{"id":188,"date":"2006-10-17T21:01:50","date_gmt":"2006-10-17T21:01:50","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/10\/17\/manifolds-and-glue\/"},"modified":"2006-10-17T21:01:50","modified_gmt":"2006-10-17T21:01:50","slug":"manifolds-and-glue","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/10\/17\/manifolds-and-glue\/","title":{"rendered":"Manifolds and Glue"},"content":{"rendered":"

So, after the last topology post, we know what a manifold is – it’s a structure where the neighborhoods of points are *locally* homeomorphic to open spheres in some ℜn<\/sup>.
\nWe also talked a bit about the idea of *gluing*, which I’ll talk about
\nmore today. Any manifold can be formed by *gluing together* subsets of ℜn<\/sup>. But what does *gluing together* mean?
\nLet’s start with a very common example. The surface of a sphere is a simple manifold. We can build it by gluing together *two* circles from ℜ2<\/sup> (a plane). We can think of that as taking each circle, and stretching it over a bowl until it’s shaped like a hemisphere. Then we glue the two hemispheres together so that the *boundary* points of the hemispheres overlap.
\nNow, how can we say that formally?<\/p>\n


\nWell, first we need to define what a sphere is in non-topological terms.
\nThe surface of a three-dimensional sphere with radius N is the set of points (x,y,z) that satisfy the equation x2<\/sup> + y2<\/sup> + z<\/sup>2<\/sup> = N2<\/sup>. For the pedantic types among us, the sphere with radius N is the set {(x,y,z) ∈ ℜ3<\/sup> : x2<\/sup> + y2<\/sup> + z2<\/sup> = N2<\/sup>}.
\nThen, since the surface of a sphere is locally two-dimensional (that is, it’s a 2-manifold) we need to define how to map from subsets of ℜ2<\/sup> to the surface of the sphere. For it to work, it needs to be an *invertible* function – that is, a function *f : X → Y* where both *f* and *f-1<\/sup>* are functions, and *∀ x : f-1<\/sup>(f(x))=x*.
\nFor a sphere that’s easy. We’ll start by describing the halves. For one half, we define a mapping from the hemisphere to a disk in ℜ2<\/sup> as a function f1<\/sub> : ℜ3<\/sup> → ℜ2<\/sup>. For each point (x,y,z) in the sphere where z≥0, we can map it onto a circle using: f1<\/sub>(x,y,z) = (x,y). For the second half, we’ll define f2<\/sub>(x,y,z) = (x,y), but this time we’ll say that f2<\/sub> is only defined for points where z≤0.
\nAnd now, for the gluing. Gluing is done using a special function called a *transition map*. A transition map for a pair of *n*-manifolds A and B is a pair of invertible functions *tA<\/sub>, tB<\/sub>*, where tA<\/sub> maps from A to an open-ball in ℜN<\/sup>, and tB<\/sub> does the same for B. The manifolds are *glued* by the transition map if for every point *x* in the overlap between A and B, there is an invertible mapping from *A* to *B* provided by *tB<\/sub>-1<\/sup>(tA<\/sub>(x))*, and vice versa.
\n\"transition-map.jpg\"
\nThe transition map for the sphere is very simple. The overlap is the *boundaries* of the two circles – the very outer perimeter, the region where z=0. The transition map is *exactly* the same functions that we used to map the two circles onto the sphere.
\nSo using that mapping, we can have a clean “glue line” between the two – there’s exactly one point of overlap, and it’s the circle where the sphere meets the XY plane. And we have a definition of the sphere as a manifold that gives us a metric – the *coordinates* on the circles can be used to describe locations on the sphere.<\/p>\n","protected":false},"excerpt":{"rendered":"

So, after the last topology post, we know what a manifold is – it’s a structure where the neighborhoods of points are *locally* homeomorphic to open spheres in some ℜn. We also talked a bit about the idea of *gluing*, which I’ll talk about more today. Any manifold can be formed by *gluing together* subsets […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[65],"tags":[],"class_list":["post-188","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-32","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/188","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=188"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/188\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=188"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=188"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}