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57
Periodic complexes and group actions
, 2001
"... In this paper we show that the cohomology of a connected CW–complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on R n × S m; we ..."
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Cited by 32 (0 self)
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In this paper we show that the cohomology of a connected CW–complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on R n × S m; we construct non–standard free actions of rank two simple groups on finite complexes Y ≃ S n × S m; and we prove that a finite p–group P acts freely on such a complex if and only if it does not contain a subgroup isomorphic to (Z/p)³.
Classification of integral lattices with large class number
 Math. Comp
, 1998
"... Abstract. A detailed exposition of Kneser’s neighbour method for quadratic lattices over totally real number fields, and of the subprocedures needed for its implementation, is given. Using an actual computer program which automatically generates representatives for all isomorphism classes in one ge ..."
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Cited by 18 (5 self)
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Abstract. A detailed exposition of Kneser’s neighbour method for quadratic lattices over totally real number fields, and of the subprocedures needed for its implementation, is given. Using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices, various results about genera of ℓelementary lattices, for small prime level ℓ, are obtained. For instance, the class number of 12dimensional 7elementary even lattices of determinant 7 6 is 395; no extremal lattice in the sense of Quebbemann exists. The implementation incorporates as essential parts previous programs of W. Plesken and B. Souvignier. 1.
Anisotropic groups of type A n and the commuting graph of finite simple groups
"... In this paper we make a contribution to the MargulisPlatonov conjecture, which describes the normal subgroup structure of algebraic groups over number fields. We establish the conjecture for inner forms of anisotropic groups of type An. We obtain information on the commuting graph of nonabelian fin ..."
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Cited by 17 (2 self)
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In this paper we make a contribution to the MargulisPlatonov conjecture, which describes the normal subgroup structure of algebraic groups over number fields. We establish the conjecture for inner forms of anisotropic groups of type An. We obtain information on the commuting graph of nonabelian finite simple groups, and consequently, using the paper by Segev, 1999, we obtain results on the normal structure and quotient groups of the multiplicative group of a division algebra. 0. Introduction. Let G be a simple, simply connected algebraic group defined over an algebraic number field K. Let T be the (finite) set of all nonarchimedean places v of K such that G is Kvanisotropic, and define G(K, T) tobe
Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable
 J. Amer. Math. Soc
"... The purpose of this paper is to prove the following. Main Theorem. Let D be a finite dimensional division algebra. Then any finite quotient of the multiplicative group D × is solvable. This result is a culmination of research done in the last several years in order ..."
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Cited by 16 (2 self)
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The purpose of this paper is to prove the following. Main Theorem. Let D be a finite dimensional division algebra. Then any finite quotient of the multiplicative group D × is solvable. This result is a culmination of research done in the last several years in order
Constructing and deconstructing group actions
, 2002
"... Given a finite group G, it is not hard to show that it can act freely on a product of spheres. A more delicate issue is the following question: what is the minimum integer k = k(G) such that G acts freely on a finite complex with the homotopy type of a product of k spheres? The study of this probl ..."
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Cited by 8 (0 self)
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Given a finite group G, it is not hard to show that it can act freely on a product of spheres. A more delicate issue is the following question: what is the minimum integer k = k(G) such that G acts freely on a finite complex with the homotopy type of a product of k spheres? The study of this problem breaks up into two distinct aspects: (1) proving bounds on k(G) in terms of subgroup data for G; and (2) constructing explicit actions on a product of k spheres. In this note we will discuss aspects of both problems, including recent progress based on the periodicity methods developed in [5]. We also describe a potential counterexample to the prevalent expectations as well as recent work on constructing geometric actions on actual products of spheres.
Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every e ..."
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Cited by 7 (3 self)
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
Group actions on central simple algebras
 J. Alg
"... Abstract. Let G be a group, F a field, and A a finitedimensional central simple algebra over F on which G acts by Falgebra automorphisms. We study the ideals and subalgebras of A which are preserved by the group action. Let V be the unique simple module of A. We show that V is a projective represe ..."
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Cited by 7 (4 self)
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Abstract. Let G be a group, F a field, and A a finitedimensional central simple algebra over F on which G acts by Falgebra automorphisms. We study the ideals and subalgebras of A which are preserved by the group action. Let V be the unique simple module of A. We show that V is a projective representation of G and A ∼ = EndD(V) makes V into a projective representation. We then prove that there is a natural onetoone correspondence between Ginvariant Dsubmodules of V and invariant left (and right) ideals of A. Under the assumption that V is irreducible, we show that an invariant (unital) subalgebra must be a simply embedded semisimple subalgebra. We introduce induction of Galgebras. We show that each invariant subalgebras is induced from a simple Halgebra for some subgroup H of finite index and obtain a parametrization of the set of invariant subalgebras in terms of induction data. We then describe invariant central simple subalgebras. For F algebraically closed, we obtain an entirely explicit classification of the invariant subalgebras. Furthermore, we show that the set of invariant subalgebras is finite if G is a finite group. Finally, we consider invariant subalgebras when V is a continuous projective representation of a topological group G. We show that if the connected component of the identity acts irreducibly on V, then all invariant subalgebras are simple. We then apply our results to obtain a particularly nice solution to the classification problem when G is a compact connected Lie group and F = C. 1.
SYMMETRIES AND EXOTIC SMOOTH STRUCTURES ON A K3 Surface
, 2008
"... Smooth and symplectic symmetries of an infinite family of distinct exotic K3 surfaces are studied, and comparison with the corresponding symmetries of the standard K3 is made. The action on the K3 lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, n ..."
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Cited by 7 (1 self)
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Smooth and symplectic symmetries of an infinite family of distinct exotic K3 surfaces are studied, and comparison with the corresponding symmetries of the standard K3 is made. The action on the K3 lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability of actions induced by a holomorphic automorphism of a prime order ≥ 7 is proved and nonexistence of smooth actions by several K3 groups is established (included among which is the binary tetrahedral group T24 which has the smallest order). Concerning symplectic symmetries, the fixedpoint set structure of a symplectic cyclic action of a prime order ≥ 5 is explicitly determined, provided that the action is homologically nontrivial.
The Chow group of the moduli space of marked cubic surfaces
, 2002
"... Naruki gave an explicit construction of the moduli space of marked cubic surfaces, starting from a toric variety and proceeding with blow ups and contractions. Using his result, we compute the Chow groups and the Chern classes of this moduli space. As an application we relate a recent result of Fre ..."
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Cited by 6 (2 self)
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Naruki gave an explicit construction of the moduli space of marked cubic surfaces, starting from a toric variety and proceeding with blow ups and contractions. Using his result, we compute the Chow groups and the Chern classes of this moduli space. As an application we relate a recent result of Freitag on the Hilbert polynomial of a certain ring of modular forms to the RiemannRoch theorem for the moduli space. Dedicated to the memory of our friend Fabio Bardelli Following on the work of Allcock, Carlson and Toledo [ACT], which identified the moduli space of marked cubic surfaces M as a ball quotient, there has been a renewed interest in moduli spaces of cubic surfaces. In particular, Allcock and Freitag [AF] found a projective embedding of M using new results of Borcherds on modular forms. This map was actually described earlier by Coble, who identified M with the moduli space of six points in P 2. We will thus call this map the CAFmap. The moduli space M is smooth except for 40 singular points, the cusps. Blowing up the cusps, one obtains a smooth projective variety C which we refer to as Naruki’s cross ratio variety. Using basic work of Cayley on cubic surfaces and associated projective invariants, certain cross
The cohomology of the McLaughlin group and some associated groups
 Math. Z
, 1997
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