The Local Structure of Algebraic K-Theory

Algebraic K-theory.- Gamma-spaces and S-algebras.- Reductions.- Topological Hochschild Homology.- The Trace K --> THH.- Topological Cyclic Homology.- The Comparison of K-theory and TC.

[1]  I. Madsen,et al.  Algebraic K-Theory of Local Number Fields: The Unramified Case , 1996 .

[2]  W. Leavitt Modules without invariant basis number , 1957 .

[3]  J. Whitehead Simplicial Spaces, Nuclei and m‐Groups , 1939 .

[4]  R. Oliver Whitehead groups of finite groups , 1988 .

[5]  Bjørn Ian Dundas,et al.  Relative K-theory and topological cyclic homology , 1997 .

[6]  Ib Madsen,et al.  On the K-theory of finite algebras over witt vectors of perfect fields , 1997 .

[7]  L. Hesselholt,et al.  Topological Hochschild homology and the Bass trace conjecture , 2013, 1305.6438.

[8]  R. Ho Algebraic Topology , 2022 .

[9]  Ib Madsen,et al.  The cyclotomic trace and algebraic K-theory of spaces , 1993 .

[10]  W. Kallen Descent for theK-theory of polynomial rings , 1986 .

[11]  C. Weibel Mayer-vietoris sequences and mod P K-theory , 1982 .

[12]  P. M. Cohn,et al.  Some remarks on the invariant basis property , 1966 .

[13]  J. McClure,et al.  ON THE TOPOLOGICAL HOCHSCHILD HOMOLOGY OF bu, I , 1993 .

[14]  S. Schwede Spectra in model categories and applications to the algebraic cotangent complex , 1997 .

[15]  V. Snaith Towards the Lichtenbaum-Quillen conjecture concerning the algebraic K-theory of schemes , 1984 .

[16]  C. Weibel Meyer vietoris sequences and module structures on NK , 1981 .

[17]  J. Adams,et al.  On the groups J(X)—IV , 1965 .

[18]  Eduardo J. Dubuc,et al.  Kan Extensions in Enriched Category Theory , 1970 .

[19]  J. Adams,et al.  Stable homotopy and generalised homology , 1974 .

[20]  D. Yao A Devissage Theorem in Waldhausen K-Theory , 1995 .

[21]  F. Waldhausen Algebraic K-theory of spaces: a manifold approach , 1982 .

[22]  D. Quillen Spectral sequences of a double semi-simplical group , 1966 .

[23]  C. Soulé Rational K-theory of the dual numbers of a ring of algebraic integers , 1981 .

[24]  B. Shipley,et al.  Algebras and Modules in Monoidal Model Categories , 1998, math/9801082.

[25]  R. McCarthy Relative algebraic K-theory and topological cyclic homology , 1997 .

[26]  Kenneth S. Brown,et al.  Cohomology of Groups , 1982 .

[27]  B. Kostant,et al.  Differential Forms on Regular Affine Algebras , 1962 .

[28]  T. Dieck Orbittypen und äquivariante Homologie II , 1972 .

[29]  W. Dwyer,et al.  Algebraic and etale $K$-theory , 1985 .

[30]  S. Betley On stableK-theory and topological Hochschild homology , 1992 .

[31]  J. Rognes K4(Z) is the trivial group , 2000 .

[32]  W. Dwyer,et al.  AlgebraicK-theory eventually surjects onto topologicalK-theory , 1982 .

[33]  J.-L. Loday $K$-théorie algébrique et représentations de groupes , 1976 .

[34]  V. Angeltveit On the algebraic K-theory of Z/p^n , 2011 .

[35]  J. Silvester Introduction to algebraic K-theory , 1981 .

[36]  Jon P. May,et al.  Generalized Tate Cohomology , 1995 .

[37]  Daniel R. Grayson The K-theory of endomorphisms , 1977 .

[38]  B. Mazur,et al.  On the van Kampen theorem , 1966 .

[39]  C. Kassel K-théorie relative d’un idéal bilatère de carré nul: étude homologique en basse dimension , 1981 .

[40]  W. Dwyer,et al.  Homotopy Limit Functors on Model Categories and Homotopical Categories , 2005 .

[41]  I. Madsen,et al.  THE STABLE MAPPING CLASS GROUP AND STABLE HOMOTOPY THEORY , 2005 .

[42]  S. Smale,et al.  On the Structure of Manifolds , 1962 .

[43]  Ib Madsen,et al.  Algebraic K-theory and traces , 1995 .

[44]  C. Weibel,et al.  TWO-PRIMARY ALGEBRAIC K-THEORY OF RINGS OF INTEGERS IN NUMBER FIELDS , 1999 .

[45]  A. K. Bousfield,et al.  Homotopy theory of Γ-spaces, spectra, and bisimplicial sets , 1978 .

[46]  D. Sullivan,et al.  The Hauptvermutung book : a collection of papers of the topology of manifolds , 1996 .

[47]  R. McCarthy The cyclic homology of an exact category , 1994 .

[48]  J. Whitehead,et al.  Simple Homotopy Types , 1950 .

[49]  W. Dwyer,et al.  Pseudo-isotopy and invariant theory—I , 1980 .

[50]  Daniel R. Grayson Higher algebraic K-theory: II , 1976 .

[51]  D. Ravenel Life after the Telescope Conjecture , 1993 .

[52]  I. Madsen,et al.  Cyclic polytopes and the $K$-theory of truncated polynomial algebras , 1997 .

[53]  G. M. Kelly Coherence theorems for lax algebras and for distributive laws , 1974 .

[54]  Eric M. Friedlander,et al.  Filtrations on the Homology of Algebraic Varieties , 1994 .

[55]  J. Rognes,et al.  The Segal conjecture for topological Hochschild homology of complex cobordism , 2010, 1010.5635.

[56]  B. Shipley Symmetric Spectra and Topological Hochschild Homology , 2000 .

[57]  R. Thomason,et al.  Higher Algebraic K-Theory of Schemes and of Derived Categories , 1990 .

[58]  A. Blumberg,et al.  Uniqueness of the multiplicative cyclotomic trace , 2011, 1103.3923.

[59]  W. Thurston,et al.  Every connected space has the homology of a K(π,1) , 1976 .

[60]  A. Blumberg,et al.  Topological Hochschild homology of Thom spectra and the free loop space , 2008, 0811.0553.

[61]  R. Staffeldt On Fundamental Theorems of Algebraic K-Theory , 1989 .

[62]  D. Popescu Letter to the editor: General Néron desingularization and approximation , 1990, Nagoya Mathematical Journal.

[63]  Hyman Bass,et al.  Algebraic K-theory , 1968 .

[64]  D. Quillen Projective modules over polynomial rings , 1976 .

[65]  L. Hodgkin,et al.  The Homotopy Type of Two-regular K-theory , 2003 .

[66]  S. Lichtenbaum Values of zeta-functions, étale cohomology, and algebraic K-theory , 1973 .

[67]  J. Milnor WHITEHEAD TORSION BY , 2007 .

[68]  J. Jardine The K -theory presheaf of spectra , 2009 .

[69]  Jon P. May,et al.  Equivariant Stable Homotopy Theory , 1986 .

[70]  John Rognes,et al.  Algebraic K-theory of topological K-theory , 2002 .

[71]  Thomas H. Geisser,et al.  The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky , 2001 .

[72]  R. Thomason Algebraic $K$-theory and etale cohomology , 1985 .

[73]  A. Suslin Algebraic K-theory of fields , 1986 .

[74]  A. J. Casson,et al.  The Hauptvermutung Book , 1996 .

[75]  F. Waldhausen Algebraic K-theory of topological spaces. I , 1978 .

[76]  A. Suslin On the equivalence of K-theories , 1981 .

[77]  T. Pirashvili POLYNOMIAL-APPROXIMATION OF EXT AND TOR GROUPS IN FUNCTOR CATEGORIES , 1993 .

[78]  M. Karoubi,et al.  $K$-théorie algébrique et $K$-théorie topologique I. , 1971 .

[79]  Viorel Costeanu On the 2-typical de Rham-Witt complex , 2007, Documenta Mathematica.

[80]  U. Shukla A relative cohomology for associative algebras , 1964 .

[81]  James E. McClure,et al.  A solution of Deligne’s Hochschild cohomology conjecture. , 2000 .

[82]  F. Waldhausen Algebraic K-theory of spaces, concordance, and stable homotopy theory , 1987 .

[83]  B. Tsygan COMMUNICATIONS OF THE MOSCOW MATHEMATICAL SOCIETY: The homology of matrix Lie algebras over rings and the Hochschild homology , 1983 .

[84]  L. Illusie Complexe de de Rham-Witt et cohomologie cristalline , 1979 .

[85]  Jon P. May Simplicial objects in algebraic topology , 1993 .

[86]  S. A. Mitchell On the Lichtenbaum-Quillen Conjectures from a Stable Homotopy-Theoretic Viewpoint , 1994 .

[87]  F. Waldhausen,et al.  An un-delooped version of algebraic K-theory , 1992 .

[88]  T. Goodwillie Relative algebraic K-theory and cyclic homology , 1986 .

[89]  Ronnie Lee The group K3(Z) is cyclic of order forty-eight , 1976 .

[90]  I. Madsen,et al.  On the de Rham–Witt complex in mixed characteristic , 2004 .

[91]  R. M. Vogt,et al.  Topological Hochschild Homology , 2000 .

[92]  G. Higman,et al.  The Units of Group‐Rings , 1940 .

[93]  On the K-Theory of the Coordinate Axes in the Plane , 2005, Nagoya Mathematical Journal.

[94]  Symmetric spectra , 1998, math/9801077.

[95]  F. Quinn,et al.  E∞ ring spaces and E∞ ring spectra , 1977 .

[96]  C. Schlichtkrull,et al.  The cyclotomic trace for symmetric ring spectra , 2009, 0903.3495.

[97]  John D. S. Jones Cyclic homology and equivariant homology , 1987 .

[98]  J.-L. Loday Cohomologie et groupe de Steinberg relatifs , 1978 .

[99]  L. Hesselholt Witt vectors of non-commutative rings and topological cyclic homology , 1997 .

[100]  Pseudo-isotopy and invariant theory, II: Rational algebraic K-theory of a space with finite fundamental group , 1980 .

[101]  W. Dwyer,et al.  Function complexes in homotopical algebra , 1980 .

[102]  B. Dundas,et al.  Integral Excision for K-Theory , 2010, 1009.3044.

[103]  C. Weibel The 2-torsion in the K-theory of the integers , 1997 .

[104]  M. Larsen,et al.  TOPOLOGICAL HOCHSCHILD HOMOLOGY AND THE CONDITION OF HOCHSCHILD-KOSTANT-ROSENBERG , 2001 .

[105]  R. Kerner,et al.  Noncommutative differential geometry of matrix algebras , 1990 .

[106]  M. Brun Topological Hochschild homology of Z/pn , 2000 .

[107]  D. M. Kan,et al.  A COMBINATORIAL DEFINITION OF HOMOTOPY GROUPS , 1958 .

[108]  S. Ferry,et al.  Novikov Conjectures, Index Theorems and Rigidity , 1995 .

[109]  T. Goodwillie Cyclic homology, derivations, and the free loopspace☆ , 1985 .

[110]  Spectra and symmetric spectra in general model categories , 2000, math/0004051.

[111]  Vladimir Voevodsky,et al.  Motivic cohomology with Z/2-coefficients , 2003 .

[112]  J. Spaliński Strong homotopy theory of cyclic sets , 1995 .

[113]  S. Shostak Philosophy of Mathematics and Natural Science , 2013 .

[114]  A. Suslin On the K-theory of local fields , 1984 .

[115]  C. Soulé On higher p-adic regulators , 1981 .

[116]  D. Burghelea,et al.  Cyclic homology and algebraic K-theory of spaces—II , 1986 .

[117]  Jean-Pierre Serre,et al.  Le théorème de Riemann-Roch , 1958 .

[118]  J. Whitehead,et al.  On Incidence Matrices, Nuclei and Homotopy Types , 1941 .

[119]  J. McClure,et al.  The chromatic convergence theorem and a tower in algebraic $K$-theory , 1993 .

[120]  Daniel R. Grayson Exterior power operations on higher K-theory , 1989 .

[121]  Paul G. Goerss,et al.  Simplicial Homotopy Theory , 2009, Modern Birkhäuser Classics.

[122]  Philippe Elbaz-Vincent,et al.  Quelques calculs de la cohomologie de ? et de la K-théorie de ?Some computations of the homology of ? and the K-theory of ? , 2002 .

[123]  T. Pirashvili On the topological hochschild homology of z/pkz , 1995 .

[124]  B. Dundas,et al.  Excision for $K$-theory of connective ring spectra , 2007, 0704.3946.

[125]  R. Vogt,et al.  Volodin K-theory of A∞-ring spaces , 1993 .

[126]  G. Segal EQUIVARIANT STABLE HOMOTOPY THEORY , 1970 .

[127]  D. Quillen Finite generation of the groups Ki of rings of algebraic integers , 1973 .

[128]  T. Pirashvili,et al.  On the MacLane cohomology for the ring of integers , 1998 .

[129]  J. Rognes Algebraic K-theory of the two-adic integers , 1999 .

[130]  M. Kervaire Multiplicateurs de Schur et K-théorie , 1970 .

[131]  I. Madsen,et al.  On the K-theory of nilpotent endomorphisms , 2003 .

[132]  S. Ferry,et al.  Novikov Conjectures, Index Theorems and Rigidity: A history and survey of the Novikov conjecture , 1995 .

[133]  John W. Gray,et al.  Formal category theory: adjointness for 2-categories , 1974 .

[134]  S. Bloch On the tangent space to Quillen K-theory , 1973 .

[135]  D. Quillen On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field , 1972 .

[136]  Jan Stienstra,et al.  A presentation for some $K_2 \left({n,R} \right)$ , 1975 .

[137]  B. Mitchell,et al.  Rings with several objects , 1972 .

[138]  A. Borel Stable real cohomology of arithmetic groups , 1974 .

[139]  D. Burghelea Some rational computations of the Waldhausen algebraic K theory , 1979 .

[140]  Daniel R. Grayson Weight filtrations via commuting automorphisms , 1995 .

[141]  Vladimir Voevodsky,et al.  On motivic cohomology with Z/l -coefficients , 2008, 0805.4430.

[142]  M. Schlichting Negative K-theory of derived categories , 2006 .

[143]  L. Lewis Is there a convenient category of spectra , 1991 .

[144]  J. M. Boardman Conditionally Convergent Spectral Sequences , 1999 .

[145]  L. Illusie Complexe cotangent et déformations II , 1971 .

[146]  Bjørn Ian Dundas,et al.  Topological Hochschild homology of ring functors and exact categories , 1996 .

[147]  May,et al.  A Concise Course in Algebraic Topology , 1999 .

[148]  M. Kervaire Smooth homology spheres and their fundamental groups , 1969 .

[149]  A. Kuku,et al.  Higher Algebraic K-Theory , 2006 .

[150]  J. Greenlees Representing Tate cohomology of G -spaces , 1987 .

[151]  H. Bass Personal Reminiscences of the Birth of Algebraic K-theory , 2003 .

[152]  G. Whitehead,et al.  Elements of Homotopy Theory , 1978 .

[153]  B. Day On closed categories of functors , 1970 .

[154]  A. Suslin Projective modules over polynomial rings are free , 1976 .

[155]  Shichirô Oka Multiplicative structure of finite ring spectra and stable homotopy of spheres , 1984 .

[156]  J. Stallings On Infinite Processes Leading to Differentiability in the Complement of a Point , 1965 .

[157]  George A. Elliott,et al.  K-theory , 1999 .

[158]  Daniel R. Grayson ADAMS OPERATIONS ON HIGHER K-THEORY , 1992 .

[159]  B. Dundas LOCALIZATION OF V-CATEGORIES , 2001 .

[160]  F. Waldhausen Algebraic K-theory of topological spaces. II , 1979 .

[161]  Jean-Pierre Serre Faisceaux algébriques cohérents , 1955 .

[162]  B. Kahn Algebraic K-theory, algebraic cycles and arithmetic geometry , 2005 .

[163]  A. Berrick An approach to algebraic K-theory , 1982 .

[164]  B. Dundas,et al.  Enriched functors and stable homotopy theory , 2003, Documenta Mathematica.

[165]  D. Quillen,et al.  Cyclic homology and the Lie algebra homology of matrices , 1984 .

[166]  Shirley Dex,et al.  JR 旅客販売総合システム(マルス)における運用及び管理について , 1991 .

[167]  Christopher L. Douglas,et al.  Higher topological cyclic homology and the Segal conjecture for tori , 2008, 0803.2745.

[168]  J. Rognes,et al.  Spaces of PL Manifolds and Categories of Simple Maps , 2013 .

[169]  C. Weibel Algebraic K-Theory of Rings of Integers in Local and Global Fields , 2005 .

[170]  W. Hsiang,et al.  A model for computing rational algebraicK-theory of simply connected spaces , 1982 .

[171]  J. Rognes Topological cyclic homology of the integers at two , 1999 .

[172]  L. Hesselholt,et al.  Topological Cyclic Homology of Schemes , 2003 .

[173]  Friedhelm Waldhausen,et al.  ALGEBRAIC K-THEORY OF SPACES I , 1978 .

[174]  T. Lambre,et al.  Quelques classes caractéristiques en théorie des nombres , 2000, math/0006237.

[175]  Peter Gabriel,et al.  Calculus of Fractions and Homotopy Theory , 1967 .

[176]  J. Lurie Higher Topos Theory , 2006, math/0608040.

[177]  On the notion of geometric realization , 2003, math/0304064.

[178]  G. Carlsson,et al.  On the algebraic $K$-theory of simply connected spaces , 1996 .

[179]  Automorphisms of manifolds , 2000, math/0012101.

[180]  A. K. Bousfield The localization of spectra with respect to homology , 1975 .

[181]  J. Lannes,et al.  Autour de la cohomologie de MacLane des corps finis , 1994 .

[182]  E. Riehl Basic concepts of enriched category theory , 2014 .

[183]  Philip S. Hirschhorn Model categories and their localizations , 2003 .

[184]  D. Quillen,et al.  Excision in bivariant periodic cyclic cohomology , 1997 .

[185]  Daniel R. Grayson Exact sequences in algebraic $K$-theory , 1987 .

[186]  F. Waldhausen Algebraic K-Theory of generalized free products, part 2 , 1978 .

[187]  J. Milnor Two Complexes Which are Homeomorphic But Combinatorially Distinct , 1961 .

[188]  W. Lueck,et al.  Detecting K‐Theory by Cyclic Homology , 2005, math/0509002.

[189]  S. Bloch Algebraic K-theory and crystalline cohomology , 1977 .

[190]  M. Lydakis Smash products and Γ-spaces , 1999, Mathematical Proceedings of the Cambridge Philosophical Society.

[191]  T. Dieck Orbittypen und äquivariante Homologie. I , 1972 .

[192]  Michael A. Mandell,et al.  Model Categories of Diagram Spectra , 2001, Proceedings of the London Mathematical Society.

[193]  On the multiplicative structure of topological Hochschild homology , 2004, math/0410367.

[194]  M. Lane Homologie des anneaux et des modules , 1979 .

[195]  F. Waldhausen Algebraic K-Theory of generalized free products, part 1 , 1978 .

[196]  David Mumford,et al.  Towards an Enumerative Geometry of the Moduli Space of Curves , 1983 .

[197]  Spiros Adams-Florou An Introduction to Algebraic K-Theory , 2011 .

[198]  M. Jibladze,et al.  Cohomology of algebraic theories , 1991 .

[199]  The homotopy type of the cobordism category , 2006, math/0605249.

[200]  T. Gunnarsson Algebraic K-theory of spaces as K-theory of monads , 1982 .

[201]  F. Farrell,et al.  Rigidity in Geometry and Topology , 2010 .

[202]  Bi-relative algebraic K-theory and topological cyclic homology , 2004, math/0409122.

[203]  J. Milnor THE STEENROD ALGEBRA AND ITS DUAL1 , 1958 .

[204]  I. Madsen,et al.  Topological Hochschild homology of number rings , 2000 .

[205]  H. O. Erdin Characteristic Classes , 2004 .

[206]  J. Rosenberg,et al.  Algebraic K-Theory and Its Applications , 1995 .

[207]  Leonard Evens,et al.  Cohomology of groups , 1991, Oxford mathematical monographs.

[208]  S. Gersten K 3 of a Ring is H 3 of the Steinberg Group , 1973 .

[209]  M. Wodzicki,et al.  Excision in algebraic $K$-theory , 1992 .

[210]  F. Keune The relativization of K2 , 1978 .

[211]  Michael A. Mandell,et al.  Equivariant Orthogonal Spectra and S-Modules , 2002 .

[212]  W. Gajda On K* (Z) and classical conjectures in the arithmetic of cyclotomic fields , 2004 .

[213]  A. Suslin On theK-theory of algebraically closed fields , 1983 .

[214]  W. Dwyer,et al.  The homotopy theory of cyclic sets , 1985 .

[215]  I. Moerdijk Bisimplicial Sets and the Group-Completion Theorem , 1989 .

[216]  C. Schlichtkrull The transfer map in topological Hochschild homology , 1998 .

[217]  B. Dundas,et al.  Covering Homology , 2007, 0706.0626.

[218]  C. Weibel The Development of Algebraic K-theory before 1980 , 2022 .

[219]  H. Hiller λ-rings and algebraic K-theory , 1981 .

[220]  A. Lindenstrauss A relative spectral sequence for topological Hochschild homology of spectra , 2000 .

[221]  Richard G. Swan,et al.  Vector bundles and projective modules , 1962 .

[222]  A. Hattori Rank Element of a Projective Module , 1965, Nagoya Mathematical Journal.

[223]  G. Carlsson,et al.  The free loop space and the algebraic k-theory of spaces , 1987 .

[224]  R. Bott The periodicity theorem for the classical groups and some of its applications , 1970 .

[225]  C. Kassel La K-théorie stable , 1982 .

[226]  Charles Kratzer λ-Structure enK-théorie algébrique , 1980 .

[227]  A. Suslin On the Grayson Spectral Sequence , 2003 .

[228]  M. Weiss,et al.  Automorphisms of Manifolds and Algebraic $K$-Theory: Part III , 2013, 1308.4047.

[229]  M. Barratt,et al.  On the homology of non-connected monoids and their associated groups , 1972 .

[230]  O. Gabber K-theory of Henselian Local Rings and Henselian Pairs , 1992 .

[231]  R. K. Dennis,et al.  K2 of discrete valuation rings , 1975 .

[232]  T. Goodwillie On the general linear group and Hochschild homology , 1985 .

[233]  J. Adams,et al.  Infinite Loop Spaces , 1978 .

[234]  G. Segal,et al.  Categories and cohomology theories , 1974 .

[235]  Filtered topological cyclic homology and relative K–theory of nilpotent ideals , 2001, math/0104240.

[236]  S. Schwede Stable homotopical algebra and Γ-spaces , 1999, Mathematical Proceedings of the Cambridge Philosophical Society.

[237]  L. Hesselholt The big de Rham–Witt complex , 2010, 1006.3125.

[238]  L. Hesselholt On thep-typical curves in Quillen'sK-theory , 1996 .

[239]  The obstruction to excision in K-theory and in cyclic homology , 2001, math/0111096.

[240]  R. Thomason Homotopy colimits in the category of small categories , 1979, Mathematical Proceedings of the Cambridge Philosophical Society.

[241]  Christian Ausoni On the algebraic K-theory of the complex K-theory spectrum , 2009, 0902.2334.

[242]  Hopf algebra structure on topological Hochschild homology , 2005, math/0502195.

[243]  B. Harris Bott periodicity via simplicial spaces , 1980 .

[244]  W. Gajda,et al.  Power maps and epicyclic spaces , 1994 .

[245]  Daniel R. Grayson,et al.  The loop space of the $Q$-construction , 1987 .

[246]  S. Lane Categories for the Working Mathematician , 1971 .

[247]  Michael A. Mandell,et al.  Localization for THH(ku) and the topological Hochschild and cyclic homology of Waldhausen categories , 2011, 1111.4003.

[248]  E. Rees NILPOTENCE AND PERIODICITY IN STABLE HOMOTOPY THEORY , 1995 .

[249]  G. Segal,et al.  Homology fibrations and the “group-completion” theorem , 1976 .

[250]  B. Mazur Relative Neighborhoods and the Theorems of Smale , 1963 .

[251]  R. Switzer The Steenrod Algebra and its Dual , 2002 .

[252]  S. Gersten K-Theory of Free Rings , 1974 .

[253]  Michael A. Mandell,et al.  Localization theorems in topological Hochschild homology and topological cyclic homology , 2008, 0802.3938.

[254]  S. Tsalidis Topological hochschild homology and the homotopy descent problem , 1998 .

[255]  A. Blumberg A discrete model of S1-homotopy theory , 2007 .

[256]  U. Tillmann On the homotopy of the stable mapping class group , 1997 .

[257]  B. Dundas On K‐Theory of Simplicial Rings and Degreewise Constructions , 1999 .

[258]  Michael A. Mandell,et al.  The localization sequence for the algebraic K-theory of topological K-theory , 2006, Acta Mathematica.

[259]  Friedhelm Waldhausen,et al.  Algebraic K-theory of spaces, localization, and the chromatic filtration of stable homotopy , 1984 .

[260]  J. Rognes,et al.  The topological Singer construction , 2010, Documenta Mathematica.

[261]  L. Hesselholt,et al.  On the $K$-theory and topological cyclic homology of smooth schemes over a discrete valuation ring , 2004 .

[262]  J. McClure,et al.  H Ring Spectra and Their Applications , 1986 .

[263]  Nobuo Shimada,et al.  Delooping symmetric monoidal categories , 1979 .

[264]  B. Shipley,et al.  Equivalences of monoidal model categories , 2002, math/0209342.

[265]  J. Rognes Two-primary algebraic K-theory of pointed spaces , 2002 .

[266]  B. Shipley A CONVENIENT MODEL CATEGORY FOR COMMUTATIVE RING SPECTRA , 2003 .

[267]  M. Kervaire Le théorème de Barden-Mazur-Stallings , 1967 .

[268]  M. Lydakis Simplicial functors and stable homotopy theory , 1998 .

[269]  C. Weibel The norm residue isomorphism theorem , 2009 .

[270]  B. Dundas,et al.  Stable K-theory and topological Hochschild homology , 1994 .

[271]  Lawrence Breen Extensions du groupe additif , 1978 .

[272]  J. Stienstra,et al.  A presentation for K2 of split radical pairs , 1977 .