{"id":79,"date":"2018-06-25T16:57:58","date_gmt":"2018-06-25T16:57:58","guid":{"rendered":"https:\/\/mathlab.io\/?p=79"},"modified":"2018-06-25T16:59:57","modified_gmt":"2018-06-25T16:59:57","slug":"spectral-sequences-ii","status":"publish","type":"post","link":"https:\/\/automathon.org\/index.php\/2018\/06\/25\/spectral-sequences-ii\/","title":{"rendered":"Spectral Sequences II"},"content":{"rendered":"

Two Stripes<\/h2>\n

The next thing to address in McCleary<\/a> is an apparent mistake on p. 9 of section 1.2. Here we again assume a first quadrant spectral sequence converging to a graded vector space \"H^*\". This is mentioned at the beginning of the section, but it’s easy to forget that when a bold-faced and titled example (1.D) seems to be presenting a reset of assumptions, rather than building upon prior discussion. Furthermore, in this example, McCleary seems to be working again with the assumption from example 1.A that \"F^p+kH^p=0\" for \"k>0\". On the other hand, this can be seen as a consequence of the fact that our spectral sequence is limited to the first quadrant, provided the filtration is finite<\/span> in the sense of Weibel’s Homological Algebra<\/em>, p. 123 (\"F^mH^s=0\" for some \"m\"). But then it would be unclear why McCleary took this as an additional assumption rather than as a consequence of prior assumptions in the first case. : \/<\/p>\n

The new part of this example is the assumption that \"E^{p,q}_2=0\" unless \"q=0\" or \"q=n\", so all terms of the spectral sequence are to be found just in two horizontal stripes. In particular \"E_\infty^{p,q}\" is only possibly non-zero in these stripes, and since these correspond to filtration quotients, the filtration takes a special form.<\/p>\n

First, we might look at the filtration on \"H^s\" where \"0\leq s \leq n-1\".\u00a0 Note that the spectral sequence terms that give information about \"H^s\" are those along the diagonal line where \"p+q=s\".\u00a0 Since \"s\leq n-1\", the only place where anything interesting might happen is when this line crosses the \"p\"-axis, i. e. when \"q=0\". This forces \"p=s\", so the only possible nonzero filtration quotient is<\/p>\n

  <\/span>   <\/span>\"\[E_\infty^{s,0}=F^sH^s/F^{s+1}H^s=F^sH^s=F^0H^s=H^s \]\"<\/p>\n

working with the assumption that \"F^{s+1}H^s=0\". So on the one hand, we get no interesting filtration of \"H^s\" for \"s<n\", but on the other hand we can see exactly what it is from the spectral sequence limit.<\/p>\n

Now we treat the case of \"H^{n+p}\", where \"p\geq 0\". I find this awkward notation again, preferring to reserve \"p\" for a pure arbitrary spectral sequence index, but since we are trying to address the mistake in this notation, we should keep it for now. The filtration of this vector space\/cohomology is interesting when \"q=n\" and \"q=0\", where the quotients are given by<\/p>\n

  <\/span>   <\/span>\"\[E^{p,n}_\infty=F^pH^{p+n}/F^{p+1}H^{p+n}\quad\text{and}\quad E^{p+n,0}_\infty=F^{p+n}H^{p+n}/0.\]\"<\/p>\n

Every where else, successive quotients are 0, meaning the filtration looks like…<\/p>\n

  <\/span>   <\/span>\"\[0\sus F^{n+p}H^{n+p}= \dots =F^{p+1}H^{n+p} \sus F^pH^{n+p}=\dots=F^1H^{n+p}\sus H^0{n+p}\]\"<\/p>\n

In the filtration on page 9, McCleary puts one of the (possibly) non-trivial quotients at \"F^nH^{n+p}\" instead of at \"F^pH^{n+p}\" where it should be.\u00a0 That’s all I’m saying.<\/p>\n

This situation is modeled on a spectral sequence for sphere bundles <\/span> i.e. bundles where the fibers are spheres of a given dimension. The stripes coincide with the fact that a sphere \"\mathbb{S}^n\" has nontrivial cohomology only at \"H^n\" and \"H^0\". This sort of computation is famous enough that it has a name: the Thom-Gysin sequence (or just Gysin sequence).<\/a><\/p>\n

As a final remark on section 1.2, McCleary says that the sequence in example 1.C is the Gysin sequence. Example 1.C doesn’t exist, we mean example 1.D : )<\/p>\n","protected":false},"excerpt":{"rendered":"

Two Stripes The next thing to address in McCleary is an apparent mistake on p. 9 of section 1.2. Here we again assume a first quadrant spectral sequence converging to a graded vector space . This is mentioned at the beginning of the section, but it’s easy to forget that when a bold-faced and titledRead more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[9],"tags":[6,7],"class_list":["post-79","post","type-post","status-publish","format-standard","hentry","category-spectral-sequences","tag-mccleary","tag-specseqs"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/automathon.org\/index.php\/wp-json\/wp\/v2\/posts\/79","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/automathon.org\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/automathon.org\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/automathon.org\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/automathon.org\/index.php\/wp-json\/wp\/v2\/comments?post=79"}],"version-history":[{"count":6,"href":"https:\/\/automathon.org\/index.php\/wp-json\/wp\/v2\/posts\/79\/revisions"}],"predecessor-version":[{"id":85,"href":"https:\/\/automathon.org\/index.php\/wp-json\/wp\/v2\/posts\/79\/revisions\/85"}],"wp:attachment":[{"href":"https:\/\/automathon.org\/index.php\/wp-json\/wp\/v2\/media?parent=79"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/automathon.org\/index.php\/wp-json\/wp\/v2\/categories?post=79"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/automathon.org\/index.php\/wp-json\/wp\/v2\/tags?post=79"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}