WILLIAM Singer [Staff/Faculty [FCRH]] | Fordham University (original) (raw)
Papers by WILLIAM Singer [Staff/Faculty [FCRH]]
Bulletin of the American Mathematical Society, Sep 1, 1970
Communicated by Michael Artin, April 3, 1970 1. Introduction. Let i£ be a fixed commutative ring ... more Communicated by Michael Artin, April 3, 1970 1. Introduction. Let i£ be a fixed commutative ring with unit. We will deal with graded algebras, coalgebras, and Hopf algebras over K as defined in Milnor-Moore [4], but we assume that the underlying X-modules are connected. Suppose Ay B are Hopf algebras; A commutative and B cocommutative. By an extension of B by A we mean a diagram of Hopf algebras and Hopf maps a 13 (1.1) A-+C-+B in which C is isomorphic to A B simultaneously as a left A-module and right £-comodule. In this paper we announce results which describe and classify all extensions by B by A. Proofs will appear in [S]. 2. Matched pairs. If B is an algebra we write r):K-+B, ti B :B®B-*B for the unit and multiplication, respectively. If A is a coalgebra we write elA->K, ypA"*A-> A ® A for the counit and comultiplication. As the first step in classifying extensions, we will show in [5] how a diagram (1.1) determines a pair of X-linear maps CTA'B ® A-» A y p B :B-*A. CA is the "action" of base on fiber that one expects in an extension problem ; pB is its dual. We prove: (a) a A gives A the structure of a left 5-module algebra; (b) PB gives B the structure of a right 4-comodule coalgebra; (c) the diagram commutes:
Journal of Algebra, Feb 1, 1996
Journal of Pure and Applied Algebra, 1980
Mathematische Zeitschrift, Dec 1, 1989
... William M. Singer* Department of Mathematics, Fordham University, Bronx, NY 10458, USA ... th... more ... William M. Singer* Department of Mathematics, Fordham University, Bronx, NY 10458, USA ... the graded s-dimensional homology of the Steenrod algebra A with coeffi-cients in M. Let P~ = F2 [ta, ..., tj be the graded polynomial algebra on s genera-tors ti, each of dimension one. ...
Transactions of the American Mathematical Society, 1973
We apply the results of the previous paper to three special cases. We obtain Steenrod operations ... more We apply the results of the previous paper to three special cases. We obtain Steenrod operations on the change-of-rings spectral sequence, on the Eilenberg-Moore spectral sequence for the cohomology of classifying spaces, and on the Serre spectral sequence. Suppose given a commutative left C-coalgebra M and a commutative left C-algebra N. First we show, by analogy with the theory of groups, that the action of B on Ext (M, N) can be described directly in terms of an action of B upon A. In this we rely heavily on [13j, [14], where it is demonstrated that an extension of Hopf algebras determines an action of base upon fiber by "conjugation". This result permits us to set up the change of rings spectral sequence converging to ExtAM, N) by analogy with Mac Lane's construction of the Lyndon spectral sequence [8, Chapter Xl]. We then consider the Steenrod operations on E.:
Journal of Pure and Applied Algebra, 1980
Mathematical proceedings of the Cambridge Philosophical Society, Sep 1, 1981
Transactions of the American Mathematical Society, 1973
We define two kinds of Steenrod operations on the spectral sequence of a bisimplicial coalgebra. ... more We define two kinds of Steenrod operations on the spectral sequence of a bisimplicial coalgebra. We show these operations compatible with the differentials of the spectral sequence, and with the Steenrod squares defined on the cohomology of the total complex. We give a general rule for computing the operations on E-. who consider the related problem of defining Steentod squares on the spectral sequence of a "mixed" bisimplicial object. .. , i.e., a functor from Ox G that is contravariant in one vatiable and covariant in the other. Rector and Smith obtain only operations of type (0.1). Whethet opetations of type (0.2) can be defined for mixed bisimplicial objects remain an open problem.
Transactions of the American Mathematical Society, Dec 1, 1983
Let A be the Steenrod algebra over the field F2. In this paper we construct for any left /(-modul... more Let A be the Steenrod algebra over the field F2. In this paper we construct for any left /(-module M a chain complex whose homology groups are isomorphic to the groups Tor;4 (F2, M). This chain complex in homological degree s is built from a ring of invariants associated with the action of the linear group GLS(F2) on a certain algebra of Laurent series. Thus, the homology of the Steenrod algebra (and so the Adams spectral sequence for spheres) is seen to have a close relationship to invariant theory. A key observation in our work is that the Adem relations can be described in terms of the invariant theory of GL2(F2). Our chain complex is not new: it turns out to be isomorphic to the one constructed by Kan and his coworkers from the dual of the lambda algebra. Thus, one effect of our work is to give an invariant-theoretic interpretation of the lambda algebra. As a consequence we find that the dual of lambda supports an action of the Steenrod algebra that commutes with the differential. The differential itself appears as a kind of "residue map". We are also able to describe the coalgebra structure of the dual of lambda using our invariant-theoretic language.
Bulletin of The London Mathematical Society, Aug 1, 2005
In his thesis (Mem. Amer. Math. Soc. 42 (1962)) A. Liulevicius defined Steenrod squaring operatio... more In his thesis (Mem. Amer. Math. Soc. 42 (1962)) A. Liulevicius defined Steenrod squaring operations Sqk on the cohomology ring of any cocommutative Hopf algebra over Z/2. Later, J. P. May showed that these operations satisfy Adem relations, interpreted so that Sq0 is not the unit but an independent operation. Thus, these Adem relations are homogeneous of length two in the generators. This paper is concerned with the bigraded algebra B that is generated by elements Sqk and subject to Adem relations; it shows that the Cartan formula gives a well‐defined coproduct on B. Also, it is shown that B with both multiplication and comultiplication should be considered neither a Hopf algebra nor a bialgebra, but another kind of structure, for which the name ‘algebra with coproducts’ is proposed in the paper. 2000 Mathematics Subject Classification 55S10 (primary), 55T15 (secondary).
Proceedings of the American Mathematical Society, Feb 1, 1991
Let Ps be the mod-2 cohomology of the elementary abelian group (Z/2Z) x • ■ • x (Z/2Z) (s factors... more Let Ps be the mod-2 cohomology of the elementary abelian group (Z/2Z) x • ■ • x (Z/2Z) (s factors). The mod-2 Steenrod algebra A acts on Ps according to well-known rules. If A C A denotes the augmentation ideal, then we are interested in determining the image of the action A ® Ps-* Ps: the space of elements in Ps that are hit by positive dimensional Steenrod squares. The problem is motivated by applications to cobordism theory [PI] and the homology of the Steenrod algebra [S]. Our main result, which generalizes work of Wood [W], identifies a new class of hit monomials. Theorem 1.1 (R. Wood, [W]). Suppose x e Ps is a monomial of degree ô, and suppose a[ô + s] > s. Then x is hit.
Transactions of the American Mathematical Society, 1975
Let A be the dual of the mod-2 Steenrod algebra. If M, N, are graded unstable .4-comodules, one c... more Let A be the dual of the mod-2 Steenrod algebra. If M, N, are graded unstable .4-comodules, one can define and compute the derived functors Coext^M, N) using unstable injective resolutions of N. Bousfield and Curtis have shown that these unstable Coext groups can be fit into a long exact "EHP sequence", an algebraic analogue of the EHP sequence of homotopy theory. Our object in the present paper is to study the relationship between the E, H, and P homomorphisms and the composition pairing Coext AN, R)® Coext¿(Af, N)-► Coext^íAÍ, R). Among our results is a formula that measures the failure of the composition product to commute. 0. Introduction. Let A be the dual of the mod-2 Steenrod algebra. If M, N are graded unstable .4-comodules (definitions below) once can define and compute the derived functors Coext^ (M, N) using unstable injective resolutions of N. These derived functors were introduced by Massey and Peterson in [9], where they were used to express the £"2-term of an unstable Adams spectral sequence. Subsequently, Bousfield and Curtis ([3], [5]) showed these unstable Coext groups can be fit into a long exact "EHP sequence", an algebraic analogue of the EHP sequence of homotopy theory. Our object in the present paper is to study the relations between the E, H, and P homomorphisms and the composition pairing Coext^A, R) ® CoexiA(M, N)-► QoextA(M, R). Our first result in this direction is, roughly, that the order in which two elements are composed does not matter if they are suspended sufficiently often. More precisely, let Sp ("p-sphere") be the unique unstable ,4-comodule for which iSp)p = Z2> iSp)n = 0 in ±p). If a E CoextA(Sp+k, Sp),ßECoextA(S"+l, S*), then (0.1) Eqa ■ Ep+kß = Epß ■ E"+la.
Algebraic & Geometric Topology, May 12, 2008
We write P˝s for the polynomial ring on s letters over the field Z=2, equipped with the standard ... more We write P˝s for the polynomial ring on s letters over the field Z=2, equipped with the standard action of † s , the symmetric group on s letters. This paper deals with the problem of determining a minimal set of generators for the invariant ring .P˝s/ † s as a module over the Steenrod algebra A. That is, we would like to determine the graded vector spaces Z=2˝A .P˝s/ † s. Our main result is stated in terms of a "bigraded Steenrod algebra" H. The generators of this algebra H, like the generators of the classical Steenrod algebra A, satisfy the Adem relations in their usual form. However, the Adem relations for the bigraded Steenrod algebra are interpreted so that Sq 0 is not the unit of the algebra; but rather, an independent generator. Our main work is to assemble the duals of the vector spaces Z=2˝A .P˝s/ † s , for all s 0, into a single bigraded vector space and to show that this bigraded object has the structure of an algebra over H.
Mathematical surveys, Aug 11, 2006
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting fo... more Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.
Journal of Pure and Applied Algebra, Dec 1, 1977
Journal of Pure and Applied Algebra, Apr 1, 1975
Journal of Pure and Applied Algebra, Dec 1, 1977
Mathematical Surveys and Monographs, 2006
arXiv (Cornell University), May 5, 2011
We examine the dual of the so-called "hit problem", the latter being the problem of determining a... more We examine the dual of the so-called "hit problem", the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as module over the Steenrod Algebra A at the prime 2. The dual problem is to determine the set of A-annihilated elements in homology. The set of A-annihilateds has been shown by David Anick to be a free associative algebra. In this note we prove that, for each k ≥ 0, the set of k partially A-annihilateds, the set of elements that are annihilated by Sq i for each i ≤ 2 k , itself forms a free associative algebra.
Mathematical Surveys and Monographs, 2006
Bulletin of the American Mathematical Society, Sep 1, 1970
Communicated by Michael Artin, April 3, 1970 1. Introduction. Let i£ be a fixed commutative ring ... more Communicated by Michael Artin, April 3, 1970 1. Introduction. Let i£ be a fixed commutative ring with unit. We will deal with graded algebras, coalgebras, and Hopf algebras over K as defined in Milnor-Moore [4], but we assume that the underlying X-modules are connected. Suppose Ay B are Hopf algebras; A commutative and B cocommutative. By an extension of B by A we mean a diagram of Hopf algebras and Hopf maps a 13 (1.1) A-+C-+B in which C is isomorphic to A B simultaneously as a left A-module and right £-comodule. In this paper we announce results which describe and classify all extensions by B by A. Proofs will appear in [S]. 2. Matched pairs. If B is an algebra we write r):K-+B, ti B :B®B-*B for the unit and multiplication, respectively. If A is a coalgebra we write elA->K, ypA"*A-> A ® A for the counit and comultiplication. As the first step in classifying extensions, we will show in [5] how a diagram (1.1) determines a pair of X-linear maps CTA'B ® A-» A y p B :B-*A. CA is the "action" of base on fiber that one expects in an extension problem ; pB is its dual. We prove: (a) a A gives A the structure of a left 5-module algebra; (b) PB gives B the structure of a right 4-comodule coalgebra; (c) the diagram commutes:
Journal of Algebra, Feb 1, 1996
Journal of Pure and Applied Algebra, 1980
Mathematische Zeitschrift, Dec 1, 1989
... William M. Singer* Department of Mathematics, Fordham University, Bronx, NY 10458, USA ... th... more ... William M. Singer* Department of Mathematics, Fordham University, Bronx, NY 10458, USA ... the graded s-dimensional homology of the Steenrod algebra A with coeffi-cients in M. Let P~ = F2 [ta, ..., tj be the graded polynomial algebra on s genera-tors ti, each of dimension one. ...
Transactions of the American Mathematical Society, 1973
We apply the results of the previous paper to three special cases. We obtain Steenrod operations ... more We apply the results of the previous paper to three special cases. We obtain Steenrod operations on the change-of-rings spectral sequence, on the Eilenberg-Moore spectral sequence for the cohomology of classifying spaces, and on the Serre spectral sequence. Suppose given a commutative left C-coalgebra M and a commutative left C-algebra N. First we show, by analogy with the theory of groups, that the action of B on Ext (M, N) can be described directly in terms of an action of B upon A. In this we rely heavily on [13j, [14], where it is demonstrated that an extension of Hopf algebras determines an action of base upon fiber by "conjugation". This result permits us to set up the change of rings spectral sequence converging to ExtAM, N) by analogy with Mac Lane's construction of the Lyndon spectral sequence [8, Chapter Xl]. We then consider the Steenrod operations on E.:
Journal of Pure and Applied Algebra, 1980
Mathematical proceedings of the Cambridge Philosophical Society, Sep 1, 1981
Transactions of the American Mathematical Society, 1973
We define two kinds of Steenrod operations on the spectral sequence of a bisimplicial coalgebra. ... more We define two kinds of Steenrod operations on the spectral sequence of a bisimplicial coalgebra. We show these operations compatible with the differentials of the spectral sequence, and with the Steenrod squares defined on the cohomology of the total complex. We give a general rule for computing the operations on E-. who consider the related problem of defining Steentod squares on the spectral sequence of a "mixed" bisimplicial object. .. , i.e., a functor from Ox G that is contravariant in one vatiable and covariant in the other. Rector and Smith obtain only operations of type (0.1). Whethet opetations of type (0.2) can be defined for mixed bisimplicial objects remain an open problem.
Transactions of the American Mathematical Society, Dec 1, 1983
Let A be the Steenrod algebra over the field F2. In this paper we construct for any left /(-modul... more Let A be the Steenrod algebra over the field F2. In this paper we construct for any left /(-module M a chain complex whose homology groups are isomorphic to the groups Tor;4 (F2, M). This chain complex in homological degree s is built from a ring of invariants associated with the action of the linear group GLS(F2) on a certain algebra of Laurent series. Thus, the homology of the Steenrod algebra (and so the Adams spectral sequence for spheres) is seen to have a close relationship to invariant theory. A key observation in our work is that the Adem relations can be described in terms of the invariant theory of GL2(F2). Our chain complex is not new: it turns out to be isomorphic to the one constructed by Kan and his coworkers from the dual of the lambda algebra. Thus, one effect of our work is to give an invariant-theoretic interpretation of the lambda algebra. As a consequence we find that the dual of lambda supports an action of the Steenrod algebra that commutes with the differential. The differential itself appears as a kind of "residue map". We are also able to describe the coalgebra structure of the dual of lambda using our invariant-theoretic language.
Bulletin of The London Mathematical Society, Aug 1, 2005
In his thesis (Mem. Amer. Math. Soc. 42 (1962)) A. Liulevicius defined Steenrod squaring operatio... more In his thesis (Mem. Amer. Math. Soc. 42 (1962)) A. Liulevicius defined Steenrod squaring operations Sqk on the cohomology ring of any cocommutative Hopf algebra over Z/2. Later, J. P. May showed that these operations satisfy Adem relations, interpreted so that Sq0 is not the unit but an independent operation. Thus, these Adem relations are homogeneous of length two in the generators. This paper is concerned with the bigraded algebra B that is generated by elements Sqk and subject to Adem relations; it shows that the Cartan formula gives a well‐defined coproduct on B. Also, it is shown that B with both multiplication and comultiplication should be considered neither a Hopf algebra nor a bialgebra, but another kind of structure, for which the name ‘algebra with coproducts’ is proposed in the paper. 2000 Mathematics Subject Classification 55S10 (primary), 55T15 (secondary).
Proceedings of the American Mathematical Society, Feb 1, 1991
Let Ps be the mod-2 cohomology of the elementary abelian group (Z/2Z) x • ■ • x (Z/2Z) (s factors... more Let Ps be the mod-2 cohomology of the elementary abelian group (Z/2Z) x • ■ • x (Z/2Z) (s factors). The mod-2 Steenrod algebra A acts on Ps according to well-known rules. If A C A denotes the augmentation ideal, then we are interested in determining the image of the action A ® Ps-* Ps: the space of elements in Ps that are hit by positive dimensional Steenrod squares. The problem is motivated by applications to cobordism theory [PI] and the homology of the Steenrod algebra [S]. Our main result, which generalizes work of Wood [W], identifies a new class of hit monomials. Theorem 1.1 (R. Wood, [W]). Suppose x e Ps is a monomial of degree ô, and suppose a[ô + s] > s. Then x is hit.
Transactions of the American Mathematical Society, 1975
Let A be the dual of the mod-2 Steenrod algebra. If M, N, are graded unstable .4-comodules, one c... more Let A be the dual of the mod-2 Steenrod algebra. If M, N, are graded unstable .4-comodules, one can define and compute the derived functors Coext^M, N) using unstable injective resolutions of N. Bousfield and Curtis have shown that these unstable Coext groups can be fit into a long exact "EHP sequence", an algebraic analogue of the EHP sequence of homotopy theory. Our object in the present paper is to study the relationship between the E, H, and P homomorphisms and the composition pairing Coext AN, R)® Coext¿(Af, N)-► Coext^íAÍ, R). Among our results is a formula that measures the failure of the composition product to commute. 0. Introduction. Let A be the dual of the mod-2 Steenrod algebra. If M, N are graded unstable .4-comodules (definitions below) once can define and compute the derived functors Coext^ (M, N) using unstable injective resolutions of N. These derived functors were introduced by Massey and Peterson in [9], where they were used to express the £"2-term of an unstable Adams spectral sequence. Subsequently, Bousfield and Curtis ([3], [5]) showed these unstable Coext groups can be fit into a long exact "EHP sequence", an algebraic analogue of the EHP sequence of homotopy theory. Our object in the present paper is to study the relations between the E, H, and P homomorphisms and the composition pairing Coext^A, R) ® CoexiA(M, N)-► QoextA(M, R). Our first result in this direction is, roughly, that the order in which two elements are composed does not matter if they are suspended sufficiently often. More precisely, let Sp ("p-sphere") be the unique unstable ,4-comodule for which iSp)p = Z2> iSp)n = 0 in ±p). If a E CoextA(Sp+k, Sp),ßECoextA(S"+l, S*), then (0.1) Eqa ■ Ep+kß = Epß ■ E"+la.
Algebraic & Geometric Topology, May 12, 2008
We write P˝s for the polynomial ring on s letters over the field Z=2, equipped with the standard ... more We write P˝s for the polynomial ring on s letters over the field Z=2, equipped with the standard action of † s , the symmetric group on s letters. This paper deals with the problem of determining a minimal set of generators for the invariant ring .P˝s/ † s as a module over the Steenrod algebra A. That is, we would like to determine the graded vector spaces Z=2˝A .P˝s/ † s. Our main result is stated in terms of a "bigraded Steenrod algebra" H. The generators of this algebra H, like the generators of the classical Steenrod algebra A, satisfy the Adem relations in their usual form. However, the Adem relations for the bigraded Steenrod algebra are interpreted so that Sq 0 is not the unit of the algebra; but rather, an independent generator. Our main work is to assemble the duals of the vector spaces Z=2˝A .P˝s/ † s , for all s 0, into a single bigraded vector space and to show that this bigraded object has the structure of an algebra over H.
Mathematical surveys, Aug 11, 2006
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting fo... more Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.
Journal of Pure and Applied Algebra, Dec 1, 1977
Journal of Pure and Applied Algebra, Apr 1, 1975
Journal of Pure and Applied Algebra, Dec 1, 1977
Mathematical Surveys and Monographs, 2006
arXiv (Cornell University), May 5, 2011
We examine the dual of the so-called "hit problem", the latter being the problem of determining a... more We examine the dual of the so-called "hit problem", the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as module over the Steenrod Algebra A at the prime 2. The dual problem is to determine the set of A-annihilated elements in homology. The set of A-annihilateds has been shown by David Anick to be a free associative algebra. In this note we prove that, for each k ≥ 0, the set of k partially A-annihilateds, the set of elements that are annihilated by Sq i for each i ≤ 2 k , itself forms a free associative algebra.
Mathematical Surveys and Monographs, 2006