Due Friday, October 2, in class<\/p>\n
Read exercises 2.9.1 – 2.9.8 (it walks you through the classification of 1-manifolds)<\/p>\n
Do exercises 2.1.1, 2.1.5, 2.1.10, 2.2.5, 2.9.18, 2.9.20, 2.9.22, and A, B, and C below<\/p>\n
I will select some problems from those to grade. Please ask me about any questions you have on the problems that are not graded.<\/p>\n
A. Let \\(Q = \\mathbb{R}^3 – \\{0\\} \/ \\sim\\) and \\(P = S^2\/ \\sim’\\) where nonzero \\(x,y \\in \\mathbb{R}^3\\) are equivalent (\\(x\\sim y\\)) if there is a nonzero \\(\\lambda \\in \\mathbb{R}\\) such that \\(x = \\lambda\\,y\\), and \\(w, z\\in S^2\\) are equivalent (\\(w\\sim’ z\\)) if \\(w = -z\\). (Give \\(Q\\) and \\(P\\) the quotient topologies). Construct a homeomorphism \\(Q\\approx P\\). We call \\(Q\\approx P\\) the real projective plane<\/em> and denote it \\(\\mathbb{R}P(2)\\).<\/p>\n B. Recall the handle decomposition of the M\u00f6bius band \\(M\\) from class: it had one 0-handle and one 1-handle with a half twist. The boundary of \\(M\\) is a circle; let \\(f:S^1 \\to \\partial M\\) be a homeomorphism. Attach a 2-handle \\(h^2 = D^2 \\times D^0\\) to \\(M\\) by a homeomorphism \\(g: \\partial D^2 \\times D^0 \\to \\partial M\\). Here I am thinking of \\(\\partial D^2\\times D^0\\) as identical to \\(S^1\\) and so \\(g\\) is really \\(f\\). Show that the handlebody \\(M\\cup h^2\\) is homeomorphic to \\(\\mathbb{R}P(2)\\).<\/p>\n C. Prove \\(\\mathbb{R}P(2)\\) is a connected, compact surface without boundary.<\/p>\n Hint for B:<\/strong><\/p>\n Let \\(D^2\\) have the equivalence relation \\(x\\sim y\\) if \\(x=-y\\) and both \\(x,y\\) are on the boundary of \\(D^2\\).<\/p>\n Prove \\(D^2\/\\sim \\approx P\\) from A.<\/p>\n Think of an annular neighborhood of the boundary of \\(D^2\\). It is a subset of \\(D^2\\) whose complement is a disc. If you can show the neighborhood of the boundary maps to a M\u00f6bius band in \\(D^2\/\\sim\\), you’re done! Why? So show it’s a M\u00f6bius band and be done.<\/p>\n","protected":false},"excerpt":{"rendered":" Due Friday, October 2, in class Read exercises 2.9.1 – 2.9.8 (it walks you through the classification of 1-manifolds) Do exercises 2.1.1, 2.1.5, 2.1.10, 2.2.5, 2.9.18, 2.9.20, 2.9.22, and A, B, and C below I will select some problems from those to grade. Please ask me about any questions you have on the problems that […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[2],"tags":[],"_links":{"self":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2015mat661\/wp-json\/wp\/v2\/posts\/24"}],"collection":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2015mat661\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2015mat661\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2015mat661\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2015mat661\/wp-json\/wp\/v2\/comments?post=24"}],"version-history":[{"count":9,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2015mat661\/wp-json\/wp\/v2\/posts\/24\/revisions"}],"predecessor-version":[{"id":36,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2015mat661\/wp-json\/wp\/v2\/posts\/24\/revisions\/36"}],"wp:attachment":[{"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2015mat661\/wp-json\/wp\/v2\/media?parent=24"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2015mat661\/wp-json\/wp\/v2\/categories?post=24"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pdhorn.expressions.syr.edu\/fall2015mat661\/wp-json\/wp\/v2\/tags?post=24"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}