"The precise and complete theory of closed simply connected topological manifolds in dimensions 4,5,6,7,... in four parts form the viewpoint of algebraic topology" A Minicourse on Topological Manifolds. Dennis Sullivan and Annibal Medina, Stony Brook December 2013 - January 2014 Motivated by some present day math intractibles it seemed opportune to try to learn something about the algebraic meaning of space by understanding/revisiting Ranicki's formulation in chain level terms of the homotopy type of closed manifolds which is subtle/complex because of the prime two, but which in the current program envisaged [numerical approximations for fluid flow] will be adapted to characteristic zero for geometry and analysis issues. The first part of the minicourse December 18th (video on http://www.maths.ed.ac.uk/~aar/surgery/minicourse1.mp4 ) described the [type I] four fold periodicity [the dimension mod four] in the algebraic topology of closed manifolds as expressed by the bilinear duality in homology due to Poincare [1900]. It also made reference to ideas of transversality: Pontryagin [1943], cobordism : thom [1953] and constructive cobordism/surgery: Milnor-Kervaire [1963-Annals]. Now we continue here with a description of some history, the further parts of the minicourse and two other courses at SB and CUNY in the spring semester 2014. The "precise and complete theory" referred to in the title of the minicourse came [late 60's plus two decades] as a direct consequence/implication of the beautiful four-fold periodicity now of [type I*] in the surgery analysis of the [KM annals1963] paper. This paper unearthed the sequence of constructive cobordism/surgery obstruction groups 0 Z/2 0 Z 0 Z/2 0 Z 0 Z/2 ........ These are related to quadratic duality as opposed to bilinear duality. The bilinear {type I} 4-fold periodicity lead instead to a sequence of groups Z/2 0 0 Z Z/2 0 0 Z Z/2 0 ...... [MorganSu Annals 1974] In part 2 of the minicourse January 9th (video on http://www.maths.ed.ac.uk/~aar/surgery/minicourse2.mp4 ) the algebraic treatment of Ranicki of local chain level poincare duality was introduced by Anibal Medina. One consequence of this Ranicki construction is eventually an understanding of the dual relationship between type I and type I* four fold periodicity. Thisunderstanding was eagerly sought unsuccessfully for a decade or two. Atiyah came close with a mod two index theorem that did some cases of the shifted Z/2.[ based on Bott's 8 fold periodicity] Later parts of the minicourse will backfill to describe in outline other efforts during two decades that led to this Ranicki construction/understanding. This came about via the use ofsurgery byNorman Levitt, Lowell Jones, Bill Browder, Frank Quinn, Greg Brumfiel, John Morgan,et al... to give an understanding of the homotopy theoretical meaning of transversality. Next semester certain courses will complement or elaborate the ideas opened up in the minicourse. Firstly, in the two sequences above the Z's are related by the famous factor of 8 in the subject of nondegenerate integral quadratic forms. Nathan Sunukjian's Math 621 will start with Rochlin's deep observation [1951] that in the context of smooth closed four manifolds the 8 has to be replaced by 16. It will then go on to contrast the theory of smooth and topological manifolds in dimension 4 ...which is a kind of amplification of Rochlin's discovery [using the signature operator d + d* of the Atiyah Singer index theory] Similarly, Math 541 will elaborate the minicourse and the differences between topological manifolds and smoother manifolds in dimensions 4 and higher. [with the same speakers as in the minicourse] For example, in dim 5,6,7,...topological manifolds admit bilipschitz charts unique up to deformation close to the identity.[ The proof of this uses etale cohomology from algebraic geometry.] Whereas in dim 4 the existence/choice of bilipschitz charts allows the gauge theory results of Donaldson [to be described in math 621] to go through using the original arguments of Donaldson. Math 621 by Nathan Sunukjian The course will be broadly interested in the difference between the theory of topological and smooth 4-manifolds. First we'll discuss the homotopy classification of simply connected 4-manifolds. This was completely worked out by Whitehead (and given its current form by Milnor). It has long been known that the theories of smooth and topological 4-manifolds should be quite different. Rochlin's theorem (1951), which places restrictions on the cohomology of smooth 4-manifolds, was the first major step in this direction. Later, gauge theory was used to give even more restrictions, and to show that certain topological 4-manifolds could admit many (infinitely many) smooth structures. Concurrently, Freedman developed a far reaching theory of topological 4-manifolds, that lead to the classification of simply connected topological 4-manifolds. No such classifications or conjectured classifications exist in the smooth realm, as opposed to higher dimensions where a mature theory exists. As a framework for introducing many of the important techniques of 4-manifolds, this course will look at several proofs of Rochlin's theorem: Rochlin's original algebraic topological proof, Kirby and Matsumoto's hands on proofs, and the proof of Kirby-Freedman, which involves many of the ideas involved in Freedman's theory of topological 4-manifolds. The original proof is a beautiful application of characteristic classes, homotopy theory, and transversality, and is closely related to the original construction of exotic 7-spheres by Milnor. Kirby and Matsumoto's proofs are beautifully geometric and will illustrate the importance of a careful understanding of embedded surfaces in 4-manifolds. Matsumoto and Kirby-Freedman prove generalizations of Rochlin's theorem. The thrust of the course, however, will be more on developing the tools of 4-manifolds, and less on this one specific theorem. Pictorial methods of Kirby Calculus will be introduced early, to give a very hands on feel to things. The progressive generalizations of Rochlin's theorem will lead up to a "utilitarian" proof of Donaldson's theorem (which places very strong constraints on the cohomology of smooth 4-manifolds) using Heegaard-Floer Homology. Depending on the interests of the class, at this point we will introduce the techniques used to construct exotic smooth structures on 4-manifolds and exotic embeddings of surfaces, as well as delve a bit deeper into Freedman's theory of topological 4-manifolds. Specific topics will depend on the interests of the class.