{"id":32,"date":"2018-10-22T10:47:49","date_gmt":"2018-10-22T14:47:49","guid":{"rendered":"http:\/\/sites.nd.edu\/jacob-landgraf\/?p=32"},"modified":"2018-10-22T10:54:13","modified_gmt":"2018-10-22T14:54:13","slug":"a-new-problem","status":"publish","type":"post","link":"https:\/\/sites.nd.edu\/jacob-landgraf\/2018\/10\/22\/a-new-problem\/","title":{"rendered":"A new problem"},"content":{"rendered":"<p>Consider a positively weighted sphere system \\(S\\) in \\(M_n = \\#_n S^1 \\times S^2\\). By Poincar\u00e9 duality, we can consider \\([S] \\in H_2(M_n)\\) as an element of \\(H^1(M_n) = H^1(F_n)\\) (where \\(F_n\\) is the free group of rank \\(n\\)). Fix some \\(\\alpha \\in H^1(F_n)\\). Then we can define the <em>cocycle complex<\/em> \\(X_\\alpha\\) to be the collection of weighted sphere systems representing \\(\\alpha\\) under this identification. <\/p>\n<p>As for the cycle complex, the question now becomes: is the cocycle complex contractible?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider a positively weighted sphere system \\(S\\) in \\(M_n = \\#_n S^1 \\times S^2\\). By Poincar\u00e9 duality, we can consider \\([S] \\in H_2(M_n)\\) as an element of \\(H^1(M_n) = H^1(F_n)\\) (where \\(F_n\\) is the free group of rank \\(n\\)). Fix some \\(\\alpha \\in H^1(F_n)\\). Then we can define the cocycle complex \\(X_\\alpha\\) to be the &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/sites.nd.edu\/jacob-landgraf\/2018\/10\/22\/a-new-problem\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;A new problem&#8221;<\/span><\/a><\/p>\n","protected":false},"author":2937,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-32","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/sites.nd.edu\/jacob-landgraf\/wp-json\/wp\/v2\/posts\/32","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.nd.edu\/jacob-landgraf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.nd.edu\/jacob-landgraf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/jacob-landgraf\/wp-json\/wp\/v2\/users\/2937"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/jacob-landgraf\/wp-json\/wp\/v2\/comments?post=32"}],"version-history":[{"count":5,"href":"https:\/\/sites.nd.edu\/jacob-landgraf\/wp-json\/wp\/v2\/posts\/32\/revisions"}],"predecessor-version":[{"id":37,"href":"https:\/\/sites.nd.edu\/jacob-landgraf\/wp-json\/wp\/v2\/posts\/32\/revisions\/37"}],"wp:attachment":[{"href":"https:\/\/sites.nd.edu\/jacob-landgraf\/wp-json\/wp\/v2\/media?parent=32"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.nd.edu\/jacob-landgraf\/wp-json\/wp\/v2\/categories?post=32"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.nd.edu\/jacob-landgraf\/wp-json\/wp\/v2\/tags?post=32"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}