Subject: Re: PC and P series Date: Fri, 24 May 2002 12:18:22 -0400 (EDT) From: "Douglas C. Ravenel" There is unlikely to be any more information of this sort. Let V1_*, V2_* and V3_* be the three graded vector spaces, so we have a long exact sequence a_i b_i c_i a_{i-1} V1_i -------> V2_i -------> V3_i -------> V1_{i-1} ---------->... Let A(t), B(t) and C(t) be Poincare series for the ranks of the graded maps a, b and c. Then we have P1(t) = C(t)/t + A(t) P2(t) = A(t) +B(t) P3(t) = B(t) +C(t) which leads to (1+t)A(t) = -t*P1(t) - P2(t) + P3(t) (1+t)B(t) = t*P1(t) - t*P2(t) - P3(t) (1+t)C(t) = t*P1(t) - t*P2(t) + t*P3(t) Setting t equal to -1 reduces each of these to your equation. The other possibility is to set t equal to a zero of either A, B, or C, but this requires knowledge of the morphisms. If C(t)=0 for all t, ie if the long exact sequence is a collection of short exact sequence, then the third equation above reduces to P2(t) = P1(t) + P3(t). as expected. Doug On Fri, 24 May 2002, Don Davis wrote: > __________________________________________________ > > Subject: Question on Poincare series for the Topology list > Date: Tue, 14 May 2002 16:38:37 +0100 > From: Martin Crossley > > In trying to find a painless way of introducing > British undergraduates to homology, I hit the > following problem: > > Suppose you have three graded vector spaces (with > Poincare series P_1, P_2, P_3) which are linked by > a long exact sequence (like the homology exact > sequence of a pair). Then you have an equation like > P_2(-1) = P_3(-1) + P_1(-1). > But this does not contain as much information as > the exact sequence, even if you suppose no knowledge > of the morphisms. > Is there a formula in the Poincare series that > catches more of the information ? > > Martin Crossley > > Douglas C. Ravenel, Chair |918 Hylan Building Department of Mathematics |drav@math.rochester.edu University of Rochester |(585) 275-4413 Rochester, New York 14627 |FAX (585) 273-4655 Personal home page: http://www.math.rochester.edu/u/drav/ Department of Mathematics home page: http://www.math.rochester.edu/ Math 443 home page: http://www.math.rochester.edu/courses/2002-spring/MTH443/ Faculty Senate home page: http://www.cc.rochester.edu/Faculty/senate/