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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 sphere
Following on from my previous post, I would like to add a little introduction to the Poincare homology sphere early on in Chapter 1. This gives an opportunity to introduce the \(E_8\) plumbing in a reasonably downtoearth context, so it … Continue reading
Posted in math, Uncategorized
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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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Manifolds = Bundles + Handles
The slogan in the title is one that I came up with for the book (or at least I think I did, though of course I may have stolen it from somewhere that I have now forgotten). In my opinion … Continue reading
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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Characteristic Classes
The next section of the book, available here, gives a very quick introduction to what the Pontrjagin classes are. This is really just a refresher – I am sure it is too terse for someone who has never seen characteristic … Continue reading
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
What Surgery Is About: Manifolds
Now I’ll begin to post material from the book proper. Here is the first section of Chapter I. I’m going to try to post stuff in bitesize pieces like this, rather than a whole chapter at a time. Of course, … Continue reading