By Barry Dayton, Charles Weibel (auth.), J. F. Jardine, V. P. Snaith (eds.)

A NATO complex learn Institute entitled "Algebraic K-theory: Connections with Geometry and Topology" was once held on the Chateau Lake Louise, Lake Louise, Alberta, Canada from December 7 to December eleven of 1987. This assembly used to be together supported by means of NATO and the common Sciences and Engineering study Council of Canada, and was once subsidized partly by means of the Canadian Mathematical Society. This publication is the quantity of lawsuits for that assembly. Algebraic K-theory is basically the research of homotopy invariants bobbing up from earrings and their linked matrix teams. extra importantly maybe, the topic has turn into imperative to the research of the connection among Topology, Algebraic Geometry and quantity concept. It attracts on all of those fields as a topic in its personal correct, however it serves in addition to an efficient translator for the applying of ideas from one box in one other. The papers during this quantity are consultant of the present country of the topic. they're, for the main half, study papers that are essentially of curiosity to researchers within the box and to these intending to be such. there's a part on difficulties during this quantity which could be of specific curiosity to scholars; it incorporates a dialogue of the issues from Gersten's famous record of 1973, in addition to a brief checklist of latest problems.

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Then with the notation above. there are exact sequences: (i) k ~. 2. 3 induces f. Then _k+1 02{f) €~. Z). oT = 01(f) f: ~(M) ~ such that - 02(f) with ffiIZ. m) The similarity between these sequences and those of Theorem is striking. and the analogy becomes transparent if we give a H. GIlLET AND C. E 41 somewhat different formulation of the differential character groups, following the article [9] of Cheeger. Suppose that M is a compact manifold; then by Poincare duality, we can compute the integral cohomology of M using chains rather than cochains: where we view C*(M) and n = dimension as a cochain complex with M.

1. 2. 3. a: AP- 1 ,P-1{Xm) ---+ ciIP(X) T} I---t (O,T}) Theorem: ([13]) There are exact sequences CHP ,p-1(X) (i) (ii) CHP ,p-1(X) ---+ CHP ,p-1{X) (iii) ~ AP- 1 ,P-1(Xm) ~ ciIP(X) ~ CHP(X) HP- 1 ,P-1(L) ~ ciIP(X) -lR ~ HP- 1 ,P-1{Xm) ---+ 0 ---+ ciIP(X) ~ ZP,p{X_} alg -lR 1 the Beilinson regulator [3]. forms in AP,P{Xm) cycles, ---+ HP,P{L) is the K1-type Chow group which is the the Quillen spectral sequence Notice that ([17],[24]). The map -lR alg E~,p-1 p ---+ 0 term of is essentially is the space of closed (p,p) representing the cohomology classes of algebraic HP,P{Xm)alg group HP'P(~), ---+ 0 ciIP{X) ZP,P{L) III CHP{X) alg -lR Here CHP ,p-1 ---+ 0 is the corresponding subspace of the cohomology and cHP(X)o cHP{X)O is defined as the kernel of consists of cycles of the form w.

Bass] Bass. : Algebraic K-theory. Benjamin. New York. 1968. : 'The Picard group of a reduced G-algebra', to appear in J. Pure Applied Algebra. [FL] Fulton, W. , Springer. 1985. : Cohomologie non abelienne, Grundlehren der Math, Springer, 1971. , 1977. , Springer, 1966. : Etale Cohomology, Princeton U. Press, Princeton, 1980. : Introduction to Algebraic K-theory, Ann. of Math. Studies 72, Princeton U. Press, Princeton, 1971. [SGA6] Berthelot, P. : Theorie des intersections et theoreme de Riemann-Roch (SGA6) , Lecture Notes in Math.