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Homotopy theory of the suspensions of the projective plane
 Memoirs AMS
"... Abstract. The homotopy theory of the suspensions of the real projective plane is largely investigated. The homotopy groups are computed up to certain range. The decompositions of the self smashes and the loop spaces are studied with some applications to the Stiefel manifolds. ..."
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Abstract. The homotopy theory of the suspensions of the real projective plane is largely investigated. The homotopy groups are computed up to certain range. The decompositions of the self smashes and the loop spaces are studied with some applications to the Stiefel manifolds.
On Maps From Loop Suspensions To Loop Spaces And The Shuffle Relations On The Cohen Groups
"... The maps from loop suspensions to loop spaces are investigated using group representations in this article. 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. Moreover the obst ..."
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Cited by 11 (9 self)
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The maps from loop suspensions to loop spaces are investigated using group representations in this article. 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. Moreover 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.
FUNCTORIAL HOMOTOPY DECOMPOSITIONS OF LOOPED COH SPACES
"... 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 coassoci ..."
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Cited by 5 (5 self)
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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 coH spaces. In this paper we use different methods which allow for the coassociative hypothesis to be removed. 1.
MODULE STRUCTURE ON LIE POWERS AND NATURAL COALGEBRASPLIT SUB HOPF ALGEBRAS OF TENSOR ALGEBRAS
"... Abstract. In this article, 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 ..."
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Cited by 3 (3 self)
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Abstract. In this article, 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. 1. introduction The algebraic question for functorial coalgebra decompositions of the tensor algebras was arising from homotopy theory described as follows. The relevant topological question is how to decompose the loop spaces. This is a classical problem in homotopy theory with applications to homotopy groups. For instance, the classical results of CohenMooreNeisendorfer [6] on the exponents of the homotopy groups of the spheres and Moore spaces were obtained from the study of the decompositions of the loop spaces of Moore spaces. The decompositions of the loop space functor Ω from plocal simply connected coHspaces to spaces was then introduced
THE DECOMPOSITION OF THE LOOP SPACE OF THE MOD 2 MOORE SPACE
"... 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 the spheres and for the Moore spaces ..."
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Cited by 1 (1 self)
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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 the spheres and for the Moore spaces at such primes. The corresponding problems for p = 2 are still open. In this paper we reduce to algebra the determination of the base indecomposable 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 2torsion in the homotopy groups of spheres. 1.
SUSPENSION SPLITTINGS AND JAMESHOPF INVARIANTS FOR RETRACTS OF THE LOOPS ON COHSPACES
"... Abstract. James constructed a functorial homotopy decomposition ΣΩΣX ≃ n=1 ΣX(n) for pathconnected, pointed CWcomplexes X. We generalize this to a functorial decomposition of ΣA where A is any functorial retract of a looped coHspace. This is used to construct Hopf invariants in a more general co ..."
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Cited by 1 (1 self)
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Abstract. James constructed a functorial homotopy decomposition ΣΩΣX ≃ n=1 ΣX(n) for pathconnected, pointed CWcomplexes X. We generalize this to a functorial decomposition of ΣA where A is any functorial retract of a looped coHspace. This is used to construct Hopf invariants in a more general context. In addition, when A = ΩY is the loops on a coHspace, we show that the wedge summands of ΣΩY further functorially decompose by using an action of an appropriate symmetric group. 1.
HOMOTOPY EXPONENTS OF SOME HOMOGENEOUS SPACES
"... Abstract. Let p be an odd prime. Using homotopy decompositions and spherical fibrations, under certain dimensional restrictions, we obtain upper bounds of the pprimary homotopy exponents of some homogeneous spaces such as generalized complex Stiefel manifolds, generlized complex Grassmann manifolds ..."
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Abstract. Let p be an odd prime. Using homotopy decompositions and spherical fibrations, under certain dimensional restrictions, we obtain upper bounds of the pprimary homotopy exponents of some homogeneous spaces such as generalized complex Stiefel manifolds, generlized complex Grassmann manifolds, SU(2n)/Sp(n), E6/F4 and F4/G2 (the latter for p = 2 and p ≥ 5). 1.
SUSPENSION SPLITTINGS AND HOPF INVARIANTS FOR RETRACTS OF THE LOOPS ON COHSPACES
"... Abstract. James constructed a functorial homotopy decomposition ΣΩΣX ≃ ..."