Martin Crossley | Swansea University (original) (raw)

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Papers by Martin Crossley

Proceedings of the Edinburgh Mathematical Society, 2001

Let A be a graded, commutative Hopf algebra. We study an action of the symmetric group Σ n on the... more Let A be a graded, commutative Hopf algebra. We study an action of the symmetric group Σ n on the tensor product of n − 1 copies of A; this action was introduced by the second author in [8] and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory [6]. We show that for a certain class of Hopf algebras the cohomology ring H * (Σ n ; A ⊗n−1) is independent of the coproduct provided n and (n − 2)! are invertible in the ground ring. Then, by choosing a sufficiently simple coproduct, we are able to deduce significant information about the Σ n invariants of A ⊗n−1 , including dimensions and algebra structure.

K-Theory, 2001

Gaussian polynomials are used to define bases with good multiplicative properties for the algebra... more Gaussian polynomials are used to define bases with good multiplicative properties for the algebra K * (K) of cooperations in K-theory and for the invariants under conjugation.

Bulletin of the London Mathematical Society, 2000

We say that a Hopf algebra is copolynomial if its dual is polynomial as an algebra. We re‐derive ... more We say that a Hopf algebra is copolynomial if its dual is polynomial as an algebra. We re‐derive Milnor's result that the mod 2 Steenrod algebra is copolynomial by means of a more general result that is also applicable to a number of other related Hopf algebras. 1991 Mathematics Subject Classification 55S10, 16W30.

Communications in Algebra, 2013

We investigate the canonical conjugation, χ, of the mod 2 dual Steenrod algebra, A * , with a vie... more We investigate the canonical conjugation, χ, of the mod 2 dual Steenrod algebra, A * , with a view to determining the subspace, A χ * , of elements invariant under χ. We give bounds on the dimension of this subspace for each degree and show that, after inverting ξ 1 , it becomes polynomial on a natural set of generators. Finally we note that, without inverting ξ 1 , A χ * is far from being polynomial.

Eprint Arxiv Math 0401414, 2004

Journal of Pure and Applied Algebra, Dec 1, 2013

Springer Undergraduate Mathematics Series, 2005

Glasgow Mathematical Journal, 2013

Malvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functio... more Malvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra177 (1995), 967–982) showed how the total symmetric group ring ⊕nZΣn could be made into a Hopf algebra with a very nice structure which admitted the Solomon descent algebra as a sub-Hopf algebra. To do this they replaced the group multiplication by a convolution product, thus distancing their structure from the group structure of Σn. In this paper we examine what is possible if we keep to the group multiplication, and we also consider the question for more general families of groups. We show that a Hopf algebra structure is not possible, but cocommutative and non-cocommutative counital bialgebras can be obtained, arising from certain diagrams of group homomorphisms. In the case of the symmetric groups we note that all such structures are weak in the sense that the dual algebras have many zero-divisors, but structures which respect descent sum...

Glasgow Mathematical Journal, 2013

Malvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functio... more Malvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra177 (1995), 967–982) showed how the total symmetric group ring ⊕nZΣn could be made into a Hopf algebra with a very nice structure which admitted the Solomon descent algebra as a sub-Hopf algebra. To do this they replaced the group multiplication by a convolution product, thus distancing their structure from the group structure of Σn. In this paper we examine what is possible if we keep to the group multiplication, and we also consider the question for more general families of groups. We show that a Hopf algebra structure is not possible, but cocommutative and non-cocommutative counital bialgebras can be obtained, arising from certain diagrams of group homomorphisms. In the case of the symmetric groups we note that all such structures are weak in the sense that the dual algebras have many zero-divisors, but structures which respect descent sum...

Arxiv preprint arXiv:1112.4961, Jan 1, 2011

Characteristic classes of fibre bundles E d+n → B n in the category of closed oriented manifolds ... more Characteristic classes of fibre bundles E d+n → B n in the category of closed oriented manifolds give rise to characteristic numbers by integrating the classes over the base. Church, Farb and Thibault [CFT] raised the question of which generalised Miller-Morita-Mumford classes have the property that the associated characteristic number is independent of the fibering and depends only on the cobordism class of the total space E. Here we determine a complete answer to this question in both the oriented category and the stably almost complex category. An MMM class has this property if and only if it is a fibre integral of a vector bundle characteristic class that satisfies a certain approximate version of the additivity of the Chern character.

Essential Topology, Jan 1, 2005

Essential Topology, Jan 1, 2005

Essential Topology, Jan 1, 2005

Essential Topology, Jan 1, 2005

Essential Topology, Jan 1, 2005

Essential Topology, Jan 1, 2005

Proceedings of the Edinburgh Mathematical Society, 2001

Let A be a graded, commutative Hopf algebra. We study an action of the symmetric group Σ n on the... more Let A be a graded, commutative Hopf algebra. We study an action of the symmetric group Σ n on the tensor product of n − 1 copies of A; this action was introduced by the second author in [8] and is relevant to the study of commutativity conditions on ring spectra in stable homotopy theory [6]. We show that for a certain class of Hopf algebras the cohomology ring H * (Σ n ; A ⊗n−1) is independent of the coproduct provided n and (n − 2)! are invertible in the ground ring. Then, by choosing a sufficiently simple coproduct, we are able to deduce significant information about the Σ n invariants of A ⊗n−1 , including dimensions and algebra structure.

K-Theory, 2001

Gaussian polynomials are used to define bases with good multiplicative properties for the algebra... more Gaussian polynomials are used to define bases with good multiplicative properties for the algebra K * (K) of cooperations in K-theory and for the invariants under conjugation.

Bulletin of the London Mathematical Society, 2000

We say that a Hopf algebra is copolynomial if its dual is polynomial as an algebra. We re‐derive ... more We say that a Hopf algebra is copolynomial if its dual is polynomial as an algebra. We re‐derive Milnor's result that the mod 2 Steenrod algebra is copolynomial by means of a more general result that is also applicable to a number of other related Hopf algebras. 1991 Mathematics Subject Classification 55S10, 16W30.

Communications in Algebra, 2013

We investigate the canonical conjugation, χ, of the mod 2 dual Steenrod algebra, A * , with a vie... more We investigate the canonical conjugation, χ, of the mod 2 dual Steenrod algebra, A * , with a view to determining the subspace, A χ * , of elements invariant under χ. We give bounds on the dimension of this subspace for each degree and show that, after inverting ξ 1 , it becomes polynomial on a natural set of generators. Finally we note that, without inverting ξ 1 , A χ * is far from being polynomial.

Eprint Arxiv Math 0401414, 2004

Journal of Pure and Applied Algebra, Dec 1, 2013

Springer Undergraduate Mathematics Series, 2005

Glasgow Mathematical Journal, 2013

Malvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functio... more Malvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra177 (1995), 967–982) showed how the total symmetric group ring ⊕nZΣn could be made into a Hopf algebra with a very nice structure which admitted the Solomon descent algebra as a sub-Hopf algebra. To do this they replaced the group multiplication by a convolution product, thus distancing their structure from the group structure of Σn. In this paper we examine what is possible if we keep to the group multiplication, and we also consider the question for more general families of groups. We show that a Hopf algebra structure is not possible, but cocommutative and non-cocommutative counital bialgebras can be obtained, arising from certain diagrams of group homomorphisms. In the case of the symmetric groups we note that all such structures are weak in the sense that the dual algebras have many zero-divisors, but structures which respect descent sum...

Glasgow Mathematical Journal, 2013

Malvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functio... more Malvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra177 (1995), 967–982) showed how the total symmetric group ring ⊕nZΣn could be made into a Hopf algebra with a very nice structure which admitted the Solomon descent algebra as a sub-Hopf algebra. To do this they replaced the group multiplication by a convolution product, thus distancing their structure from the group structure of Σn. In this paper we examine what is possible if we keep to the group multiplication, and we also consider the question for more general families of groups. We show that a Hopf algebra structure is not possible, but cocommutative and non-cocommutative counital bialgebras can be obtained, arising from certain diagrams of group homomorphisms. In the case of the symmetric groups we note that all such structures are weak in the sense that the dual algebras have many zero-divisors, but structures which respect descent sum...

Arxiv preprint arXiv:1112.4961, Jan 1, 2011

Characteristic classes of fibre bundles E d+n → B n in the category of closed oriented manifolds ... more Characteristic classes of fibre bundles E d+n → B n in the category of closed oriented manifolds give rise to characteristic numbers by integrating the classes over the base. Church, Farb and Thibault [CFT] raised the question of which generalised Miller-Morita-Mumford classes have the property that the associated characteristic number is independent of the fibering and depends only on the cobordism class of the total space E. Here we determine a complete answer to this question in both the oriented category and the stably almost complex category. An MMM class has this property if and only if it is a fibre integral of a vector bundle characteristic class that satisfies a certain approximate version of the additivity of the Chern character.

Essential Topology, Jan 1, 2005

Essential Topology, Jan 1, 2005

Essential Topology, Jan 1, 2005

Essential Topology, Jan 1, 2005

Essential Topology, Jan 1, 2005

Essential Topology, Jan 1, 2005