m:... m:sil... (m:, m:~).. a B, aa' a(8,). Ab, Ab t B"(O), B(G) B2P (11') 0(0, Z)... Oomp, OomPo. drp 8rp d*. d' List of notation

Transcription

1 Bibliography [1] Atiyah, M. F., Characters and cohomology of finite groups, Inst. des Hautes Etudes Scient. Publ. Math. 9 (1961), [2] Atiyah, M. F., and T. C. Wall, Cohomology of groups, Proceedings on Algebr. Number Theory, J. W. Cassels and A. Frohlich ed.; Thompson Book Co., Washington, D. C., 1967, [3] Borel, A., Sur la cohomologie des espaces fibres principaux et des espaces homogimes de groupes de Lie compacts, Ann. Math. 54 (1953), L4] Borel, A. (ed.), Seminar on Transformation Groups, Annals of Math. Studies No. 46, Princeton University Press, Princeton [5] Bourbaki, N., Elements de MatMmatique, Livre II, Chap. 3 (Algebre multilineaire), 2me Cd. 1958; Livre II, Chap. 2 (Algebre lineaire), 3me Cd. 1962, Hermann, Paris. [6] Bourbaki, N., Elements de MatMmatique, Livre II, Chap. 6 (Modules sur les anneaux principaux), Hermann, Paris [7] Bourgin, N., Modern Algebraic Topology, MacMillan Co., New York [8] Bredon, G. E., Sheaf Theory, McGraw-Hill, New York [9] Cartan, H., La transgression dans un groupe de Lie et dans un espace fibre principal, Colloque de Topologie, Bruxelles 1950, [10] Cartan, H., Algebres d'eilenberg-maclane et Homotopie, Seminaire Henri Cartan, 7e annee 1954/55, Paris. [11] Cartan, H., and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton [12] Dold, A. E., and R. K. Lashof, Principal quasifibrations and fibre homotopy equivalences; Ill. J. Math. 3 (1959), [13] Eilenberg, S., and G. M. Kelly, Closed categories, Proc. Conf. on Categorical Algebra, La Jolla 1965, Springer-Verlag, New York 1966.

2 230 Bibliography [14] Eilenberg, S., and S. MacLane, On the groups H(ll, n) I, II, III, Ann. of Math. 58 (1953), ; 60 (1954), ; 60 (1954), [15] Eilenberg, S., and J. C. Moore, Homological algebra and fibrations, Colloque de Topologie, Bruxelles 1964, [16] Eilenberg, S., and J. C. Moore, Homological algebra and fibrations I, Comment. Math. Helv. 40 (1966), [17] Evens, L., The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), ; Erratum: ibid 102 (1922), 545. [18] Fuchs, L., Abelian Groups, Pergamon Press, New York [19] Hofmann, K. H., Introduction to the Theory of Compact Groups I, Lecture Notes, Tulane University, [20] Hofmann, K. H., Categories with convergence, exponential functors, and the cohomology of compact abelian groups, Math. Z. 104 (1968), [21] Hofmann, K. H., Tensorprodukte lokal kompakter abelscher Gruppen, J. reine angew. Math. 216 (1964), [22] Hofmann, K. H., Der Schur'sche Multiplikator topologischer Gruppen, Math. Z. 79 (1962), [23] Hofmann, K. H., and P. S. Mostert, Elements of Compact Semigroups, Chas. E. Merrill Books, Columbus [24] Hofmann, K. H., and P. S. Mostert, The cohomology of compact abelian groups, Bull. Amer. Math. Soc. 74 (1968), [25] Hofmann, K. H., and P. S. Mostert, About the cohomology ring of a finite abelian group, Bull. Amer. Math. Soc. 75 (1969), [26] Hu, S. T., Homotopy Theory, Academic Press, New York [27] Huppert, B., Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg New York [28] Husemoller, D., Fibre Bundles, McGraw-Hill, New York [29] Koszul, J. L., Sur un type d'algebres differentielles en rapport avec la transgression, Colloque de Topologie, Bruxelles 1950, (30] Lang, S., Rapport sur la Cohomologie des Groupes, W. A. Benjamin, New York [31] Leray, J., L'anneau spectral et l'anneau filtre d'homologie d'un espace localement compact et d'une application continue, J. Math. Pure Appl. 29 (1950), [32] MacLane, S., Homology, Springer-Verlag, Berlin-Gottingen-Heidelberg [33] MacLane, S., Categorical algebra, Bull. Amer. Math. Soc. 71 (1965), [34] McCord, M. E., Classifying spaces and infinite symmetric products, Trans. Amer. Math. Soc. 146 (1969), [35] Mitchell, B., Theory of Categories, Academic Press, New York [36] Moscowitz, M., Homological algebra in locally compact abelian groups, Trans. Amer. Math. Soc. 127 (1967),

3 Bibliography 231 [37] Palermo, F. P., The cohomology ring of product complexes, 'l'rans. Amer. Math. Soc. 86 (1957), [38] Petrie, T., The Eilenberg-Moore, Rothenberg-Steenrod spectral sequence for K-theory, Proc. Amer. Math. Soc. 19 (1968), [39] Priddy, S., Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), [40] Rothenberg, M., and N. E. Steenrod, The cohomology of classifying spaces of H-spaces. Bull. Amer. Math. Soc. 71 (1965), [41] Shafer, T., On the homology ring of an abelian group, Dissertation, University of Chicago [42] Spanier, E. H., Algebraic Topology, McGraw-Hill, New York [43] Steenrod, N., Milgram's classifying space of a topological group, Topology '1 (1968), [44] Steenrod, N., and D. B. A. Epstein, Cohomology Operations, Ann. of Math. Studies, No. 50, Princeton [45] Williams, R. F., The construction of certain O-dimensional transformation groups, Trans. Amer. Math. Soc. 129 (1967),

4 List of notation m:.... m:sil... (m:, m:~).. a B, aa' a(8,). Ab, Ab t B"(O), B(G) B2P (11') 0(0, Z)... Oomp, OomPo. drp 8rp d*. d' ~. D. D/. E t, E2 (exponential functors) E" E 2, Ea, E, (11'), E 2 (1P), Ea(lP) E:(IP), E:(IP) E 2 (ip). E;(1P1, E;(IP) E,[IP], E,{IP} E(f) EfI(G), E(G) E~, E~I, e~, e~i. f.... V-1, V-4 IV-1 after IV , V V-2 IV-1.5, V-1, V , IV-1, V II , , 2.26, 2.39, , 2.29, 2.30, 2.38, II-3.1 if.

5 List of notation 233 j,l. F.. f!. (p). G,G. G t, G1 Go.. B a1g B Slng BopJ~, BOpJab ~ 11,(-, -) i..~, la. Kanj(-) Lie, Lieo Liei.. LIM Lim.. Lim, Lim M.... IDl.... IDl(A), IDl (m i ) [p].. p... P (Ri' IX) Q.»t. R. R/Z 8. 8(q) 8 (Ri' IX) (8, T) (comma category) - x.. X* Y X(tI) Zi Z.. Z(m). Zp Z(pOO) a,p 'Y II-3.8 after II-2.1 II-3.9 II-3.8 V-5.4 IV-2.6 V-1.10 V-1.9 IV-2 II-1.9ff., IV IV-1.13 IV-1.5 V-2.1 IV-1 IV-1 IV -2.8 (preceding) II-3,IV ,1-2.4,1-3.3, II-2.1 IV-2 III the rationals reals reals mod integers V IV after II-2.1 II-1.1 II-1.1 II-2.1 integers cyclic group (ring) of order m (ring) of p-adic integers Priifer group for prime p 1-1.2,

6 2M List of llotatwi!l. F(G). o.. 0, Of.,1*,,1* 'fj "). ltnex) p.. fjj rp*, rp~, rp*' Ii "P. "Pr;.,p ~.. (Ii, O'[;n L'(q) 'l" W 1\. h (character group). L (annihilator group). * (opposite LdualJ category). <:.... TheoFem II (II -1) 1-3.1, II-2.1 II ift' ft' , 2.21, 3.3, IV-2 II-1.5 Theorem I (1-2) IV , 2.44, , III-3.12 after V t , ff., , 2.37, , III ,.2.6 after V-4.4 II II-2.13 II-2.13 I-1,IV V-5.4 III-2.1, IV-1.1 VI

7 Index bar resolution 124 basis (of an R-module) 50 Bockstein diagram 87 - differential on h (G, R/aR) 166 classifying space up to n 158 coexponent 129 cofibration 161 comma category 177 commutative Hopf algebra 189 compatible functor natural transformation 26 completeness of a category 174 coretraction 81 cup product (on the co chain level) 104 derived limit functor 161 divided powers 37 Dold-Lashof construction 158 ~-complete 174 'I)-continuous 176 'I)-dense range 180 'I)-extension 181 edge algebra 44 - morphism 44 elementary morphism 49 exponent 129 exponential functor 23, 25 extendable functor 181 exterior algebra 1\ 25 free action of mono ids 225 freely generated category 194 functor of Hopf algebras 189 generating degree 139 geometric realization 167 integer 48 integral element 48 - group ring 108 join 156 Kan extension 185 Milnor spectrum of universal (classifying) spaces for G 158 monoidal category 24 - functor 25

8 236 Index multiplicative category 23 - functor 25 n-model167 orbit 224 Poincare series 132 polynomial algebra with divided powers 37 pre-bockstein diagramm 87 projective resolution 98 retraction 81 simplicial mapping object 167 spectral algebra 43 spectrum of classifying spaces for G of universal spaces for G 157 standard Bockstein diagram 87 - resolution 78 strictly ;I)-dense category ;I)-dense range 184 subadditive functor 26 submultiplicative functor 26 symmetric algebra P 25 - divided algebra 38 telescope 159 universal space up to n 157 weakly principal ideal ring 48 z-constituent 130