# Algebraic K-Theory and Algebraic Topology by Robert Boltje, G.-Martin Cram, V. P. Snaith (auth.), P. G.

By Robert Boltje, G.-Martin Cram, V. P. Snaith (auth.), P. G. Goerss, J. F. Jardine (eds.)

A NATO complex learn Institute entitled "Algebraic K-theory and Algebraic Topology" was once held at Chateau Lake Louise, Lake Louise, Alberta, Canada from December 12 to December sixteen of 1991. This e-book is the quantity of court cases for this assembly. The papers that seem listed here are consultant of lots of the lectures that got on the convention, and for this reason current a "snapshot" of the country ofthe K-theoretic paintings on the finish of 1991. The underlying target of the assembly used to be to debate contemporary paintings regarding the Lichtenbaum-Quillen complicated of conjectures, fro~ either the algebraic and topological issues of view. The papers during this quantity take care of a number subject matters, together with motivic cohomology theories, cyclic homology, intersection homology, larger type box concept, and the previous telescope conjecture. This assembly used to be together funded through delivers from NATO and the nationwide technological know-how Foun dation within the usa. i need to take this chance to thank those companies for his or her aid. i'd additionally wish to thank the opposite contributors of the organizing com mittee, specifically Paul Goerss, Bruno Kahn and Chuck Weibel, for his or her assist in making the convention winning. This used to be the second one NATO complicated research Institute to be held during this venue; the 1st used to be in 1987. The good fortune of either meetings owes a lot to the professionalism and helpfulness of the management and employees of castle Lake Louise.

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Table of Contents

I. common TOPOLOGY.

1. Set concept and Logic.

2. Topological areas and non-stop Functions.

three. Connectedness and Compactness.

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five. The Tychonoff Theorem.

6. Metrization Theorems and Paracompactness.

7. whole Metric areas and serve as Spaces.

eight. Baire areas and measurement Theory.

II. ALGEBRAIC TOPOLOGY.

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The middle homology of the top horizontal complex is H'j-/(X,Kd) and the middle homology of the middle horizontal complex is Hd- 2 (Zp,Kd-d· The variety Eisa smooth quadric of dimension d - 2 :;::: 1, hence satisfies Ao( E) = 0 (this is a general result on quadrics, valid over any ground field; however in the case under consideration, since any quadric over a finite field has a rational point, the smooth quadric E is birational over its ground field to projective space and the result follows from the birational invariance of the group A0 on smooth projective varieties, together with the well-known vanishing of that group on projective space).

Lu)), since D? = U fori = 1, ... ,p. (1v;) + pind{f(1u) = O, i=O as an easy calculation in H IU 9:! Cp x Cp shows. This completes the proof of part (a). And part (b) is an obvious consequence of part (a). 1) the following statements are equivalent: (i) (cH)n. E fi are as follows: H ~ G' ~ G with IG' I HI = p, N := ker>.

2), so that the canonical extension maps CHIN and the inductive cover maps CHIN are defined. Note that, since aH commutes with inflation (Thm. e. 17) for all N