Operations on Manifolds and Minimal Presentations 1 Siddhartha Kanungo 1 Introduction First we describe various ways of gluing manifolds together: connected sum, connected sum along the boundary, attachment of handles, etc. A brief discussion of the effect of these operations on homology prepares the ground for the more precise results to follow. Then we describe a way to build some highly connected manifolds; it turns out later that this method is, in a sense, generic. That is, all highly connected manifolds can be constructed in this way. Along the way, an important result describes the situation when two successive at- tachments of handles produces no change: The second handle destroys the first. This is Smale’s Cancellation Lemma. The main idea here, concerns the existence of the handle presentation with the min- imal number of handles determined by its homology groups. The following example should explain the importance of this idea. The minimal number of handles necessary to build an n-dimensional sphere is two: two n-discs glued along boundaries. If we suceed in proving that a homotopy sphere admits a presentation with the minimal number of handles determined by its homology, then it must admit a presentation with two handles. In turn, this implies that it is homeomorphic to the sphere, i.e., the Poincare conjecture. 2 Operations on manifolds Given two connected m-dimensional manifolds M 1 ,M 2 , let h i : R m → M i ,i =1, 2, be two imbeddings. Let α : (0, ∞) → (0, ∞) be an arbitrary orientation reversing diffeomorphism. We define α m : R m - 0 → R m - 0 by α m (v)= α(|v|) v |v| (2.1) The connected sum M 1 #M 2 is the space obtained from the (disoint) union of M 1 - h 1 (0) and M 2 - h 2 (0) by identifying h 1 (v) with h 2 (α m (v)). It turns out that M 1 #M 2 1 Survey paper written for S. Simi´ c’s Math 213: Differential Geometry, San Jose State University, Fall 2007 1