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Category Archives: Discussion
Torus trickery 201, part 2
So how are appropriate versions of Facts 1 and 2 (from the previous post) to be proved, and what has this all got to do with the torus? The Euclidean space \({\mathbb R}^2\) that appears is a little piece of … Continue reading
Torus trickery 201
One of the things I really want to do with the Surgery for Amateurs project is to make a reasonably plausible presentation of surgery theory for topological manifolds, as well as for smooth and/or PL. But this requires one to … Continue reading
The Poincare conjecture
The next few sections of Chapter 1 are intended to introduce some key examples of constructions with manifolds: The highdimensional Poincaré conjecture and the hcobordism theorem Milnor’s exotic spheres Variation of Pontrjagin classes within a homotopy type In the version … Continue reading
Posted in Discussion, math
Tagged cobordism, exotic sphere, Poincare conjecture, Poincare duality
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Steenrod squares
The Steenrod squares are discussed at some length in Chapter 5 of the book (so we’re getting a bit out of sequence here). But they are also, of course, closely related to characteristic classes: Milnor and Stasheff’s book defines the … Continue reading
Posted in Discussion, math
Tagged characteristic class, cohomology, Steenrod squares
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Manifold Sins and Wickedness
In the previous post I mentioned the two categories of manifolds with which we’re going to be mostly concerned: smooth manifolds, equipped with an atlas whose transition maps are \(C^\infty\), and topological manifolds (locally Euclidean spaces). Smooth manifolds are, of … Continue reading