Abstract Nonsense

Crushing one theorem at a time

Smooth Maps and the Category of Smooth Manifolds (Pt. III)


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September 3, 2012 Posted by | Manifold Theory, Topology | , , , , , , | Leave a comment

Smooth Maps and the Category of Smooth Manifolds (Pt. II)


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September 3, 2012 Posted by | Manifold Theory, Topology | , , , , , | 1 Comment

Smooth Maps and the Category of Smooth Manifolds (Pt. I)


Point of Post: In this post we define what it means for a map between two manifolds to be smooth.

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Motivation

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We literally defined smooth manifolds to be the topological spaces where we will have a relatively sound meaning of what a “smooth map” is. Thus, it would seem that the first order of business is to fully define and explore this notion of smooth map. The basic idea though is precisely what we have said before. A map between smooth manifold will be smooth if it is smooth locally around each point and its image–when we think about the space locally as Euclidean space.

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The interesting part is that once we define smooth map we will then be able to define the category of (finite dimensional) smooth manifolds. We will then be able to discuss the functor which takes a smooth manifold to it’s algebra of smooth functions (for us, function will mean a map into \mathbb{R}). We will then be able to make sense of the following statement: the smooth structure of a manifold  is largely encoded in its algebra of smooth functions.

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September 3, 2012 Posted by | Manifold Theory, Topology, Uncategorized | , , , , , | 2 Comments

Smooth Manifolds (Pt. III)


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August 30, 2012 Posted by | Manifold Theory, Topology | , , , , | Leave a comment

Smooth Manifolds (Pt. II)


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August 30, 2012 Posted by | Topological Groups, Topology | , , , , | 3 Comments

Smooth Manifolds (Pt. I)


Point of Post: In this post we define the notion of smooth manifolds, give ample examples, and prove some fundamental results about the construction of smooth manifolds.

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Motivation

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In previous posts we have defined the notion of a topological manifold as being the end result of a progression through nicer and nicer spaces. In this post we take this progression one step further and describe a class of spaces that, in some sense, are some of the nicest spaces one can reasonably hope to deal with. The spaces come up when we ask ourselves “now that we have spaces that locally look like \mathbb{R}^n, what can we do with them?” Thus, one starts introspecting: what aspect of \mathbb{R}^n, besides being the most natural space for us lowly humans to think of, makes it so special?

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August 30, 2012 Posted by | Manifold Theory, Topology | , , , , , | 2 Comments

Topological Manifolds (Pt. II)


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August 30, 2012 Posted by | Manifold Theory, Topology | , , , , , | 2 Comments

Topological Manifolds (pt. I)


Point of Post: In this post we define the notion of a topological manifolds, discuss some of their importance, and prove some elementary topological facts about them.

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Motivation

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Anyone who has taken a general topology course before can tell you the following simple fact: topological spaces can get messy. It is not hard to create topological spaces which completely break our intuition for what a topological space should be! It’s definitively possible that a topological space doesn’t locally look like connected space (i.e. isn’t locally connected), doesn’t locally look like a compact space (i.e. isn’t locally compact), and (in a more dramatic way) it doesn’t even have to be able to distinguish points (i.e. doesn’t have to be Kolomogorov)!

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August 30, 2012 Posted by | Manifold Theory, Topology | , , , , , | 2 Comments

The Fundamental Groupoid and Group (Pt. IV)


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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , , , , | 4 Comments

The Fundamental Groupoid and Group (Pt. III)


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August 30, 2012 Posted by | Algebraic Topology, Topology | , , , , , , , , | 1 Comment

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