Reinhard Schultz | University of California, Riverside (original) (raw)
Papers by Reinhard Schultz
Lecture Notes in Mathematics, 1984
American Journal of Mathematics, 1974
Inventiones Mathematicae, 1969
The general problem of classifying up to orientation-preserving diffeomorphism those smooth manif... more The general problem of classifying up to orientation-preserving diffeomorphism those smooth manifolds homeomorphic to a given manifold is probably too complicated to be treated in any effective uniform manner. The first case to be treated was of course the sphere, where everything has been established modulo computation of the Adams spectral sequence. The next case considered was a product of two spheres, where the author and DeSapio independently reduced the classification to homotopy theory with the exception of determining the action of ~P4k + 2 on some (4 k + 1)-dimensional products. It is completely straightforward to extend this classification to the k-sphere bundles associated to (k+l)-plane bundles over S" which have nowhere zero cross sections and satisfy k < n; this was done in the author's thesis and in [1]. The latter paper also announced results in the case n<k<_n+2. We shall more generally give results in the case k > n which apparently diverge from [1] in some respects. We make two remarks for completeness. A classification of smoothings of bundles with k < n (not necessarily having cross sections) may be derived from [3, w 5]. Finally, throughout this paper the phrases "combinatorially equivalent" and "homeomorphic" are interchangeable by the Hauptvermutung for simply connected closed manifolds with torsion free homology and dimension at least 5.
Lecture Notes in Mathematics, 1990
Inventiones Mathematicae, 1989
In geometric topology it is important to have useful criteria for recognizing special types of ma... more In geometric topology it is important to have useful criteria for recognizing special types of manifolds. For example, it is often necessary to determine whether a connected compact manifold with boundary W can be expressed as a product V• [0, 1]. If W is two-dimensional, then the classification of surfaces shows that some obvious necessary conditions are sufficient. Specifically, the boundary must split into two components, and the inclusion of either must be a homotopy equivalence. A manifold that satisfies these conditions is called an h-cobordism, if the inclusions are simple homotopy equivalences, the manifold is called an s-cobordism. Every product Vx [0, 1] is an s-cobordism, and the s-cobordism theorem of D. Barden, B. Mazur and J. Stallings shows that the converse is also true in all dimensions greater than or equal to six. This result holds in each of standard categories of manifolds. In other words, if the manifold W is taken to be smooth or PL, then the isomorphism with Wo x [0, 1] may also be taken to be smooth or PL. For many years it has been known that the s-cobordism theorem extends to certain specific situations in dimension 3, 4, and 5 (compare Lawson [27], Shaneson [48]) but it was also known that the s-cobordism theorem does not generalize completely in at least one of these dimensions (Siebenmann [49], Lawson [27]). In fact, it was known that failures occur in each of the three manifold categories (topological, PL, and smooth manifolds). Recent advances in low-dimensional topology have improved our understanding of the extent to which the s-cobordism theorem can and cannot be generalized. The results of M. Freedman yield a topological s-cobordism theorem in dimension five provided the fundamental group of W is relatively small [16, 39]. On the other hand, Freedman's results and earlier work of T. Matumoto and L. Siebenmann [32] (cf. [23]) show that the topological s-cobordism theorem fails in dimension four; one can even insist that both boundary components are homeomorphic (specifically to S 1 x RP2).
Osaka Journal of Mathematics, 1994
The Rochlin invariant for 3-manifolds is extended to higher dimensions using a result of Ochanine... more The Rochlin invariant for 3-manifolds is extended to higher dimensions using a result of Ochanine. Given a semifree differentiable S-action on a closed mod 2 homology sphere M with fixed point set F , generalized Rochlin invariants are definable for both M and F , and one result of this paper states that these two invariants are equal. This yields restrictions on the types of semifree differentiable S-actions that some homology spheres can support and the fixed point sets of actions on homology (8k + 7)-spheres. An action of a group G on a space X is said to be semifree if for each x ∈ X either X is fixed under every element of G or else x is not fixed by any element of G except the identity. During the nineteen sixties and seventies it became apparent that the techniques of differential topology had numerous applications to differentiable actions of compact Lie groups (cf. [Bro2], [BP], [RS]). In particular, these and previously developed techniques yielded considerable information...
Proceedings of the Conference at the University of Newcastle upon Tyne, August 1976
Contemporary Mathematics, 1985
... A. Necochea," Borsuk-Ulam theorems for prime periodic transformation groups"... more ... A. Necochea," Borsuk-Ulam theorems for prime periodic transformation groups" (p. 544). ... LIST OF PARTICIPANTS DR Anderson (Syracuse) K. A. Assadi (Virginia) R. R. Ball (Purdue) B. E. Barbanel (Mass.) L. D. Burghelea (Ohio State) P. EC Cho (Rutgers) B. F. Connolly (Notre ...
Illinois Journal of Mathematics, 1997
Inventiones Mathematicae, 1992
In the past two decades the geometrization theorems and conjectures of W. Thurston (see [Th1]) ha... more In the past two decades the geometrization theorems and conjectures of W. Thurston (see [Th1]) have played a crucial role in the study of 3-manifolds. One of the first advances in this direction was the solution of the Smith Conjecture, which states that an orientation-preserving periodic diffeomorphism of the 3-sphere is equivalent to an orthogonal transformation (cf. [Mor1]). There are natural analogs of the Smith Conjecture for other symmetry phenomena in 3-dimensional topology. One of these is the following linearity question: Problem. Is every smooth action of a compact Lie group G on the sphere S 3 , the disk D 3 , or Euclidean space R 3 equivalent to an orthogonal action? In [MY] W. Meeks and S.-T. Yau proved that a smooth orientation-preserving action of a finite group G on R 3 is equivalent to an orthogonal action provided G is not isomorphic to the alternating group A 5 (which is isomorphic to the group of orientation-preserving symmetries of a regular icosahedron). The results of [MY] also yield orthogonality theorems for smooth actions of the same groups on D 3 and also smooth actions of these groups on S 3 with nonempty fixed sets. The purpose of this paper is to prove the analog of the Meeks-Yau result for A 5 : Theorem A. Let Φ be a smooth action of A 5 on R 3 , D 3 , or S 3 ; in the latter case, assume the fixed set is nonempty. Then Φ is differentiably equivalent to a linear action. Since the analogous results for smooth actions of positive-dimensional compact Lie groups on R 3 , D 3 , and S 3 are well known (cf. [Ra]), we have the following general statement. Theorem B. Every smooth action of a compact Lie group G on R 3 or D 3 is differentiably equivalent to a linear action. A similar conclusion holds for smooth actions on S 3 with nonempty fixed point sets. Remark. Examples of Bing [Bi] show that the analog of Theorem B for topological actions is false. A preprint of W. Thurston from 1982 announces many strong results about smooth group actions on 3-manifolds ([Th2]; cf. the footnote on p. 55 in [Mor2]), and the orthogonality of smooth A 5 −actions on R 3 , D 3 , and S 3 is included in the results stated there. Our approach involves some developments that have taken place since the appearance of [Th2]-specifically, the formulation of invariant gauge theory for certain 4-manifolds with smooth group actions (cf. [BKS]) and the The authors were partially supported by NSF Grants DMS 8901583 and DM 8902543, respectively. The second named author would like to thank Northwestern University for access to its facilities during portions of this work.
Given two nonhomeomorphic topological spaces X and Y , it is often interesting or important to sp... more Given two nonhomeomorphic topological spaces X and Y , it is often interesting or important to specify necessary or sufficient conditions for X ×R and Y ×R to be homeomorphic, where as usual R denotes the real line. More generally, it is also useful to have criteria for determining whether X×R and Y ×R are homeomorphic for some k > 1. If X and Y are closed manifolds, the following result, which is due to B. Mazur in the smooth and piecewise linear (PL) categories [16, 17], provides an abstract answer. and to R. Kirby and L. Siebenmann in the topological category [12], provides an abstract answer. In this result CAT refer to the category of smooth, piecewise linear or topological manifolds and a CAT-isomorphism is a diffeomorphism, piecewise linear homeomorphism or homeomorphism respectively:
Proceedings of the American Mathematical Society
Given a manifold M M of dimension at least 4 whose universal covering is homeomorphic to a sphere... more Given a manifold M M of dimension at least 4 whose universal covering is homeomorphic to a sphere, the main result states that a compact manifold W W is isomorphic to a cylinder M × [ 0 , 1 ] M\times [0,1] if and only if W W is homotopy equivalent to this cylinder and the boundary is isomorphic to two copies of M M ; this holds in the smooth, PL and topological categories. The result yields a classification of smooth, finite group actions on homotopy spheres (in dimensions ≥ 5 \geq 5 ) with exactly two singular points.
Lecture Notes in Mathematics, 1990
Duke Mathematical Journal
Topology, 1988
0. INTRODUCTION GIVEN AN action of a group G on a sphere, it is natural to ask if the action can ... more 0. INTRODUCTION GIVEN AN action of a group G on a sphere, it is natural to ask if the action can in some sense be realized as a single or multiple suspension of an action on a lower-dimensional sphere. In many cases the answer to such questions have extremely far-reaching implications for problems of independent interest. For example, the entire classification theory of free involutions on spheres is built around the desuspendability question (compare Lopez de Medrano [25], Browder-Livesay [3]). In the theory of free involutions the group action on a suspension is chosen so that the involution interchanges the north and south poles of the suspension space. Of course there is another way of suspending a group action; one can choose the group action so that the north and south poles become isolated fixed points. The purpose of this paper is to consider group actions on spheres that resemble the lattersort of suspension for free actions on spheres. Specifically, we shall consider actions on spheres that have two isolated fixed points and are free on the complement of the fixed point set. We are mainly interested in actions of this type on S4, for if n 24 then one can give fairly standard necessary and sufficient conditions for a group action on S" to be the suspension of a free action on S"-'. More precisely, one can define two algebraic K-theoretic obstructions with values in the projective class group of H[G], and the action is a suspension if and only if both of these obstructions vanish (e.g., see Theorem A below). In contrast, one of our main results is that the sufficient conditions for other dimensions do not suffice for S4; in fact, we construct an infinite family of counterexamples. The proof that our examples do not desuspend requires a recent result of A. Casson on the vanishing of the Rochlin invariant of a homotopy 3-sphere [S] (see [16] for a description of Casson's results and some further information). However, we show that a semifree action on S4 with two fixed points does have many of the properties of a suspension if the algebraic K-theoretic conditions are satisfied. Here is a formal statement of our conclusions: WEAK DESUSPENSION THEOREM. Let G be aJinite cyclic group acting semifreely on S4 with two fixed points, and assume that the projective obstructions in l?,(Z[G]) associated to both jixed points are trivial. Then the complement of thejixed point set W is a union of G-invariant compact cobordisms (Wj; d, Wj, 8, Wj) where-CC <j< CO, such that the following hold: (1) Win W,=4 ifj#k or k= f 1. (2) wj~wj+,=a,wj=a,wj+,.
Mathematical Proceedings of the Cambridge Philosophical Society, 2004
Mathematical Proceedings of the Cambridge Philosophical Society, 2002
In this paper we prove that R7 admits smooth periodic maps with no fixed points for every period ... more In this paper we prove that R7 admits smooth periodic maps with no fixed points for every period that is not a prime power. Results of P. A. Smith show that such examples do not exist in any lower dimensions.
Bulletin of the American Mathematical Society, 1984
Lecture Notes in Mathematics, 1984
American Journal of Mathematics, 1974
Inventiones Mathematicae, 1969
The general problem of classifying up to orientation-preserving diffeomorphism those smooth manif... more The general problem of classifying up to orientation-preserving diffeomorphism those smooth manifolds homeomorphic to a given manifold is probably too complicated to be treated in any effective uniform manner. The first case to be treated was of course the sphere, where everything has been established modulo computation of the Adams spectral sequence. The next case considered was a product of two spheres, where the author and DeSapio independently reduced the classification to homotopy theory with the exception of determining the action of ~P4k + 2 on some (4 k + 1)-dimensional products. It is completely straightforward to extend this classification to the k-sphere bundles associated to (k+l)-plane bundles over S" which have nowhere zero cross sections and satisfy k < n; this was done in the author's thesis and in [1]. The latter paper also announced results in the case n<k<_n+2. We shall more generally give results in the case k > n which apparently diverge from [1] in some respects. We make two remarks for completeness. A classification of smoothings of bundles with k < n (not necessarily having cross sections) may be derived from [3, w 5]. Finally, throughout this paper the phrases "combinatorially equivalent" and "homeomorphic" are interchangeable by the Hauptvermutung for simply connected closed manifolds with torsion free homology and dimension at least 5.
Lecture Notes in Mathematics, 1990
Inventiones Mathematicae, 1989
In geometric topology it is important to have useful criteria for recognizing special types of ma... more In geometric topology it is important to have useful criteria for recognizing special types of manifolds. For example, it is often necessary to determine whether a connected compact manifold with boundary W can be expressed as a product V• [0, 1]. If W is two-dimensional, then the classification of surfaces shows that some obvious necessary conditions are sufficient. Specifically, the boundary must split into two components, and the inclusion of either must be a homotopy equivalence. A manifold that satisfies these conditions is called an h-cobordism, if the inclusions are simple homotopy equivalences, the manifold is called an s-cobordism. Every product Vx [0, 1] is an s-cobordism, and the s-cobordism theorem of D. Barden, B. Mazur and J. Stallings shows that the converse is also true in all dimensions greater than or equal to six. This result holds in each of standard categories of manifolds. In other words, if the manifold W is taken to be smooth or PL, then the isomorphism with Wo x [0, 1] may also be taken to be smooth or PL. For many years it has been known that the s-cobordism theorem extends to certain specific situations in dimension 3, 4, and 5 (compare Lawson [27], Shaneson [48]) but it was also known that the s-cobordism theorem does not generalize completely in at least one of these dimensions (Siebenmann [49], Lawson [27]). In fact, it was known that failures occur in each of the three manifold categories (topological, PL, and smooth manifolds). Recent advances in low-dimensional topology have improved our understanding of the extent to which the s-cobordism theorem can and cannot be generalized. The results of M. Freedman yield a topological s-cobordism theorem in dimension five provided the fundamental group of W is relatively small [16, 39]. On the other hand, Freedman's results and earlier work of T. Matumoto and L. Siebenmann [32] (cf. [23]) show that the topological s-cobordism theorem fails in dimension four; one can even insist that both boundary components are homeomorphic (specifically to S 1 x RP2).
Osaka Journal of Mathematics, 1994
The Rochlin invariant for 3-manifolds is extended to higher dimensions using a result of Ochanine... more The Rochlin invariant for 3-manifolds is extended to higher dimensions using a result of Ochanine. Given a semifree differentiable S-action on a closed mod 2 homology sphere M with fixed point set F , generalized Rochlin invariants are definable for both M and F , and one result of this paper states that these two invariants are equal. This yields restrictions on the types of semifree differentiable S-actions that some homology spheres can support and the fixed point sets of actions on homology (8k + 7)-spheres. An action of a group G on a space X is said to be semifree if for each x ∈ X either X is fixed under every element of G or else x is not fixed by any element of G except the identity. During the nineteen sixties and seventies it became apparent that the techniques of differential topology had numerous applications to differentiable actions of compact Lie groups (cf. [Bro2], [BP], [RS]). In particular, these and previously developed techniques yielded considerable information...
Proceedings of the Conference at the University of Newcastle upon Tyne, August 1976
Contemporary Mathematics, 1985
... A. Necochea," Borsuk-Ulam theorems for prime periodic transformation groups"... more ... A. Necochea," Borsuk-Ulam theorems for prime periodic transformation groups" (p. 544). ... LIST OF PARTICIPANTS DR Anderson (Syracuse) K. A. Assadi (Virginia) R. R. Ball (Purdue) B. E. Barbanel (Mass.) L. D. Burghelea (Ohio State) P. EC Cho (Rutgers) B. F. Connolly (Notre ...
Illinois Journal of Mathematics, 1997
Inventiones Mathematicae, 1992
In the past two decades the geometrization theorems and conjectures of W. Thurston (see [Th1]) ha... more In the past two decades the geometrization theorems and conjectures of W. Thurston (see [Th1]) have played a crucial role in the study of 3-manifolds. One of the first advances in this direction was the solution of the Smith Conjecture, which states that an orientation-preserving periodic diffeomorphism of the 3-sphere is equivalent to an orthogonal transformation (cf. [Mor1]). There are natural analogs of the Smith Conjecture for other symmetry phenomena in 3-dimensional topology. One of these is the following linearity question: Problem. Is every smooth action of a compact Lie group G on the sphere S 3 , the disk D 3 , or Euclidean space R 3 equivalent to an orthogonal action? In [MY] W. Meeks and S.-T. Yau proved that a smooth orientation-preserving action of a finite group G on R 3 is equivalent to an orthogonal action provided G is not isomorphic to the alternating group A 5 (which is isomorphic to the group of orientation-preserving symmetries of a regular icosahedron). The results of [MY] also yield orthogonality theorems for smooth actions of the same groups on D 3 and also smooth actions of these groups on S 3 with nonempty fixed sets. The purpose of this paper is to prove the analog of the Meeks-Yau result for A 5 : Theorem A. Let Φ be a smooth action of A 5 on R 3 , D 3 , or S 3 ; in the latter case, assume the fixed set is nonempty. Then Φ is differentiably equivalent to a linear action. Since the analogous results for smooth actions of positive-dimensional compact Lie groups on R 3 , D 3 , and S 3 are well known (cf. [Ra]), we have the following general statement. Theorem B. Every smooth action of a compact Lie group G on R 3 or D 3 is differentiably equivalent to a linear action. A similar conclusion holds for smooth actions on S 3 with nonempty fixed point sets. Remark. Examples of Bing [Bi] show that the analog of Theorem B for topological actions is false. A preprint of W. Thurston from 1982 announces many strong results about smooth group actions on 3-manifolds ([Th2]; cf. the footnote on p. 55 in [Mor2]), and the orthogonality of smooth A 5 −actions on R 3 , D 3 , and S 3 is included in the results stated there. Our approach involves some developments that have taken place since the appearance of [Th2]-specifically, the formulation of invariant gauge theory for certain 4-manifolds with smooth group actions (cf. [BKS]) and the The authors were partially supported by NSF Grants DMS 8901583 and DM 8902543, respectively. The second named author would like to thank Northwestern University for access to its facilities during portions of this work.
Given two nonhomeomorphic topological spaces X and Y , it is often interesting or important to sp... more Given two nonhomeomorphic topological spaces X and Y , it is often interesting or important to specify necessary or sufficient conditions for X ×R and Y ×R to be homeomorphic, where as usual R denotes the real line. More generally, it is also useful to have criteria for determining whether X×R and Y ×R are homeomorphic for some k > 1. If X and Y are closed manifolds, the following result, which is due to B. Mazur in the smooth and piecewise linear (PL) categories [16, 17], provides an abstract answer. and to R. Kirby and L. Siebenmann in the topological category [12], provides an abstract answer. In this result CAT refer to the category of smooth, piecewise linear or topological manifolds and a CAT-isomorphism is a diffeomorphism, piecewise linear homeomorphism or homeomorphism respectively:
Proceedings of the American Mathematical Society
Given a manifold M M of dimension at least 4 whose universal covering is homeomorphic to a sphere... more Given a manifold M M of dimension at least 4 whose universal covering is homeomorphic to a sphere, the main result states that a compact manifold W W is isomorphic to a cylinder M × [ 0 , 1 ] M\times [0,1] if and only if W W is homotopy equivalent to this cylinder and the boundary is isomorphic to two copies of M M ; this holds in the smooth, PL and topological categories. The result yields a classification of smooth, finite group actions on homotopy spheres (in dimensions ≥ 5 \geq 5 ) with exactly two singular points.
Lecture Notes in Mathematics, 1990
Duke Mathematical Journal
Topology, 1988
0. INTRODUCTION GIVEN AN action of a group G on a sphere, it is natural to ask if the action can ... more 0. INTRODUCTION GIVEN AN action of a group G on a sphere, it is natural to ask if the action can in some sense be realized as a single or multiple suspension of an action on a lower-dimensional sphere. In many cases the answer to such questions have extremely far-reaching implications for problems of independent interest. For example, the entire classification theory of free involutions on spheres is built around the desuspendability question (compare Lopez de Medrano [25], Browder-Livesay [3]). In the theory of free involutions the group action on a suspension is chosen so that the involution interchanges the north and south poles of the suspension space. Of course there is another way of suspending a group action; one can choose the group action so that the north and south poles become isolated fixed points. The purpose of this paper is to consider group actions on spheres that resemble the lattersort of suspension for free actions on spheres. Specifically, we shall consider actions on spheres that have two isolated fixed points and are free on the complement of the fixed point set. We are mainly interested in actions of this type on S4, for if n 24 then one can give fairly standard necessary and sufficient conditions for a group action on S" to be the suspension of a free action on S"-'. More precisely, one can define two algebraic K-theoretic obstructions with values in the projective class group of H[G], and the action is a suspension if and only if both of these obstructions vanish (e.g., see Theorem A below). In contrast, one of our main results is that the sufficient conditions for other dimensions do not suffice for S4; in fact, we construct an infinite family of counterexamples. The proof that our examples do not desuspend requires a recent result of A. Casson on the vanishing of the Rochlin invariant of a homotopy 3-sphere [S] (see [16] for a description of Casson's results and some further information). However, we show that a semifree action on S4 with two fixed points does have many of the properties of a suspension if the algebraic K-theoretic conditions are satisfied. Here is a formal statement of our conclusions: WEAK DESUSPENSION THEOREM. Let G be aJinite cyclic group acting semifreely on S4 with two fixed points, and assume that the projective obstructions in l?,(Z[G]) associated to both jixed points are trivial. Then the complement of thejixed point set W is a union of G-invariant compact cobordisms (Wj; d, Wj, 8, Wj) where-CC <j< CO, such that the following hold: (1) Win W,=4 ifj#k or k= f 1. (2) wj~wj+,=a,wj=a,wj+,.
Mathematical Proceedings of the Cambridge Philosophical Society, 2004
Mathematical Proceedings of the Cambridge Philosophical Society, 2002
In this paper we prove that R7 admits smooth periodic maps with no fixed points for every period ... more In this paper we prove that R7 admits smooth periodic maps with no fixed points for every period that is not a prime power. Results of P. A. Smith show that such examples do not exist in any lower dimensions.
Bulletin of the American Mathematical Society, 1984