arXiv subjects

Brayton Gray

Publications and source records attributed to Brayton Gray.

6 recordsLinked to original sources

arXiv2014

Abelian properties of Anick spaces

The Anick spaces play a key role in an unstable filtration of the stable homotopy of V(0) to produce secondary EHP sequences. This work establishes that the Anick spaces are homotopy associative and homotopy commutative H-spaces, and that they have a universal mapping property for maps into a homotopy commutative and homotopy associative H-space with a (necessary) condition on the growth of the p torsion. This is utilized to establish an unstable composition theory. The techniques involve generalized Whitehead products based on co-H spaces, calculations in a congruence category that lies between the unstable category and the stable category, and a controlled version of the extension theorem for principal fibrations.

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arXiv2011

Universal Abelian H-spaces

The question of the existence of Universal homotopy commutative and homotopy associative H-spaces (called Abelian H-spaces) is studied. Such a space T(X) would prolong a map from X into an Abelian H-space to a unique H-map from T into X. Examples of such pairs (X,T) are given and conditions are discussed which limit the possible spaces X for which such a T can exist. The Anick spaces are shown not to be universal Abelian H-spaces for the corresponding Moore spaces, but conditions are discussed which could lead to a universal property with respect to a more limited range of targets.

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arXiv2009

On Generalized Whitehead Products

A symmetric monoidal pairing is defined among simply connected co-H spaces and this is used to generalize the Whitehead product map S(X ^ Y) --> SX v SY to co-H spaces.

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arXiv2009

Filtering the fiber of the pinch map

A new type of Hopf invariant is described for the fiber of the pinch map from the mapping cone of a map from A to X onto to the suspension of A; this is then used to study the boundary map in the fibration sequence of Cohen, Moore and Neisendorfer in the case that the mapping cone is an odd dimensional Moore space. The components of the boundary map are then shown to be compatible with Hopf invariants and a filtered splitting of the loops on the fiber is obtained.

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arXiv2008

Decompositions involving Anick's spaces

Recently Stephen Theriault and I found an elementary construction of Anick's spaces and proved their main properties(arXiv:0710.1024).In this work the fundamental fibration is decomposed. This is useful in studying maps out of Anick's spaces and will be needed in order to determine it's universal properties.

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arXiv2007

An elementary construction of Anick's fibration

Cohen, Moore, and Neisendorfer's work on the odd primary homotopy theory of spheres and Moore spaces, as well as the first author's work on the secondary suspension, predicted the existence of a p-local fibration S^2n-1 --> T --> ΩS^2n+1 whose connecting map is degree p^r. In a long and complex monograph, Anick constructed such a fibration for p>= 5 and r>= 1. Using new methods we give a much more conceptual construction which is also valid for p=3 and r>= 1. We go on to establish several properties of the space T.

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