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Kono A., Tamaki D. — Generalized Cohomology
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Название: Generalized CohomologyАвторы: Kono A., Tamaki D. Аннотация: In the 1950s, Eilenberg and Steenrod presented their famous characterization of homology theory by seven axioms. Somewhat later, it was found that keeping just the first six of these axioms (all except the condition on the "homology" of the point), one can obtain many other interesting systems of algebraic invariants of topological manifolds, such as $K$-theory, cobordisms, and others. These theories come under the common name of generalized homology (or cohomology) theories. The purpose of the book is to give an exposition of generalized (co)homology theories that can be read by a wide group of mathematicians who are not experts in algebraic topology. It starts with basic notions of homotopy theory and then introduces the axioms of generalized (co)homology theory. Then the authors discuss various types of generalized cohomology theories, such as complex-oriented cohomology theories and Chern classes, $K$-theory, complex cobordisms, and formal group laws. A separate chapter is devoted to spectral sequences and their use in generalized cohomology theories. The book is intended to serve as an introduction to the subject for mathematicians who do not have advanced knowledge of algebraic topology. Prerequisites include standard graduate courses in algebra and topology, with some knowledge of ordinary homology theory and homotopy theory.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2006Количество страниц: 272Добавлена в каталог: 23.05.2008Операции: Положить на полку |
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117 196 79 79 17 17 19 35 225 Adams conjecture 89 102 Adams operation 79 81 Adams spectral sequence 153 Adams — Novikov spectral sequence 159 Additivity axiom 30 Algebraic K-theory 105 114 Algebraic K-theory, higher 110 Atiyah — Hirzebruch spectral sequence 164 Bar-type Eilenberg — Moore spectral sequence 166 Base space 7 Becker — Gottlieb transfer 90 92 Blakers — Massey theorem 12 Bott periodicity 20 73 Brown functor 33 Brown — Peterson spectrum 199 BU 20 Canonical line bundle 46 49 Cap product 41 Cell complex 4 Cellular approximation theorem 5 Cellular decomposition 4 Change-of-rings isomorphism 184 CHARACTER 96 Characteristic class 43 Chern character 71 Chern class 54 Chern class, total 55 Classifying map 51 54 67 Classifying space 16 165 Classifying space of a Brown functor 34 Cobar-type Eilenberg — Moore spectral sequence 174 Cobordant, framed 21 Codegeneracy operator 207 Coefficient group 26 Coface operator 207 Cofi bration 3 Cofi bration of cosimplicial space 213 Cofi bration of spectrum 228 Cofiber of map of spectra 228 Cofibrant 161 Cohomology spectral sequence of filtered space 129 Cohomology spectral sequence of simplicial space 161 Cohomology theory 23 Cohomology theory, axioms for 23 Cohomology theory, complex-oriented 46 Cohomology theory, multiplicative 40 Cohomology theory, ordinary 26 Cohomology theory, reduced 27 Cohomotopy group 210 Colimit 217 Compactly generated 1 complete 139 Completion, E-nilpotent 157 Complex cobordism 62 Complex genus 196 Complex K-group 67 Complex K-theory 67 Complex K-theory, connective 77 Complex K-theory, periodic 77 Complex orientable 47 Complex orientation 47 Conner-Floyd isomorphism 195 Converge 140 Converge, conditionally 141 Converge, strongly 140 Converge, weakly 140 Convergence 139 Core 157 Cosimplicial object 207 Cosimplicial space 207 213 Coskeleton 213 Covering homotopy property 6 Covering homotopy property for spectra 228 Cross product 40 Cup product 40 CW-complex 4 CW-pair 23 Degeneracy operator 207 Delooping 38 Derived couple 137 Differential of spectral sequence 117 Dimension axiom 26 DR pair 2 E-nilpotent completion 157 Eilenberg — Mac Lane space 17 Eilenberg — Mac Lane spectrum 37 Elliptic cohomology 198 ' Elliptic genus 197 Etale homotopy theory 114 Euler characteristic 91 Euler class 65 Exact couple 135 Exact couple, Expanded form of 137 Exactness axiom 24 28 Excision axiom 25 Exhaust 139 Face operator 207 Fat wedge 27 Fiber 7 Fiber homotopy equivalent 87 Fiber of map of spectra 228 Fiberwise join 87 Fibrant, cosimplicial space 168 214 Fibrant, topological space 223 Fibration 6 Fibration of cosimplicial space 213 Fibration of spectrum 228 Fibration, Hurewicz 6 Fibration, locally trivial 6 Fibration, Serre 6 Filtration 119 Formal group law 190 Formal group law, additive 191 Formal group law, homomorphism of 191 Formal group law, isomorphism of 192 Formal group law, multiplicative 191 Formal group law, strict isomorphism of 192 Frame 112 Framed submanifold 21 Framing 21 Freudenthal suspension map 12 Freudenthal suspension theorem 11 Functor, homotopy 33 Functor, representable- 33 Generalized cohomology theory 23 Generalized homology theory 26 Genus 196 Genus, 196 Genus, complex 196 Genus, elliptic 197 Genus, Todd 196 Geometric bar construction 165 Geometric cobar construction 174 Geometric realization 160 222 Grothendieck construction 80 Grothendieck group 67 Gysin sequence 64 Hausdorff filtration 139 Hausdorff, weak 1 Homology spectral sequence of cosimplicial space 173 Homology spectral sequence of filtered space 125 Homology spectral sequence of simplicial space 161 Homology theory 26 Homology theory, generalized 26 Homotopical algebra 117 Homotopy associative 15 Homotopy axiom 25 28 Homotopy colimit 126 221 Homotopy excision 12 Homotopy extension property 2 Homotopy extension property for spectra 228 Homotopy functor 33 Homotopy lifting problem 6 Homotopy limit 221 Homotopy set 11 Homotopy spectral sequence 149 Homotopy spectral sequence of cosimplicial space 169 Homotopy spectral sequence of fibration tower 149 Hopf algebra 14 Hopf algebroid 154 Hopf space 13 Hurewicz map 9 Hurewicz Theorem 9 Ind uced homomorphism 24 Index theorem 77 Infinite cycle 124 Infinite loop machine 110 Infinite loop space 19 38 J-equivalent 87 J-homomorphism 87 J.H.C.Whitehead Theorem 10 K-theory 20 67 K-theory, Morava 177 199 KO-group 77 L-spectrum 231 Landweber Exact Functor Theorem 196 Lazard ring 193 Leray — Hirsh theorem 43 Leray — Serre spectral sequence 75 132 167 LIMIT 217 localization 17 Logarithm 192 Loop space 11 Mapping cylinder 3 Mapping track 8 Maximal torus 96 Mayer — Viet oris exact sequence 25 Mayer — Vietoris axiom 33 Milnor exact sequence 31 Mittag — Leffler exact sequence 219 Model category 117 Model category of cosimplicial spaces 214 Model category, Strom 4 Monoid 67 Monoid, commutative 67 Monoid, topological 16 Moore complex 209 Morava K-theory 177 199 Multiplication of cohomology theory 39 Multiplication of spectrum 40 Multiplication, external 40 Multiplication, internal 40 Natural transformation of cohomology theories 30 ND R structure 2 NDR pair 2 NDR representation 2 Newton's formula 70 Nondegenerate base point 12 Operad 230 Pairing of cohomology theories 39 Pairing of spectra 40 Permanent cycle 124 Plus construction 108 Pontrjagin — Thom construction 21 Postnikov decomposition 148 Prespectrum 226 primitive 14 Quaternionic vector bundle 65 Reduced homology theory 28 Reduced suspension 27 Representable functor 33 Representation 95 Ring 96 Ring spectrum 23 236 Ring, induced 97 S-algebra 236 S-module 234 Simplicial group 206 Simplicial object 205 Simplicial set 206 Simplicial space 160 206 Skeletal filtration 4 160 Skeleton 4 Skeleton of cosimplicial space 213 Smash product 11 Smash product of pointed spaces 11 Smash product of spectra 152 225 231 Smash product of spectrum and space 227 Smash product, external 229 Solid ring 157 Spectral sequence 115 Spectral sequence with entering differential 143 Spectral sequence with exiting differential 143 Spectral sequence, Adams 153 Spectral sequence, Adams — Novikov 159 Spectral sequence, Atiyah — Hirzebruch 164 Spectral sequence, bar-type Eilenberg — Moore 166 Spectral sequence, co bartype Eilenberg — Moore 174 Spectral sequence, convergence of 139 Spectral sequence, homotopy 149 169 Spectral sequence, Leray — Serre 75 132 167 Spectral sequence, unstable Adams 159 172 Spectrum 23 35 225 Spectrum in the sense of EKMM 226 Spectrum, 19 35 225 Spectrum, Brown — Peterson 199 Spectrum, Eilenberg — MacLane 37 Spectrum, L- 231 Spectrum, sphere 37 227 Spectrum, suspension 37 227 Sphere spectrum 37 227 Splitting principle 48 Stable cohomology operation 30 Stable homotopy category 229 Stable homotopy group 38 Stably equivalent 69 Steenrod algebra 182 Steenrod algebra, dual of 182 Suspension 11 Suspension axiom 28 Suspension spectrum 37 Suspension, reduced 27 Suspension, spectrum 227 Telescope 222 Thorn class 45 Thorn class for quaternionic vector bundle 66 Thorn complex 22 Thorn isomorphism 45 60 Todd genus 196 Topological monoid 16 TOT 168 Total Chern class 55 Total space 7 Totalization functor 168 222 Tower of fibrations 147 Tower of fibrations for fibrant cosimplicial space 168 TRANSFER 90 92 Triad 25 Triple 25 Unstable Adams spectral sequence 159 Weak equivalence 5 Weak equivalence of cosimplicial space 213 Weak equivalence of L-spectrum 232
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