{"id":194,"date":"2006-10-26T08:56:06","date_gmt":"2006-10-26T08:56:06","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/10\/26\/building-manifolds-with-products\/"},"modified":"2006-10-26T08:56:06","modified_gmt":"2006-10-26T08:56:06","slug":"building-manifolds-with-products","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/10\/26\/building-manifolds-with-products\/","title":{"rendered":"Building Manifolds with Products"},"content":{"rendered":"

Time to get back to some topology, with the new computer. Short post this morning, but at least it’s something. (I had a few posts queued up, just needing diagrams, but they got burned with the old computer. I had my work stuff backed up, but I don’t let my personal stuff get into the company backups; I like to keep them clearly separated. And I didn’t run my backups the way I should have for a few weeks.)
\nLast time, I started to explain a bit of patchwork: building manifolds from other manifolds using *gluing*. I’ll have more to say about patchwork on manifolds, but first, I want to look at another way of building interesting manifolds.
\nAt heart, I’m really an algebraist, and some of the really interesting manifolds can be defined algebraically in terms of topological product. You see, if you’ve got two manifolds **S** and **T**, then their product topology **S×T** is also a manifold. Since we already talked about topological product – both in classic topological terms, and in categorical terms, I’m not going to go back and repeat the definition. But I will just walk through a couple of examples of interesting manifolds that you can build using the product.
\nThe easiest example is to just take some lines. Just a simple, basic line. That’s a 1 dimensional manifold. What’s the product of two lines? Hopefully, you can easily guess that: it’s a plane. The standard cartesian metric spaces are all topological products of sets of lines: ℜn<\/sup> is the product of *n* lines.
\nTo be a bit more interesting, take a circle – the basic, simple circle on a cartesian plane. Not the *contents* of the circle, but the closed line of the circle itself. In topological terms, that’s a 1-sphere, and it’s also a very simple manifold with no boundary. Now take a line, which is also a simple manifold.
\nWhat happens when you take the product of the line and the circle? You get a hollow cylinder.
\n\"cylinder.jpg\"
\nWhat about if you take the product of the circle with *itself*? Thing about the definition of product: from any point *p* in the product **S×T**, you should be able to *project* an image of
\n**S** and an image of **T**. What’s the shape where you can make that work right? The torus.
\n\"torus.jpg\"
\nIn fact the torus is a member of a family of topological spaces called the toroids. For any dimensionality *n*, there is an *n*-toroid which the the product of *n* circles. The 1-toroid is a circle; the 2-toroid is our familiar torus; the 3-toroid is a mess. (Beyond the 2-toroid, our ability to visualize them falls apart; what kind of figure can be *sliced* to produce a torus and a circle? The *concept* isn’t too difficult, but the *image* is almost impossible.)<\/p>\n","protected":false},"excerpt":{"rendered":"

Time to get back to some topology, with the new computer. Short post this morning, but at least it’s something. (I had a few posts queued up, just needing diagrams, but they got burned with the old computer. I had my work stuff backed up, but I don’t let my personal stuff get into the […]<\/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-194","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-38","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/194","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=194"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/194\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=194"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=194"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}