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Kono A., Tamaki D. — Generalized Cohomology
Kono A., Tamaki D. — Generalized Cohomology

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Название: Generalized Cohomology

Авторы: 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: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 272

Добавлена в каталог: 23.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$E^r$-term      117
$\hat{A}$-genus      196
$\lambda$-ring      79
$\lambda$-semiring      79
$\mathbb{P}$-local      17
$\mathbb{P}$-localization      17
$\Omega$-spectrum      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, $\hat{A}$-      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, $\Omega$-      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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