LOOP SPACE HOMOLOGY OF SPACES OF SMALL CATEGORY

Only little is known concerning H(OMEGAX; k) , the loop space homolog y of a finite CW complex X with coefficients in a field k. A space X i s called an r-cone if there exists a filtration = X0 subset-of X1 su bset-of ... subset-of X(r) = X, such that X(i) has the homotopy type o f the cofibre of a map from a wedge of sphere into X(i-1). Denote by A (X) the sub-Hopf algebra image of H(OMEGAX1). We prove then that for a graded r-cone, r less-than-or-equal-to 3, there exists an isomorphis m A(X) x T(U) --> congruent-to H(OMEGAX).