continuous deformation<\/em> stuff means in strictly formal terms.<\/p>\n So, let’s dive in and hit the formalism.<\/p>\n
Suppose we’ve got two topological spaces, S<\/b> and T<\/b>, and two continuous functions f,g:S<\/b>→T<\/b>. A homotopy is a function h<\/em> which associates every value in the unit interval [0,1] with a function from S<\/b> to T<\/b>. So we can treat h<\/em> as a function from S<\/b>×[0,1]→T<\/b>, where ∀x:h(x,0)=f(x) and h(x,1)=g(x). For any given value x, then, h(x,·) is a curve from f(x) to g(x). <\/p>\n Thus – expressed simply, the homotopy is a function that precisely describes the transformation between the two homotopical functions. Homotopy defines an equivalence relation<\/em> between continuous functions: continuous functions between topological spaces are topologically equivalent if there is a homotopy between them. (This paragraph originally included an extremely confusing typo – in the first sentence, I repeatedly wrote “homology” where I meant “homotopy”. Thanks to commenter elspi for the catch!)<\/em> <\/p>\n We can also define a type of homotopy equivalence between topological spaces. Suppose again that we have two topological spaces S<\/b> and T<\/b>. S<\/b> and T<\/b> are homotopically equivalent if there are continuous functions f:S<\/b>→T<\/b> and g:T<\/b>→S<\/b> where gºf is homotopic to the identity function for T, 1T<\/b><\/sub>, and fºg is homotopic to the identity function for S, 1S<\/b><\/sub>. The functions f and g are called homotopy equivalences<\/em>. <\/p>\n This gives us a nice way of really formalizing the idea of continuous deformation of spaces<\/em> in homeomorphism – every homeomorphism is also a homotopy equivalence. But it’s not both ways – there are homotopy equivalences that are not<\/em> homeomorphisms. <\/p>\n The reason why is interesting: if you look at our homotopy definition, the equivalence is based on a continuous deformations – including<\/em> contraction. So, for example, a ball is not homeomorphic to a point – but it is<\/em> homotopically equivalent. The contraction all the way from the ball to the point doesn’t violate anything about the homotopical equivalence. In fact, there’s a special name for the set of topological spaces that are homotopically equivalent to a single point: they’re called contractible<\/em> spaces. (Originally, I erroneously wrote “sphere” instead of “ball” in this paragraph. I can’t even blame it on a typo – I just screwed up. Thanks to commenter John Armstrong for the catch.<\/em><\/p>\n Addendum:<\/b> Commenter elspi mentioned another wonderful example of a homotopy that isn’t a homeomorphism, and I thought it was a good enough example that I wish I’d included it in the original post, so I’m promoting it here. The mobius band is homotopically equivalent to a circle – compact the band down to a line, and the twist “disappears” and you’ve got a circle. But it’s pretty obvious that the mobius is not<\/em> homeomorphic to a circle!. Thanks again, elspi – great example!<\/p>\n","protected":false},"excerpt":{"rendered":"I’ve been working on a couple of articles talking about homology, which is an interesting (but difficult) topic in algebraic topology. While I was writing, I used a metaphor with a technique that’s used in homotopy, and realized that while I’ve referred to it obliquely, I’ve never actually talked about homotopy. When we talked about […]<\/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":true,"_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-330","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/s4lzZS-homotopy","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/330","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=330"}],"version-history":[{"count":1,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/330\/revisions"}],"predecessor-version":[{"id":3319,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/330\/revisions\/3319"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=330"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=330"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=330"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}