By Ams-Ims-Siam Joint Summer Research Conference on Groupoids in analysis, Jean Renault, Arlan Ramsay

Groupoids frequently take place whilst there's symmetry of a nature now not expressible when it comes to teams. different makes use of of groupoids can contain whatever of a dynamical nature. certainly, a few of the major examples come from workforce activities. it may even be famous that during many occasions the place groupoids were used, the most emphasis has now not been on symmetry or dynamics concerns. for instance, a foliation is an equivalence relation and has one other groupoid linked to it, referred to as the holonomy groupoid. whereas the implicit symmetry and dynamics are suitable, the groupoid documents ordinarily the constitution of the gap of leaves and the holonomy.More ordinarily, using groupoids is particularly a lot with regards to a number of notions of orbit equivalence. the viewpoint that groupoids describe 'singular areas' are available within the paintings of A. Grothendieck and is conventional within the non-commutative geometry of A. Connes. This booklet offers the lawsuits from the Joint summer time examine convention on 'Groupoids in research, Geometry, and Physics' held in Boulder, CO. The booklet starts with an advent to ways that groupoids enable a extra finished view of symmetry than is noticeable through teams. subject matters diversity from foliations, pseudo-differential operators, $KK$-theory, amenability, Fell bundles, and index conception to quantization of Poisson manifolds. Readers will locate examples of vital instruments for operating with groupoids. This ebook is geared to scholars and researchers. it truly is meant to enhance their knowing of groupoids and to inspire them to seem extra whereas studying in regards to the instruments used

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Let pa and pp be the projections of the componentwise normal bundles. 4(z) there is an isovariant diagram preserving homotopy rel Sing(X) from / to a map f\ such DIAGRAM COHOMOLOGY AND ISOVARIANT H O M O T O PY 33 that fipa — ppfi on S(a). We claim there is also an isovariant homotopy from D(a) to D(/3) rel S(a) U Sing(X) from f\ to a length preserving map. First of all, there is an isovariant homotopy rel S(a) U Sing(X) from / i to a map J2 such that the map $2 sends ^D(a) into ^D(j3) for every positive integer n; this is true because (i) one can find a sequence of positive real numbers 6n < 1 such that lim<5n = 0 and f\ sends 6nD(a) into ^D(/3) for all n, (ii) there is an ambient isotopy of X rel S(a) U Sing(X) that maps ^D(a) fiber preservingly into SnD(a) for all n.

9. Let G be a finite group, and let X and Y be compact locally linear G-manifolds. Let Qx and QY be regular G-invariant quasistratihcations and let B(QFX) and ^4(QF y ) be defined as before. Then the forgetful map G — isovariant G — isovariant homotopy classes of homotopy classes of continuous isovariant continuous isovariant diagram morphisms maps of spaces B(QFX)->A(QFY) X —>Y is an isomorphism. 9. 10. Let G be a finite group, and let smooth G-manifolds with treelike isotropy structure. 4], and let B(QFX) and before.

Of course, this includes the case where ~ / ( / ) is empty and / is an isovariant map that determines a morphism of diagrams as above. The name "almost isovariant" suggests that isovariant maps should be almost isovariant. However, the relationship is not quite that simple because almost isovariance requires the existence of invariant quasistratifications and specific choices of such structures on the domain and codomain. 2. A). 1, let B$(QFX) be the diagram of closed subspaces associated 27 28 GIORA DULA AND REINHARD SCHULTZ to the quasistratification Qx(6) f°r <$ > 0, and let f : X —* Y be a continuous isovariant map.