[last updated 2000-2-16] n-stage trees, n-fibonacci numbers, and the cohomology of k(a,n), part 2 in part 1 i suggested that the cohomology of the eilenberg-maclane space k(a,n) could be understood in a way involving n-stage trees and n-fibonacci numbers. here i demonstrate this phenomenon in the special case of the cohomology groups of k(z/2,2) with coefficients in the field z/2; i describe a certain cochain complex whose cohomology groups are the desired cohomology groups. the dimensions of the vector spaces in the cochain complex are the fibonacci numbers 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. thus the co-boundary matrixes are approximate golden rectangles, which can be constructed according to a certain recursive scheme not described here, or by a method involving 2-stage trees as follows: the matrixes are indexed by the non-degenerate 2-stage trees with given numbers of edges, and the (j,k) entry of the co-boundary matrix is zero unless k is a 2-stage tree with x and y adjacent to each other and j is the 2-stage tree with one less edge than k obtained by joining together the x branch with the y branch, in which case the matrix entry is the binomial coefficient ((x+y)!/(x!*y!)) modulo 2. i hope to eventually give a more thorough explanation of how and why this method of calculating the cohomology groups of k(a,n) works but for now i just list for the particular example considered the first few co-boundary matrixes and their ranks, and the dimensions of the cohomology groups calculated from them. (note: unlike in part 1, it seems convenient here for some reason to count the empty list as a 2-stage tree with no edges.) number of 2-stage trees n with n edges 0 1 {empty list} 1 0 {} 2 1 {1} 3 1 {2} 4 2 {11,3} 5 3 {21,12,4} 6 5 {111,31,22,13,5} 7 8 {211,121,41,112,32,23,14,6} 8 13 {1111,311,221,131,51,212,122,42,113,33,24,15,7} empty| list| | rank of co-boundary matrix = 0 dimension of 0th cohomology = 1 - 0 - 0 = 1 1 _____ rank of co-boundary matrix = 0 dimension of 1th cohomology = 0 - 0 - 0 = 0 2 _____ | 1 | 0 | rank of co-boundary matrix = 0 dimension of 2th cohomology = 1 - 0 - 0 = 1 11 3 __________ | 2 | 0 0 | rank of co-boundary matrix = 0 dimension of 3th cohomology = 1 - 0 - 0 = 1 21 12 4 _______________ | 11 | 0 0 0 | 3 | 1 1 0 | rank of co-boundary matrix = 1 dimension of 4th cohomology = 2 - 0 - 1 = 1 111 31 22 13 5 _________________________ | 21 | 0 0 0 0 0 | 12 | 0 0 0 0 0 | 4 | 0 0 0 0 0 | rank of co-boundary matrix = 0 dimension of 5th cohomology = 3 - 1 - 0 = 2 211 121 41 112 32 23 14 6 ________________________________________ | 111 | 0 0 0 0 0 0 0 0 | 31 | 1 1 0 0 0 0 0 0 | 22 | 0 0 0 0 0 0 0 0 | 13 | 0 1 0 0 0 0 0 0 | 5 | 0 0 1 0 0 0 1 0 | rank of co-boundary matrix = 3 dimension of 6th cohomology = 5 - 0 - 3 = 2 1111 311 221 131 51 212 122 42 113 33 24 15 7 _________________________________________________________________ | 211 | 0 0 0 0 0 0 0 0 0 0 0 0 0 | 121 | 0 0 0 0 0 0 0 0 0 0 0 0 0 | 41 | 0 0 0 0 0 0 0 0 0 0 0 0 0 | 112 | 0 0 0 0 0 0 0 0 0 0 0 0 0 | 32 | 0 0 0 0 0 1 1 0 0 0 0 0 0 | 23 | 0 0 1 0 0 1 0 0 0 0 0 0 0 | 14 | 0 0 0 0 0 0 0 0 0 0 0 0 0 | 6 | 0 0 0 0 0 0 0 1 0 0 1 0 0 | rank of co-boundary matrix = 3 dimension of 7th cohomology = 8 - 3 - 3 = 2