Geometric Applications of Homotopy Theory II: Proceedings, by J. F. Adams (auth.), M. G. Barratt, M. E. Mahowald (eds.)

By J. F. Adams (auth.), M. G. Barratt, M. E. Mahowald (eds.)

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Extra resources for Geometric Applications of Homotopy Theory II: Proceedings, Evanston, March 21–26, 1977

Example text

From indeed, ~i=I by i n d u c t i o n (i) v n is divisible (ii) pnw n lies i wiv~-1° ~i=n-1 Vn = PWn t i in degree on the logarithmic described is another 2(pi-1). 2 provides the information we need about from a general free-free spectrum. the map r completely. maps to and We also need the following lemma about BP. ~4 Suppose a is a graded module concentrated ~nduees finitel~ ~enerat@~ free i__nnde£ree n, and that f : G ~ B P - - ~ G ~ B P the identity Z(p)- i__ssany map f, = I : , n ( G @ B P ) - - ~ , n ( G @ B P ) .

The way we assembled 5 to form MU was by no means obvious with Chern classes and the associated much more direct way the spectra BP(p) in section to anyone algebraic not well acquainted theory. 20 of [6]. ~efinition ring 6~I spectrum and BP(M)p rational Given any set M of primes, we define BP(M) as having the localizations = V(M-p)~BP(p) the M-local BP(M) 0 : K(V(M)0) , for each prime p in M, using the isomorphisms ~ p : B P ( M ) p - - ~ B P ( M ) o obvious defined on homotopy groups by ,,(BP(M) p) : V ( M - p ) ~ V ( p ) p In particular, p, and BP(O) primes, = K(Q).

F(x,O) of bundles yield immediately = x, and F(x,F(y,z)) ordinary trum, cohomology, formula in two called Properties the identities = F(F(x,y),z). 3) reasons, is the specsimple 33 cH(~®~) We are i n t e r e s t e d mula by c h a n g i n g spectrum Chern in the p o s s i b i l i t y the choice it is an a l g e b r a i c class whose However, Consider = cH(~) + cH(~). of Chern class. exercise formal group E^H, from E and c H from H, of w h i c h know from MacLane) H,(E) this general spectrum, = ~,(EAH), theory and that so that in favorable is Z M - f r e e cases which inherits is its h o m o t o p y mal power class series; c E.

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