Get Algebraic K-Theory: Proceedings of a Conference Held at PDF

By Anthony Bak (auth.), R. Keith Dennis (eds.)

ISBN-10: 0387119663

ISBN-13: 9780387119663

ISBN-10: 3540119663

ISBN-13: 9783540119661

ISBN-10: 3540395563

ISBN-13: 9783540395560

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Extra resources for Algebraic K-Theory: Proceedings of a Conference Held at Oberwolfach, June 1980 Part II

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Note that, with H = e, this says that the underlying nonequivariant homo- topy type of B(G,~) is just B~. Again, we are interested in increasing sequences {~n } with union ~. While the third author has conbtructed an equivariant plus construction for G-connected Gspaces X, namely those X for which all X H are path connected, we shall not use it. Instead, we shall prove that, in the cases of interest to us, equivalent to a topological G-monoid B ~(G). J~n B(G,~n ) is G- This means that B ~ (G) is a G-space and a topological monoid such that its unit is a fixed point and its product is a G- 28 map.

For M as above, application of the "group completion theorem" fixed point spaces implies that ~ :M ÷ ~ I is a group completion. [22, 26, 27] to To e~plolt this fact, we need the following notion. 1. A cofinal sequence in a Hopf G-space X is a sequence of points a t ~ X G, with a 0 the unit, such that at+ 1 is in the same path component of X G as bta t for some b t ~ X G and such that, for each H C G and c ~ X H, there exists d ~X H and t ~ 0 such that dc and a t are in the same path component of X H.

X) i x (ii) for x EH p Define a homomorphism vP:R0(~) + [R(H;R) @ Ro(RP)] ~ [R(H;C) @ R(RP)] @ [R(H;H) @ RSp(RP)I similarly, where R(H;F) is the free Abelian group generated by those irreducible real representations of H with associated division ring F. 5. with I [B~,K] i The following diagram commutes, where R(H) ~ K(BH p) is identified and the products run over R(K) p c R+(H,H): (vP) ~ x R(H) @ R(~ p) 0 H KGB(G,E) ' v With the evident modifications, • x [BH~, x K] 0 i ~ x R(H) @ K(B~ p) p the analogous diagram commutes in the real case.

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Algebraic K-Theory: Proceedings of a Conference Held at Oberwolfach, June 1980 Part II by Anthony Bak (auth.), R. Keith Dennis (eds.)


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