A nonnilpotent 1-2-presented graded Hopf algebra whose Hilbert series converges in the unit circle
The author indicates applications of some examples of a graded Lie algebra \(\eta\) to the theory of homotopy groups of finite CW-complexes. The enveloping algebra \(R^!\) of \(\eta\) is a Hopf algebra and has the properties indicated in the title of this paper: \[ R^!= \frac{k\langle X,Y,Z,U,V\rangle} {\binom{X^2,\;[X,Y],\;[X,Z],\;[Y,Z]- [X,U],\;Z^2+[Y,U],} {Z^2+ [X,V],\;[Z,U]- [Y,V],\;[U,V],\;V^2}}, \] where \(k\langle X,Y,Z,U,V\rangle\) is the free associative algebra in variables \(X,Y,Z,U,V\). The above properties are the answer to a 15 year-old existence question by Anick and Löfwall. In the last section, the authors show how the Lie algebra considered makes it possible to construct a finite simply-connected CW-complex with 12 cells: 5 cells of dimension 2 and 7 cells of dimension 4, whose integral loop space homology groups have torsion of all orders.
scientific article; zbMATH DE number 1076163
scientific article