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Papers by William browder
Transactions of the American Mathematical Society, 1999
In this paper we will study the cohomology of a family of p-groups associated to Fp-Lie algebras.... more In this paper we will study the cohomology of a family of p-groups associated to Fp-Lie algebras. More precisely, we study a category BGrp of p-groups which will be equivalent to the category of Fp-bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group G in this category, its Fp-cohomology is that of an elementary abelian p-group if and only if it is associated to a Lie algebra. We then proceed to study the exponent of H * (G; Z) in the case that G is associated to a Lie algebra L. To do this, we use the Bockstein spectral sequence and derive a formula that gives B * 2 in terms of the Lie algebra cohomologies of L. We then expand some of these results to a wider category of p-groups. In particular, we calculate the cohomology of the p-groups Γ n,k which are defined to be the kernel of the mod p reduction GLn(Z/p k+1 Z) mod −→ GLn(Fp).
Illinois Journal of Mathematics, 1960
Illinois Journal of Mathematics, 1963
... in dimension 1, 3, or 7. The remainder of the paper is devoted to the proof of (1.1). 2. The ... more ... in dimension 1, 3, or 7. The remainder of the paper is devoted to the proof of (1.1). 2. The space E1 X Let X be an H-space with multiplication m. Denote by CX the (reduced) cone on X, which we think of as the space obtained from [0, 1] XXby identify-Page 3. 494 WILLIAM ...
Bulletin of the American Mathematical Society, 1959
Proceedings of the Conference on Transformation Groups, 1968
Recent progress in differential topology has developed many powerful techniques for the study of ... more Recent progress in differential topology has developed many powerful techniques for the study of differentiable manifolds, for classification of manifolds, embeddings, diffeomorphisms, etc. These methods have been applied to the study of differentiable transformation groups with some notable successes. I will try to outline in this paper some results of the theory of surgery (or spherical modification) which have given many good results about manifolds, and show how these results may be applied to study actions of groups, in particular actions of S 1 and S 3.
Bulletin of the American Mathematical Society, 1966
Mathematica Scandinavica
ABSTRACT
Contemporary Mathematics, 1985
Bulletin of the American Mathematical Society, 1966
In this note we announce some results extending results of S. P. Novikov ([6] and [7]), the autho... more In this note we announce some results extending results of S. P. Novikov ([6] and [7]), the author [2], and C. T. C. Wall [8]. In the above papers it is shown how to characterize the homotopy type of 1-connected smooth closed manifolds of dimension w^S,«^2 mod 4, and how to reduce the diffeomorphy classification of such manifolds to homotopy theory, with similar results for bounded manifolds in [8], We show how to adapt these techniques to manifolds with TT\-Z and get analogous results. By studying the "mapping torus" using these results one may obtain results on existence and pseudo-isotopy of diffeomorphisms, (see [3]). One has for example the situation of a closed smooth manifold M n and a map ƒ : M n->X, such that the normal bundle v of M in S n+k is induced by ƒ from a bundle £ over X. One does surgery on M with respect to the map ƒ, i.e. if W is the cobordism determined by the surgery, then ƒ extends to a map F: W-+X such that the normal bundle of W in S n+k Xl is induced from £ by F. In case M is simply connected many conditions facilitate the surgery, such as the Whitney embedding theorem, and the Hurewicz theorem, so that, with appropriate hypothesis on X and £, it is often possible to do surgery to create a manifold homotopy equivalent to X. The case of a nonzero fundamental group poses many problems, but if itxM-Z, one can reduce the situation to the simply connected case by using extra geometrical structure. The idea is to consider a 1-connected manifold U n with two 1-connected boundary components, dU-Ao^JAi, with ƒ: A 0-*A\ a diffeomorphism, and consider the identification space M n of U with aÇzAo identified to ƒ (a) £-41. Then M n is closed and connected with TT\M = Z, and it can be shown using surgery that any smooth connected M n with wiM = Z, n^5 can be represented this way. One may then study U and A 0 , A\ using the techniques of surgery on 1-connected manifolds and then use this to obtain information about M. In §1 we deal with closed manifolds and in §2, with manifolds with boundary. In §2 we examine in particular the case of homology circles, which gives certain results on the complements of higher dimensional knots (e.g. Corollaries 2.3 and 2.4).
Bulletin of the American Mathematical Society, 1971
Bulletin of the American Mathematical Society, 1971
Essays on Topology and Related Topics, 1970
An action of S1 on a manifold is semifree, (resp. quasi-free), if the action has just two isotrop... more An action of S1 on a manifold is semifree, (resp. quasi-free), if the action has just two isotropy groups 0 and S1 (resp. 0 and Zn). All actions considered here will be assumed smooth, i.e., S 1 × M→M is a smooth map.
Lecture Notes in Mathematics, 1979
Transactions of the American Mathematical Society, 1963
Differential Hopf algebras arise in several contexts in algebraic topology. The Bockstein spectra... more Differential Hopf algebras arise in several contexts in algebraic topology. The Bockstein spectral sequence of an //-space is one example that has been investigated by many authors [3; 1; 7; 8]. Borel [3] and Araki [1] proved algebraic theorems about the structure of differential Hopf algebras of special kinds. These special theorems enabled them to determine the odd torsion in the cohomology of the exceptional Lie groups. If X and Tare //-spaces and/:X-> Tis a fibre map which is multiplicative, then the spectral sequence of / is a spectral sequence of Hopf algebras. This situation was first discussed by J. C. Moore [17], and later by the author [5; 6]. The techniques of [5] were later extended by the author (in unpublished work) to prove theorems about the homology and cohomology suspensions, i.e., when X is the space of paths of the //-space Y. The proofs rested upon a general theorem about the structure of this spectral sequence. Some of these suspension theorems had been proved by Moore using a different spectral sequence of Hopf algebras [12]. In this paper we make a study of differential Hopf algebras, and prove general theorems on the structure of their homology. These theorems generalize the results of Borel and Araki. Applied to the case of multiplicative fibre maps, we obtain a general theorem about the structure of the spectral sequence (even in the nonacyclic case), which, in particular, yields simple proofs of the suspension theorems mentioned above. Applied to the Bockstein spectral sequence, we get information on torsion in //-spaces. This study of differential Hopf algebras depends on two spectral sequences which may be defined in different circumstances. If one of them is defined, then the terms of that spectral sequence satisfy the conditions necessary for the other to be defined. Thus we get a spectral sequence for the term of the other spectral sequence(2), and this spectral sequence has a very simple form which makes it easy to calculate the form of its homology. Thus the structure of the limit
Topology, 1988
WE STUDY actions of elementary abelian p-groups (i.e. G = fi E/p, p prime), on manifolds or near-... more WE STUDY actions of elementary abelian p-groups (i.e. G = fi E/p, p prime), on manifolds or near-manifolds X (i.e. X is an n-manifold off of a singularity set of dimension <n-2). The usual approach to actions of G on finite dimensional spaces Y is that pioneered by Bore1 [2], and refined by Quillen [l 11 and others, and shows that one can calculate the homology of the fixed set YG from the homology of the Bore1 construction Y x E, (all with [F, G coefficients). One may roughly describe this approach by saying that if one can calculate a great deal about Y x E,, then one may calculate a great deal about YG. In this paper we G take a different point of view which says that if one can calculate a little bit about X x E, G and its relation to X, then one gets a little bit of information about Xc, where X is an nmanifold, or some generalization of manifold. The little bit of information we seek about Y x E, is the fate of the top dimensional cohomology H"(X) in H"(X x EG) (with Z coecfficients, which will tell us if Xc # 0. G For example let G = fi Z/p act, preserving orientation on an m-manifold M, and let j: M+M; E, denote inclusion, and let E=IHm(M)/j*Hm(M;
Tohoku Mathematical Journal, 1973
The Mathematical Intelligencer, 1987
Transactions of the American Mathematical Society, 1999
In this paper we will study the cohomology of a family of p-groups associated to Fp-Lie algebras.... more In this paper we will study the cohomology of a family of p-groups associated to Fp-Lie algebras. More precisely, we study a category BGrp of p-groups which will be equivalent to the category of Fp-bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group G in this category, its Fp-cohomology is that of an elementary abelian p-group if and only if it is associated to a Lie algebra. We then proceed to study the exponent of H * (G; Z) in the case that G is associated to a Lie algebra L. To do this, we use the Bockstein spectral sequence and derive a formula that gives B * 2 in terms of the Lie algebra cohomologies of L. We then expand some of these results to a wider category of p-groups. In particular, we calculate the cohomology of the p-groups Γ n,k which are defined to be the kernel of the mod p reduction GLn(Z/p k+1 Z) mod −→ GLn(Fp).
Illinois Journal of Mathematics, 1960
Illinois Journal of Mathematics, 1963
... in dimension 1, 3, or 7. The remainder of the paper is devoted to the proof of (1.1). 2. The ... more ... in dimension 1, 3, or 7. The remainder of the paper is devoted to the proof of (1.1). 2. The space E1 X Let X be an H-space with multiplication m. Denote by CX the (reduced) cone on X, which we think of as the space obtained from [0, 1] XXby identify-Page 3. 494 WILLIAM ...
Bulletin of the American Mathematical Society, 1959
Proceedings of the Conference on Transformation Groups, 1968
Recent progress in differential topology has developed many powerful techniques for the study of ... more Recent progress in differential topology has developed many powerful techniques for the study of differentiable manifolds, for classification of manifolds, embeddings, diffeomorphisms, etc. These methods have been applied to the study of differentiable transformation groups with some notable successes. I will try to outline in this paper some results of the theory of surgery (or spherical modification) which have given many good results about manifolds, and show how these results may be applied to study actions of groups, in particular actions of S 1 and S 3.
Bulletin of the American Mathematical Society, 1966
Mathematica Scandinavica
ABSTRACT
Contemporary Mathematics, 1985
Bulletin of the American Mathematical Society, 1966
In this note we announce some results extending results of S. P. Novikov ([6] and [7]), the autho... more In this note we announce some results extending results of S. P. Novikov ([6] and [7]), the author [2], and C. T. C. Wall [8]. In the above papers it is shown how to characterize the homotopy type of 1-connected smooth closed manifolds of dimension w^S,«^2 mod 4, and how to reduce the diffeomorphy classification of such manifolds to homotopy theory, with similar results for bounded manifolds in [8], We show how to adapt these techniques to manifolds with TT\-Z and get analogous results. By studying the "mapping torus" using these results one may obtain results on existence and pseudo-isotopy of diffeomorphisms, (see [3]). One has for example the situation of a closed smooth manifold M n and a map ƒ : M n->X, such that the normal bundle v of M in S n+k is induced by ƒ from a bundle £ over X. One does surgery on M with respect to the map ƒ, i.e. if W is the cobordism determined by the surgery, then ƒ extends to a map F: W-+X such that the normal bundle of W in S n+k Xl is induced from £ by F. In case M is simply connected many conditions facilitate the surgery, such as the Whitney embedding theorem, and the Hurewicz theorem, so that, with appropriate hypothesis on X and £, it is often possible to do surgery to create a manifold homotopy equivalent to X. The case of a nonzero fundamental group poses many problems, but if itxM-Z, one can reduce the situation to the simply connected case by using extra geometrical structure. The idea is to consider a 1-connected manifold U n with two 1-connected boundary components, dU-Ao^JAi, with ƒ: A 0-*A\ a diffeomorphism, and consider the identification space M n of U with aÇzAo identified to ƒ (a) £-41. Then M n is closed and connected with TT\M = Z, and it can be shown using surgery that any smooth connected M n with wiM = Z, n^5 can be represented this way. One may then study U and A 0 , A\ using the techniques of surgery on 1-connected manifolds and then use this to obtain information about M. In §1 we deal with closed manifolds and in §2, with manifolds with boundary. In §2 we examine in particular the case of homology circles, which gives certain results on the complements of higher dimensional knots (e.g. Corollaries 2.3 and 2.4).
Bulletin of the American Mathematical Society, 1971
Bulletin of the American Mathematical Society, 1971
Essays on Topology and Related Topics, 1970
An action of S1 on a manifold is semifree, (resp. quasi-free), if the action has just two isotrop... more An action of S1 on a manifold is semifree, (resp. quasi-free), if the action has just two isotropy groups 0 and S1 (resp. 0 and Zn). All actions considered here will be assumed smooth, i.e., S 1 × M→M is a smooth map.
Lecture Notes in Mathematics, 1979
Transactions of the American Mathematical Society, 1963
Differential Hopf algebras arise in several contexts in algebraic topology. The Bockstein spectra... more Differential Hopf algebras arise in several contexts in algebraic topology. The Bockstein spectral sequence of an //-space is one example that has been investigated by many authors [3; 1; 7; 8]. Borel [3] and Araki [1] proved algebraic theorems about the structure of differential Hopf algebras of special kinds. These special theorems enabled them to determine the odd torsion in the cohomology of the exceptional Lie groups. If X and Tare //-spaces and/:X-> Tis a fibre map which is multiplicative, then the spectral sequence of / is a spectral sequence of Hopf algebras. This situation was first discussed by J. C. Moore [17], and later by the author [5; 6]. The techniques of [5] were later extended by the author (in unpublished work) to prove theorems about the homology and cohomology suspensions, i.e., when X is the space of paths of the //-space Y. The proofs rested upon a general theorem about the structure of this spectral sequence. Some of these suspension theorems had been proved by Moore using a different spectral sequence of Hopf algebras [12]. In this paper we make a study of differential Hopf algebras, and prove general theorems on the structure of their homology. These theorems generalize the results of Borel and Araki. Applied to the case of multiplicative fibre maps, we obtain a general theorem about the structure of the spectral sequence (even in the nonacyclic case), which, in particular, yields simple proofs of the suspension theorems mentioned above. Applied to the Bockstein spectral sequence, we get information on torsion in //-spaces. This study of differential Hopf algebras depends on two spectral sequences which may be defined in different circumstances. If one of them is defined, then the terms of that spectral sequence satisfy the conditions necessary for the other to be defined. Thus we get a spectral sequence for the term of the other spectral sequence(2), and this spectral sequence has a very simple form which makes it easy to calculate the form of its homology. Thus the structure of the limit
Topology, 1988
WE STUDY actions of elementary abelian p-groups (i.e. G = fi E/p, p prime), on manifolds or near-... more WE STUDY actions of elementary abelian p-groups (i.e. G = fi E/p, p prime), on manifolds or near-manifolds X (i.e. X is an n-manifold off of a singularity set of dimension <n-2). The usual approach to actions of G on finite dimensional spaces Y is that pioneered by Bore1 [2], and refined by Quillen [l 11 and others, and shows that one can calculate the homology of the fixed set YG from the homology of the Bore1 construction Y x E, (all with [F, G coefficients). One may roughly describe this approach by saying that if one can calculate a great deal about Y x E,, then one may calculate a great deal about YG. In this paper we G take a different point of view which says that if one can calculate a little bit about X x E, G and its relation to X, then one gets a little bit of information about Xc, where X is an nmanifold, or some generalization of manifold. The little bit of information we seek about Y x E, is the fate of the top dimensional cohomology H"(X) in H"(X x EG) (with Z coecfficients, which will tell us if Xc # 0. G For example let G = fi Z/p act, preserving orientation on an m-manifold M, and let j: M+M; E, denote inclusion, and let E=IHm(M)/j*Hm(M;
Tohoku Mathematical Journal, 1973
The Mathematical Intelligencer, 1987