Brown-Peterson Homology: An Introduction and Sampler (Cbms by W. Stephen Wilson

By W. Stephen Wilson

This e-book is basically directed to graduate scholars drawn to the sphere and to algebraic topologists who desire to research whatever approximately BP. starting with the geometric history of complicated bordism, the writer is going directly to a dialogue of formal teams and an creation to BP-homology. He then offers his view of the key advancements within the box within the final decade (the calculation of the homology of Eilenberg-MacLane areas during this part might be worthwhile in educating complicated algebraic topology courses). The ebook concludes with a piece on risky operations with reviews on the place purposes might come from sooner or later.

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1 1 (LOCAL VERSION). As an algebra where the tensor product is over all I, J as above and the coproduct follows by Hopf ring ' properties from the a s and {3's. 1 1 rapidly enough. We give this example to demonstrate the computation techniques in a familiar setting. 5. 40), a i) o a i) = 0. The relation e o e 1 = {31 is ( ( 1 obvious. K 1 and H* CP '"' . There are no additional relations in 8 . 5 follows. 13) From 8 . 1 1 we have (8 1 4) . 15 (MILNOR [Mi2 ] ) . The coproduct on the Hop[ algebra H* H is given by n 1/l (�n ) = L �0 · ���� ® �� and 1/I (Tn ) = n L ���� ® T; + �0 · Tn ® 1.

In E * CP"" ( (s, t] ] , {J(s){3(t) = {3(s +F t). o PROOF. Write {31{3i = 1:k ck {3k and (x 1 +p x 2 )k = 1:1, i a 1�x {x� . Then c• � � � c, x ) c,. ) P;)> � a1j . BROWN PETERSON HOMOLOGY P(s) P( t) = 'E;,A s ipli = 'E;,j, Jca;1s itipk P(s +p t). 30 (RW. J . P(s)" = P( (n ) p(s)) . 0 On the surface , this appears to be another nonsense formula involving totally inacces­ sible coefficients. However, in the important case E = BP we can extract an explicit for­ mula. 3 1 ) I t i s fairly easy t o see that the other P's are decomposable.

15 , pmn = F v1tl L > i O j�O 'I;O

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