Homomorphism complexes were introduced by Lovász to study topological obstructions to graph colorings. We show that folding in the second parameter of the homomorphism complex yields a homotopy equivalence. We study how the Mycielski construction changes the homotopy type of the homomorphism complex. We construct graphs showing that the topological bound obtained by odd cycles can be arbitraril...

A crucial observation in Quillen’s definition of higher algebraic K-theory was that the right way to proceed is to define the higer K-groups as the homotopy groups of a space ([21]). Quillen gave two different space level models, one via the plus construction and the other via the Q-construction. The Q construction version allowed Quillen to prove a number of important formal properties of the ...

This announcement describes new methods in the study and classification of differentiable, PI or topological manifolds with infinite fundamental group. If, for example, Y + l is a closed manifold with n^ = Gx *H G2, n> 4, there is a decomposition Y~YtUx Y2 with ir1(Yx) = G1,irl(Yl) = G2, itt(X) = H. For large classes of fundamental groups, the codimension one splitting theorems of [C3], extendi...

This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loose pairs of homotopy classes. We give a brief (and necessarily very incomplete) survey of some old and new ...

Given a manifold M and a proper sub-bundle ∆ ⊂ TM , we study homotopy properties of the horizontal base-point free loop space Λ, i.e. the space of absolutely continuous maps γ : S → M whose velocities are constrained to ∆ (for example: legendrian knots in a contact manifold). A key technical ingredient for our study is the proof that the base-point map F : Λ → M (the map associating to every lo...

The branching space of a flow is the topological space of germs of its nonconstant execution paths beginning in the same way. However, there exist weakly Shomotopy equivalent flows having non weakly homotopy equivalent branching spaces. This topological space is then badly behaved from a computer-scientific viewpoint since weakly S-homotopy equivalent flows must correspond to higher dimensional...

Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of “geometric” models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact,...

Concurrency, i.e. the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of \geometric" models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact, ...