In the context of Algebraic Topology, an analogous notion to the one of size function has been developed under the name of size

homotopy group (Frosini and Mulazzani, 1999).

1] is a discrete

homotopy group, to be defined below.

In this section we explain briefly how Kenzo can compute the

homotopy group [[pi].

It had been shown previously by Babson [2] and independently by Bjorner that that the discrete fundamental group of the permutahedron is isomorphic to the classical

homotopy group of the real complement of the k-equal arrangement, [M.

0]) one can associate the n-th

homotopy group [[pi].

0](X) about degrees of the rational

homotopy group of X.

Especially, there is a sequence of subgroups in the m-th

homotopy group [[pi].

ii) A more interesting example is when each of X and Y has only two nontrivial

homotopy groups (see [section]3.

Since MacPherson's work, some progress on this question has been made, most notably by Anderson [And99], who obtained results on

homotopy groups of the matroid Grassmannian, and by Anderson and Davis [AD02], who constructed maps between the real Grassmannian and the matroid Grassmannian--showing that philosophically, there is a splitting of the map from topology to combinatorics--and thereby gained some understanding of the mod 2 cohomology of the matroid Grassmannian.

We had no previous experience about homology or

homotopy groups, no opportunity to calculate even simple homology or

homotopy groups before.

Homotopy groups of graphs have been defined in Benayat and Kadri (1997) and Babson et al.

Behrens (mathematics, Massachusetts Institute of Technology) describes the relationship between two machines for computing the 2-primary unstable

homotopy groups of spheres: the EHP spectral sequence and the Goodwillie tower of the identity.