{"id":2934,"date":"2014-04-02T14:41:29","date_gmt":"2014-04-02T18:41:29","guid":{"rendered":"http:\/\/www.goodmath.org\/blog\/?p=2934"},"modified":"2014-04-02T14:41:29","modified_gmt":"2014-04-02T18:41:29","slug":"manifold-are-the-manifolds","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2014\/04\/02\/manifold-are-the-manifolds\/","title":{"rendered":"Manifold are the Manifolds"},"content":{"rendered":"

In the stuff I’ve been writing about topology so far, I’ve been talking about topologies mostly algebraically. I’m not sure why, but for me, algebraic topology is both the most interesting and easy to understand bit of topology. Most people seem to think I’m crazy; in general, people seem to think that algebraic topology is hard. It must say something about me that I find other things much harder, but I’ll leave it to someone else to figure out what.<\/p>\n

Despite the fact that I love the algebraic side, there’s a lot of interesting stuff in topology that you need to talk about, which isn’t purely algebraic. Today I’m going to talk about one of the most important ones: manifolds.<\/p>\n

The definition we use for topological spaces is really abstract. A topological space is a set of points with a structural relation. You can define that relation either in terms of neighborhoods, or in terms of open sets – the two end up being equivalent. (You can define the open sets in terms of neighborhoods, or the neighborhoods in terms of open sets; they define the same structure and imply each other.)<\/p>\n

That abstract definition is wonderful. It lets you talk about lots of different structures using the language and mechanics of topology. But it is very<\/em> abstract. When we think about a topological space, we’re usually thinking of something much more specific than what’s implied by that definition. The word space<\/em> has an intuitive meaning for us. We hear it, and we think of shapes<\/em> and surfaces. Those are properties of many, but not<\/em> all topological spaces. For example, there are topological spaces where you can have multiple distinct points with identical neighborhoods. That’s definitely not part of what we expect!<\/p>\n

The things that we think of as a spaces are really are really a special class of topological spaces, called manifolds<\/em>.<\/p>\n

Informally, a manifold is a topological space where its neighborhoods for a surface that appears to be<\/em> euclidean if you look at small sections. All euclidean surfaces are manifolds – a two dimensional plane, defined as a topological space, is a manifold. But there are also manifolds that aren’t really euclidean, like a torus, or the surface of a sphere – and they’re the things that make manifolds interesting.<\/p>\n

The formal definition is very much like the informal, with a few additions. But before we get there, we need to brush up on some definitions.<\/p>\n