By W. Stephen Wilson
This e-book is basically directed to graduate scholars drawn to the sector and to algebraic topologists who desire to study anything approximately BP. starting with the geometric historical past of advanced 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 most important advancements within the box within the final decade (the calculation of the homology of Eilenberg-MacLane areas during this part could be valuable in educating complicated algebraic topology courses). The ebook concludes with a bit on volatile operations with reviews on the place purposes may perhaps come from sooner or later.
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Additional resources for Brown-Peterson homology: an introduction and sampler
A bipolynomial Hopf algebra is one where both it and its dual are polynomial . of · · · ' , THEOREM 1 0. 1 ( [W1 ) ; SECOND PROOF [RW1 ) ; C HAN S PROOF [Ch] ). H* BP2 k + l ' is an exterior algebra and H* BP; k is bipoly nomial. We have, for k + n > 0 , rank Qllk + n BP� = rank 7Tk BP = rank HkBP. o S KETCH OF C HAN S PROOF. The proof is by induction. To ground the induction we ' have which proves the result for degree 1 = k + � i < k, all n, and Hn + kBP� . the bar spectral sequence Hn +iBP� , 0 n.
11R(vn) = vn mod In. o PRooF. Inductively we can assume In is invariant and we have InBP*BP = BP*BPin n so working modulo In makes sense. 8 follows. 7 inductively. 8 is overpowering. 9 (LANDWEBER [L2) , MORAVA [Mo1) ) . n <,oo. o SKETCH OF THE PROOF. We know In is invariant and prime . Assume I is an invariant prime ideal. Inductively assume In I -In. If we could show C I; we show that if In :/::I, then In+ 1 C I. 10) then we would have v! EI and since I is prime , vn as well; thus In+t C I. A proof of (4.
Finding generators X; is j ust a matter of dividing o n - I (v� ) by vn- l as much as possible. We give a simple example . 26 (MILLER-WILSON [MW] , RAVENEL'S PROOF [R1 ] ). 11R (vn ) == vn + vn _ 1t( -1 - v�_ 1 t 1 mod /n- t • o This is a good example of skimming off information from the inductive formulas. This formula contains a tremendous amount of information and can be pushed a long way for results ; see [MW] . PROOF. 6. 6 is vn _ 1 + F vn _ 1t r +F vn which is vn + vn_ 1 t f l . o s Now we can present our example of some dividing by vn _ 1 • Write k = ap , a =F 0 (p).
Brown-Peterson homology: an introduction and sampler by W. Stephen Wilson