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Category Archives: math
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 down-to-earth 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 high-dimensional Poincaré conjecture and the h-cobordism 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 bite-size pieces like this, rather than a whole chapter at a time. Of course, … Continue reading
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