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A typical course in quantum field theory begins with a thorough examination of a "toy model", usually the φ 4 theory. Our purpose here is to provide a detailed description of a "toy model" for topological quantum field theory, suitable for use as a foundation for more sophisticated developments. We carry through all the steps of the path integral quantization: start with a lagrangian, construct the classical action, construct a measure, and do the integral. When the gauge group is finite the "path integral" reduces to a finite sum. This remark makes it clear that the analytical difficulties simplify enormously, and that there should be no essential problem in carrying out the process. Many interesting features remain, however. The algebraic and topological structure are essentially unchanged, and are much clearer when not overshadowed by the analysis. And even the analysis does not entirely disappear: the details of the construction of the state spaces requires a much more precise formulation of the classical theory than is usually given, and reveals some incompleteness in the understanding of the classical theory for continuous Lie groups [F1].Chern-Simons theory with finite gauge group was introduced by Dijkgraaf and Witten [DW], who essentially cataloged the possible lagrangians and gave some sample calculations. Special cases were considered by Segal [S2] and Kontsevich [K]. More abstract and mathematically-oriented versions are considered in [Q1], [Q2], and connections with the representation-theoretic approach of Reshetikhin and Turaev are described by Ferguson [Fg] and Yetter [Y].Let Γ denote the gauge group, which we assume to be finite. In §1 and §2 we carry out the quantization: describe the lagrangians, classical actions, and path integrals. The fields in this version are regular covering spaces P → X whose group of deck transformations is Γ. The action is a (torsion) characteristic number associated to a class in H 3 (BΓ; R/Z). We represent this class in singular cohomology by a singular cocycle, and use this write the action (1.6) as an integral over X. (The 1 cochain plays the role of the lagrangian.) We caution that the resulting theory depends in a subtle way on this choice of cocycle: a different choice gives a theory which is isomorphic in an appropriate sense, but the isomorphism between the two is not canonical: it depends on further choices. We must also make careful sense of integration of singular cocycles over manifolds with boundary; this is explained in Appendix B. The classical theory is somewhat unusual in that the action on manifolds with boundary is not a number, but rather an element in a complex line determined by the restriction of the field to the boundary. These lines, which properly belong to the hamiltonian theory, are the source of much of the structure in the theory. We defer some of the details of the classical theory to [F1] and [F3], which explores the classical Chern-Simons theory for arbitrary compact gauge groups. The quantization in §2 is straightforward. Of general ...

Abstract. Versions of the finiteness obstruction and simple homotopy theory "within ~ over X" are developed. This provides a setting for obstructions to the map analogs of the end and s-cobordism theorems for manifolds. These are applied to study equivariant mapping cylinder neighborhoods in topological group actions, triangulations of locally triangulable spaces, and block bundle structures on approximate fibrations.

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The use of virtual reality (VR) in the training of operative dentistry is a recent innovation and little research has been published on its efficacy compared to conventional training methods. Two groups of dental students, with no experience in operative dentistry, were trained solely by either VR or conventional training in the preparation of conventional class 1 cavities. The subjects all used the same operative armamentarium and phantom heads, and were allocated the same duration of practice periods. At the completion of these training periods, both groups produced two class 1 cavities on the lower left first molar, which were subsequently coded and blindly scored for the traditional assessment criteria of outline form, retention form, smoothness, cavity depth and cavity margin angulation. An ordinal score of 0-3 or 0-4 was assigned for each assessment criterion: the higher the score, the worse the evaluation. After initial independent scoring, the two examiners discussed any notable differences until an agreed score was reached. Once the codes were broken, non-parametric analyses were performed on the data. Wilcoxon Tests for the semiquantitative scores indicated significant differences between the VR and conventional training groups for outline form, depth and smoothness but not for retention or cavity margin angulation at P < 0.05 level, with the VR group receiving the higher, i.e. worse, scores. Cavity margin angulation approached significance with a P-value of 0.0536. The results indicated that VR-based skills acquisition is unsuitable for use as the sole method of feedback and evaluation for novice students.

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The use of virtual reality (VR) in the training of operative dentistry is a recent innovation and little research has been published on its efficacy compared to conventional training methods. To evaluate possible benefits, junior undergraduate dental students were randomly assigned to one of three groups: group 1 as taught by conventional means only; group 2 as trained by conventional means combined with VR repetition and reinforcement (with access to a human instructor for operative advice); and group 3 as trained by conventional means combined with VR repetition and reinforcement, but without instructor evaluation/advice, which was only supplied via the VR-associated software. At the end of the research period, all groups executed two class 1 preparations that were evaluated blindly by 'expert' trainers, under traditional criteria (outline, retention, smoothness, depth, wall angulation and cavity margin index). Analyses of resulting scores indicated a lack of significant differences between the three groups except for scores for the category of 'outline form', for group 2, which produced significantly lower (i.e. better) scores than the conventionally trained group. A statistical comparison between scores from two 'expert' examiners indicated lack of agreement, despite identical written and visual criteria being used for evaluation by both. Both examiners, however, generally showed similar trends in evaluation. An anonymous questionnaire suggested that students recognized the benefits of VR training (e.g. ready access to assessment, error identification and how they can be corrected), but the majority felt that it would not replace conventional training methods (95%), although participants recognized the potential for development of VR systems in dentistry. The most common reasons cited for the preference of conventional training were excessive critical feedback (55%), lack of personal contact (50%) and technical hardware difficulties (20%) associated with VR-based training.

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