Projective modules and class groups

Let M n + 3 {M^{n + 3}} be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold K 3 {K^3} with an effective circle action of hyperbolic type. Assume that the fiber N n {N^n} in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set S TOP ( N n × D k , ∂ ) {S_{\text {TOP}}}\,({N^n} \times {D^k},\partial ) of topological surgery vanishes for sufficiently large k k . Then the projective class and Whitehead groups of M M vanish and S TOP ( M n + 3 × D k , ∂ ) = 0 {S_{\text {TOP}}}\,({M^{n + 3}}\, \times \, {D^k},\partial ) = 0 if n + k ⩾ 3 n + k \geqslant 3 or if K 3 {K^3} is closed and n = 2 n = 2 . The UNil \text {UNil} groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that UNil j {\text {UNil}}_j vanishes in spite of the failure of Cappell’s sufficient condition.

Chapter 26 - Wall's Finiteness Obstruction

We give a characterization of k-dimensional (k > 1) Menger manifolds admitting boundaries in the sense of Chapman and Siebenmann.In [5] Chapman and Siebenmann considered the problem of putting a boundary on a Hilbert cube manifold (similar problems in the cases of smooth and piecewise linear manifolds were considered in [2,14]; a parametrical version of the above problem was considered in [13]).It was proved there that if a Q-manifold M satisfies certain minimal necessary homotopy-theoretic conditions (finite type and tameness at oo ), then there are two obstructions o0O(M) and Too(M) to M having a boundary.The first one is an element of the group lim{J?67ri(M -A) : A c M compact}, where %nx is the projective class group functor.If Ooo(M) = 0, then the second obstruction can be defined as an element of the first derived limit of the inverse system lim {W hnx (M -A) : A M compact} , where Whnx is the Whitehead group functor.Further, it was proved in [5] that the different boundaries that can be put on M constitute a whole shape class and that a classification of all possible ways of putting boundaries on M can be done in terms of the group lim {Whnx (M -A) : Ac.M compact}.In the present paper we carry out a similar program for the problem of putting boundaries on pn+x -manifolds, where pn+x denotes the (n + 1 )-dimensional universal Menger compactum (a p"+x -manifold M admits a boundary if there exists a compact p"+x-manifold N such that M = N-Z , where Z is a Z-set in N ; in this case we shall say that N is a compactification of M corresponding to the boundary Z , and conversely, Z is a boundary of M corresponding to the compactification N. We recall also that a closed subset Z of a space X is said to be a Z-set if for each open cover ^ e cov(X) there is a map f:X-*X-Z ^-close to the identity map of X ).Having in mind a deep analogy between the theories of pn+x -manifolds and Q-manifolds [1,[6][7][8][9] it is not surprising that the corresponding results are valid in the case of pn+xmanifolds as well.However, it should be observed that the situation in the last case is much simpler.For example, the analogies of the above-described

The Projective Class Group and the Whitehead Group

This paper discusses research investigating to what extent, if any, group work influences the students’ learning experience in Fundamentals of Electronics. Two classes of second year students from the College of Engineering at American University of Sharjah were involved. At least one class project is an established assignment in the class and is usually done in groups. For the purposes of this research, one class completed the project in groups and the other class completed it individually. The students were then given a short test specific to the material covered by the project. They also filled out a short survey and participated in focus group interviews to discuss their experience. The results indicate that although many of the students stated they enjoyed working in groups, a number of them admitted they did not gain a full understanding of the course material through the group work activity.

Group projects have long been an important element of higher education classes. Class projects involve additional cooperation and coordination among students. Student perceptions are an important factor in evaluating the effectiveness of projects. This exploratory study used a 39-item questionnaire to examine undergraduate student perceptions of class projects in courses in recreation and sport management at the conclusion of the projects. Students were highly favorable of class projects as a learning experience and identified increased skill development in time management, planning, and organizing and working with others. Presence of a noncontributing group member had no affect on student perceptions of class project benefits (t (38) = 1.7, p = 0.09). These findings provide support for the use of class projects in recreation and sport management courses as a valuable teaching tool. Further research of student perceptions of class projects across multiple universities is needed.

A group decision support system (GDSS) exercise employing the GE Matrix for international market evaluation was conducted with an Executive M.B.A. class. A postexercise survey of subjects indicated support for using this technology to elicit ideas and evaluations from groups. This technology is recommended for marketing class exercises and group projects.

Let K be the field of algebraic functions on an elliptic curve that can be described by an equation of the form y 2 = f ( x ) {y^2} = f(x) where f ( x ) f(x) is a quartic polynomial over a field k. Moreover, assume that the Riemann surface for K contains no points rational over k. When k is the field of real numbers it is well known that K may also be expressed as a quadratic extension of a function field L = k ( u , v ) L = k(u,v) of algebraic functions on a conic whose Riemann surface also contains no points rational over k. We extend this result to p-adic ground fields k. Moreover, we describe the various subfields of index two and genus zero (conic subfields) in terms of the k-rational points on the Jacobian of K. This is done for arbitrary ground fields. In particular, the embedding of the projective class group of K (over k) is seen to describe exactly those conic subfields that possess k-rational points.

The following results of Whitehead and Wall are well-known applications of the algebraic K-theoretic functors K0 and K1 to basic homotopy questions in topology.THEOREM 1 [20]. If f : X → Y is a homotopy equivalence between compact CW complexes, then there is a torsion τ(ƒ) in the algebraically-defined Whitehead group Wh π1(Y) which vanishes if and only if f is a simple homotopy equivalence.THEOREM 2 [18]. If X is an arbitrary space which is finitely dominated (i.e., homotopically dominated by a compact polyhedron), then there is an obstruction σ(X) in the algebraically-defined reduced projective class group which vanishes if and only if X is homotopy equivalent to some compact polyhedron.If we direct sum over components, then the above statements make good sense even if the spaces involved are not connected.

Let M be a compact connected Riemannian manifold whose sectional curvature values are all nonpositive. Let T denote the fundamental group of M. We prove that any homotopy equivalence : N -> M from a compact closed manifold TV is homotopic to a homeomorphism, provided that m > 5 where m = dim M . We show that the surgery L-group L k+m (T, w { ) is isomorphic to the set of homotopy classes of k k

This article explores practical ways for effectively training international business students in Hofstede's individualism and collectivism dimension. It details the behaviors arising from this dimension that frequently interfere with the success of group projects in international business classes as well as the implications arising from these behaviors for international business relations between those from individualist and collectivist cultures. Too often, projects aimed at fostering collectivism fail. Conventional theories of motivation derive from and apply to North American individualist culture, but may not have the same application in collectivist situations. Tools for adapting group projects which are given to individuals from individualist cultures to help them develop a more collectivist orientation are explored here. These tools-group autonomy in excluding non-performers (loafers), and self and group evaluations empower groups while engendering collectivist concepts.

The study is an evaluation of a series of five linked projects commissioned and run by the South Bank Centre in London in collaboration with six secondary schools. Students worked with a range of professional musicians over three years, commencing at age 11, Year 7. The methodology included interviews with the participants, observation at the Centre and in schools and an attitude inventory, for participating students and ‘control classes’, which were administered following each project. There were significantly different attitude scores to music in school between project and control groups. The project classes also maintained significantly less variance as the projects took place. There was a strong individual school effect. Interview and observational data support the interpretation that the attitudes of students towards music in school and the musical confidence of teachers were enhanced.

Reparative Public Musicology: Empowering and Centering Community Knowledge Production through Counter-Storytelling Practice

We had sung a program of songs and parodies at the Biometric Society's Eastern North American Region (ENAR) 1986 spring meeting in Atlanta, and at the 60th anniversary meeting of the Washington Statistical Society (WSS) in 1987, and it was suggested that we prepare a program for the ASA Sesquicentennial. We mentioned it to Executive Director Barbara Bailar, who, after conferring with President Janet Norwood, responded, Consider yourself volunteered. We should mention that we had reassured them via some credentials that we are professional (and semi-professional) singers. Dwight Brock is one of the 16 members of the Paul Hill Chorale soloists of Washington, D.C., a group that performs as Washington Singers, and he and Mary Brock and Treva Whitmore had performed in oratorios, cabaret musicals, and regional theater groups in light classical and musical comedy projects. Paul Minton had a good deal of amateur experience. Here, of course, we performed as amateurs, a contribution to the ASA Sesquicentennial celebration. Norwood and Bailar had planned a formal banquet with a distinguished speaker, and our would be appropriate as entertainment following the banquet. Dwight, Mary, and Paul had originally met in Dallas some years ago, when Paul was Chairman of the Department of Statistics at Southern Methodist University, and Dwight was a Ph.D. candidate. At the time, Dwight was a paid soloist at the Episcopal Church of the Incarnation, where Paul had sung in the boys' choir as a child. three recognized their mutual interest in serious music, and later continued their personal association after Dwight was employed by the National Center for Health Statistics in Bethesda (later becoming Chief of Biometry at the National Center on Aging), and Paul moved to Richmond, Virginia, as Dean of the School of Arts and Sciences at Virginia Commonwealth University (later Director of the Institute of Statistics, and since retired). Mary continued and expanded her music and worked as an accountant and in a professional society office in biology, as their two children grew up in the Gaithersburg, Maryland, area. Paul has commuted from Richmond regularly to join Dwight and Mary in many of their oratorio performances for a suburban Maryland church, where Dwight is a featured soloist, and Mary and Paul have also had solo assignments. Treva is also a member of this group, which performs large works of classical sacred music with a small orchestra and imported conductors of stature, and, at times, imported soloists from such areas as the Washington Opera. came about through Mike Kutner's desire to have entertainment after the traditional banquet. Mike was the Local Arrangements Chair, a key role in the spring meetings with a great deal of responsibility and autonomy. Bill Schucany had mentioned Dwight to Mike, and Dwight suggested that he and Mary should arrange a program with Paul. Mike arranged the entertainment at some risk, however, since there was not general agreement that professional societies should have entertainment. There were other acts besides ours, and Mike checked on the quality of each beforehand. We began planning for the show with the idea that we should first establish that we are singers. Thus, the first several numbers should be straight, from well-known Broadway shows or musical comedy songs. Mary would open the show with Art Is Calling Me, Paul would follow with Every Day Is Ladies' Day With Me, Dwight and Mary would do a medley from The Desert Song, with solos for each, and then the three would sing a trio from Kismet called Is My Beloved. After demonstrating that we were singers, we wanted to begin bringing in some parodies, using songs everyone would recognize. This turned out to be a difficult task. We began by reviewing the titles of such songs, brainstorming for anything that we could connect even remotely with statistics. Paul remembers throwing out some on the telephone, thinking they were not very good, but we set to work improving them. Paul had a principle of starting with our favorite songs, as more personally rewarding to ourselves. One of Paul's favorite songs is All Things Are, which begins You are . . .; this could be turned into ENAR .... This went over well, and was redone for the ASA meeting as:

UTILIZATION OF THE LOCAL MEDIA IN AN INTRODUCTORY STATISTICS COURSE