Subject: Re: question abt Eil-Mac From: "W. Stephen Wilson" Date: Wed, 10 Aug 2005 21:21:47 -0400 (EDT) >>Jesus Gonzalez wrote >> >> It is known that the Eilenberg-MacLane space K(Z/2^e,2) is uglier than >> expected in the sense that it has infinite BP-projective dimension (at >> least for e>1). Yet: is it true that multiplication by v_2 is monic? I don't have it at home, but in @ARTICLE{JW:EM, AUTHOR = {D. C. Johnson and W. S. Wilson}, TITLE = { The projective dimension of the complex bordism of {Eilenberg-Mac\,Lane} spaces}, JOURNAL = {Osaka Journal of Mathematics}, VOLUME = {14}, PAGES = {533--536}, YEAR = {1977 } } I think we proved infinite hom dim even for e=1. From @ARTICLE{JW2, AUTHOR = {D. C. Johnson and W. S. Wilson}, TITLE = {Projective dimension and {Brown-Peterson} homology}, JOURNAL =Top, VOLUME = {12}, PAGES = {327--353}, YEAR = {1973 } } v_2 cannot be injective. To find an element killed by v_2 it is probably enough to look at our proof. >> If one replaces K(Z/2^e,2) by K(Z/2^e,1) \times K(Z/2^e,1), is it known: >> 1. if v_2 multiplication is monic ? >> 2. hom.dim = 2 ?? >> >> Jesus Gonzalez My memory (about what I'm supposed to know here) fails me although I think this is known. Certainly several people have tried hard to understand such things. Steve ________________________________________________________________ Subject: Re: question abt Eil-Mac From: johnson@ms.uky.edu Date: Thu, 11 Aug 2005 10:00:44 -0400 (EDT) Jesus and Steve, My comments only refer to the e = 1 case. I suspect that Jesus’ question really was directed to harder e > 1 cases. Steve is correct that the projective dimension of BP_*(K(Z/2,2)) is infinite. Multiplication by the generator v_2 is not monic in BP<2>_*(K(Z/2,2)) but v_2 may or may not be monic in BP_*(K(Z/2,2)). I just don’t know and Steve’s and my paper (s) has (have) nothing to say about this. Given a complex X, no matter how bad BP_*(X) is, one can construct a cofibre sequence W -> A -> X where BP_*(A) is free over BP_* and maps onto BP_*(X). This means that BP_*(W) injects into a free Z_{(p)} module and thus is free itself OVER Z_{(p)}. Thus multiplication by any v_n in BP_*(W) is monic. But BP_*(W) is only one degree less ugly than BP_*(X) – say only one Botox treatment better. Now let P = K(Z/2,1) as is customary. Study of P \times P reduces to P#P (the smash product of two copies of P). Peter Landweber’s thesis gave us an even-odd splitting of BP_*(P#P). The odd-dimensional (Tor) part looks like BP_*(P) which is nice. The even-dimensional part is the double tensor product. I think Peter (and Bob Stong) knew all about this. Steve may not remember this, but he and I proved (Amer. J) that the n-fold BP_* tensor product of BP_*(P)’s has a Landweber presentation which is free over BP_*/I_n. The power behind the proof comes from Steve’s and Doug Ravenel’s solution of the Conner-Floyd conjecture. Thus the BP_* projective dimension of BP_*(P \times P) is indeed two. All the best, Dave Johnson