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.

Size

homotopy groups for computation of natural size distances.

Ranging from the later 1950s and into the later 1960s, these papers and include the "exotic spheres," including a procedure for killing

homotopy groups of differentiable manifolds; expository lectures on topology, differentiable structures, and smooth manifolds with boundary based on "Variedades diferenciables con frontera", papers on relations with algebraic topology, and a series on cobiordism that is evidence of a staggering level of work done in a very short time.

Although in Kan [KD55,561] the development of

homotopy groups associated to homotopy system is done through cubical complexes, we have found easier to use simplicial sets and their standard

homotopy groups in order to associate

homotopy groups to a homotopy.

Fix a prime number p, and consider the Adams spectral sequence of the stable

homotopy groups of the sphere at p.

It was also shown by Fadell and Neuwirth [12] that the higher

homotopy groups of the complement are trivial.

The papers, in English and French, include such subjects as invariants of combinatorial line arrangements and Rybnikov's example, time averaged optimization of dynamic inequalities on a circle, Thom polynomial computing strategies, quasi-convex decomposition in o-minimal structures,

homotopy groups of complements to ample divisions, Massey products of complex hypersurface complements, weighted homogeneous polynomials and blow-analytic equivalence, an infinitesimal criterion for topological triviality of families of sections of analytical variants, valuations and local uniformization, and finite Dehn surgery along A'Campo's divide knots.

Roman Mikhailov and Jie Wu, in their very nice paper [34], apply different Algebraic Topology techniques to get concrete results about

homotopy groups of suspended classifying spaces.