{"title":"Homotopic Hopf-Galois extensions: foundations and examples","authors":"K. Hess","doi":"10.2140/GTM.2009.16.79","DOIUrl":"https://doi.org/10.2140/GTM.2009.16.79","url":null,"abstract":"Hopf‐Galois extensions of rings generalize Galois extensions, with the coaction of a Hopf algebra replacing the action of a group. Galois extensions with respect to a group G are the Hopf‐Galois extensions with respect to the dual of the group algebra of G . Rognes recently defined an analogous notion of Hopf‐Galois extensions in the category of structured ring spectra, motivated by the fundamental example of the unit map from the sphere spectrum to MU . This article introduces a theory of homotopic Hopf‐Galois extensions in a monoidal category with compatible model category structure that generalizes the case of structured ring spectra. In particular, we provide explicit examples of homotopic Hopf‐Galois extensions in various categories of interest to topologists, showing that, for example, a principal fibration of simplicial monoids is a homotopic Hopf‐ Galois extension in the category of simplicial sets. We also investigate the relation of homotopic Hopf‐Galois extensions to descent. 16W30, 55U35; 13B05, 55P42, 57T05, 57T30","PeriodicalId":115248,"journal":{"name":"Geometry and Topology Monographs","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125192911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The spectra ko and ku are not Thom spectra: an approach using THH","authors":"V. Angeltveit, Michael Hill, T. Lawson","doi":"10.2140/GTM.2009.16.1","DOIUrl":"https://doi.org/10.2140/GTM.2009.16.1","url":null,"abstract":"The construction of various bordism theories as Thom spectra served as a motivating example for the development of highly structured ring spectra. Various other examples of Thom spectra followed; for instance, various Eilenberg–MacLane spectra are known to be constructed in this way (see Mahowald [5]). However, Mahowald [6] proved that the connective K–theory spectra ko and ku are not the 2–local Thom spectra of any vector bundles, and that the spectrum ko is not the Thom spectrum of a spherical fibration classified by a map of H-spaces. Rudyak [7] later proved that ko and ku are not Thom spectra p–locally at odd primes p .","PeriodicalId":115248,"journal":{"name":"Geometry and Topology Monographs","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114549300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On excess filtration on the Steenrod algebra","authors":"A. Yamaguchi","doi":"10.2140/gtm.2007.10.423","DOIUrl":"https://doi.org/10.2140/gtm.2007.10.423","url":null,"abstract":"In this note, we study some properties of the filtration of the Steenrod algebra defined from the excess of admissible monomials. We give several conditions on a cocommutative graded Hopf algebra A which enable us to develop the theory of unstable A ‐modules. 55S10","PeriodicalId":115248,"journal":{"name":"Geometry and Topology Monographs","volume":"250 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115589701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Milnor operations and the generalized Chern character","authors":"T. Torii","doi":"10.2140/gtm.2007.10.383","DOIUrl":"https://doi.org/10.2140/gtm.2007.10.383","url":null,"abstract":"We have shown that the n‐th Morava K ‐theory K .X/ for a CW‐spectrum X with action of Morava stabilizer group Gn can be recovered from the system of some height‐.nC 1/ cohomology groups E .Z/ with GnC1 ‐action indexed by finite subspectra Z . In this note we reformulate and extend the above result. We construct a symmetric monoidal functor F from the category of E _ .E/‐precomodules to the category of K .K/‐comodules. Then we show that K .X/ is naturally isomorphic to the inverse limit of F.E .Z// as a K .K/‐comodule. 55N22; 55N20, 55S05 Dedicated to Professor Nishida on the occasion of his 60th birthday","PeriodicalId":115248,"journal":{"name":"Geometry and Topology Monographs","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129334106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"INTERACTIONS OF STRINGS AND EQUIVARIANT HOMOLOGY THEORIES","authors":"S. Okuyama, Kazuhisa Shimakawa","doi":"10.2140/gtm.2007.10.333","DOIUrl":"https://doi.org/10.2140/gtm.2007.10.333","url":null,"abstract":"We introduce the notion of the space of parallel strings with par- tially summable labels, which can be viewed as a geometrically constructed group completion of the space of particles with labels. We utilize this to con- struct a machinery which produces equivariant generalized homology theories from such simple and abundant data as partial monoids.","PeriodicalId":115248,"journal":{"name":"Geometry and Topology Monographs","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123852911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the homotopy groups of E.n/-local spectra with unusual invariant ideals","authors":"Hirofumi Nakai, K. Shimomura","doi":"10.2140/gtm.2007.10.319","DOIUrl":"https://doi.org/10.2140/gtm.2007.10.319","url":null,"abstract":"Let E.n/ and T.m/ for nonnegative integers n and m denote the Johnson‐Wilson and the Ravenel spectra, respectively. Given a spectrum whose E.n/ ‐homology is E.n/ .T.m//=.v1;:::;vn 1/, then each homotopy group of it estimates the order of each homotopy group of LnT.m/. We here study the E.n/‐based Adams E2 ‐term of it and present that the determination of the E2 ‐term is unexpectedly complex for odd prime case. At the prime two, we determine the E1 ‐term for .L2T.1/=.v1//, whose computation is easier than that of .L2T.1// as we expect. 55Q99","PeriodicalId":115248,"journal":{"name":"Geometry and Topology Monographs","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117324801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the E 1 -term of the gravity spectral sequence","authors":"D. Tamaki","doi":"10.2140/gtm.2007.10.347","DOIUrl":"https://doi.org/10.2140/gtm.2007.10.347","url":null,"abstract":"The author constructed a spectral sequence strongly converging to h_*(Omega^n Sigma^n X) for any homology theory in [Topology 33 (1994) 631-662]. In this note, we prove that the E^1-term of the spectral sequence is isomorphic to the cobar construction, and hence the spectral sequence is isomorphic to the classical cobar-type Eilenberg-Moore spectral sequence based on the geometric cobar construction from the E^1-term. Similar arguments can be also applied to its variants constructed in [Contemp Math 293 (2002) 299-329].","PeriodicalId":115248,"journal":{"name":"Geometry and Topology Monographs","volume":"213 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121846071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Noncoherence of some lattices in Isom.H n","authors":"M. Kapovich, L. Potyagailo, E. Vinberg","doi":"10.2140/gtm.2008.14.335","DOIUrl":"https://doi.org/10.2140/gtm.2008.14.335","url":null,"abstract":"We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic n-space for n greater than 3. For instance, every nonuniform arithmetic lattice in SO(n,1) is noncoherent, provided that n is at least 6.","PeriodicalId":115248,"journal":{"name":"Geometry and Topology Monographs","volume":"82 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121014662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Odd-primary homotopy exponents of compact simple Lie groups","authors":"D. M. Davis, S. Theriault","doi":"10.2140/gtm.2008.13.195","DOIUrl":"https://doi.org/10.2140/gtm.2008.13.195","url":null,"abstract":"We note that a recent result of the second author yields upper bounds for odd-primary homotopy exponents of compact simple Lie groups which are often quite close to the lower bounds obtained from v1-periodic homotopy theory. 1. Statement of results The homotopy p-exponent of a topological space X, denoted expp(X), is the largest e such that some homotopy group πi(X) contains a Z/p-summand. In work dating back to 1989, the first author and collaborators have obtained lower bounds for expp(X) for all compact simple Lie groups X and all primes p by using v1-periodic homotopy theory. Recently, the second author ([11]) proved a general result, stated here as Lemma 2.1, which can yield upper bounds for homotopy exponents of spaces which map to a sphere. In this paper, we show that these two bounds often lead to a quite narrow range of values for expp(X) when p is odd and X is a compact simple Lie group. Our first new result, which will be proved in Section 2, combines Lemma 2.1 with a classical result of Borel-Hirzebruch. Theorem 1.1. Let p be odd. a. If n < p + p, then expp(SU(n)) ≤ n− 1 + νp((n− 1)!). b. If n ≥ p + 1, then expp(SU(n)) ≤ n + p− 3 + (bn−2 p−1 c−p+2 2 ) . Here and throughout, νp(−) denotes the exponent of p in an integer, p is an odd prime, and bxc denotes the integer part of x . All spaces are localized at p. It is Date: January 17, 2006.","PeriodicalId":115248,"journal":{"name":"Geometry and Topology Monographs","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126074252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"String topology of Poincaré duality groups","authors":"Hossein Abbaspour, R. Cohen, Kate Gruher","doi":"10.2140/gtm.2008.13.1","DOIUrl":"https://doi.org/10.2140/gtm.2008.13.1","url":null,"abstract":"where the sum is taken over conjugacy classes of elements in G . In this paper we construct a multiplication on LG directly in terms of intersection products on the centralizers. This multiplication makes LG a graded, associative, commutative algebra. When G is the fundamental group of an aspherical, closed oriented n‐ manifold M , then .LG/ D HC n.LM/, where LM is the free loop space of M . We show that the product on LG corresponds to the string topology loop product on H .LM/ defined by Chas and Sullivan. 55P35; 20J06 Dedicated to Fred Cohen on the occasion of his 60th birthday","PeriodicalId":115248,"journal":{"name":"Geometry and Topology Monographs","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121681924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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