=R, C or H.
(Note. There is a connection between W(RP^{¥} ÙX) and billiards (cyclic
configurations) by considering the wordlength 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 bialgebras was investigated
in [6]. An answer to Moore's old problem on minimal
simplicial groups was given in [16].
F. R. Cohen, and J. Wu, On braid groups, free
groups, and the loop space of the 2sphere, Progress in
Mathematic Techniques215(2003), Algebraic Topology:
Categorical Decompositions, 93105.
Jie Wu, Murray Gerstenhaber and Jim Stasheff, On
the Hodge decomposition of differential graded bialgebras,
J. of Pure and Appl. Algebra162(2001),
10325.
Paul Selick and Jie Wu, On natural decompositions
of loop suspensions and natural coalgebra decompositions of tensor
algebras, Memoirs AMS148(2000), No. 701.