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 anything approximately BP. starting with the geometric heritage 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 foremost advancements within the box within the final decade (the calculation of the homology of Eilenberg-MacLane areas during this part should be precious in educating complicated algebraic topology courses). The publication concludes with a bit on risky operations with reviews on the place functions may possibly come from sooner or later.

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1 0) , 11R(a) ¥ 1L (a) mod In . o This makes H kBP*/In into an FP [vn J module . D. we know that H kBP* /ln is a direct sum of free copies of Fp [vn ] and cyclic modules, Fp [vn ] /(v! )- This 1 develops a craving for the next step, H BP*/ln , which , for reasons that will be clear in a moment, seems accessible. Be fore we move on to the higher Ext groups we should have some method of computation [M] . Le t (4. 1 7) n c o p ie s 26 W. STEPHEN WILSON be the cobar resolution with (4. 1 9) 1/I('Y) = L 'Y ' ® 'Y " and 1/l(a) = L a ' ® a " .

56) W. 57) Both of the end groups are in known stable stems isomorphic to Z/(p). The only question is if -y 1 is hit by the left-hand Z/(p ) . The stable homotopy experts knew that this group was generated by 13f [To2 ] . The filtration is thus greater than two so it cannot hit -y1. The complexity of the computation of H 2BP* necessitates some new techniques. 60) called the chromatic spectral sequence [MRW] . components. Since H *p -I BP* � Q, we have It breaks H *BP* up into its vn -periodic ..

5 (GLOBAL VERSION). K 1 , and H* CP "" C H. K2, subject to the relation that e 1 o e = {3 . o 1 = H* (Zfp) , 1 Denote the height p truncated polynomial algebra and the exterior algebra respectively by E(x) = Z/(p) [x ] /(x 2 ). 9) • · · · ) , i k � 0, defme (8 . 1 0) (3< 0 > = [ I ] - [O J . THEOREM 8 . 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.

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