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...

There are many applications in motion planning where it is important to consider and distinguish between different homotopy classes of trajectories. Two trajectories are homotopic if one trajectory can be continuously deformed into another without passing through an obstacle, and a homotopy class is a collection of homotopic trajectories. In this paper we consider the problem of robot explorati...

We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. This has two main applications. First, by considering inverse diagrams in Voevodsky...

Integral symplectic 4–manifolds may be described in terms of Lefschetz fibrations. In this note we give a formula for the signature of any Lefschetz fibration in terms of the second cohomology of the moduli space of stable curves. As a consequence we see that the sphere in moduli space defined by any (not necessarily holomorphic) Lefschetz fibration has positive “symplectic volume”; it evaluate...

We explain how the notion of homotopy colimits gives rise to that of mapping spaces, even in categories which are not simplicial. We apply the technique of model approximations and use elementary properties of the category of spaces to be able to construct resolutions. We prove that the homotopy category of any monoidal model category is always a central algebra over the homotopy category of Sp...

Simple-homotopy for simplicial and CW complexes is a special kind of topological homotopy constructed by elementary collapses and expansions. In this paper we introduce graph homotopy for graphs and Graham homotopy for hypergraphs, and study the relation between these homotopies and the simplehomotopy for simplicial complexes. The graph homotopy is useful to describe topological properties of d...

Quantum field theory is the most universal method in physics, applied to all the area from condensed-matter physics to high-energy physics. The standard tool to deal with quantum field theory is the perturbation method, which is quite useful if we know the vacuum of the system, namely the starting point of our analysis. On the other hand, sometimes the vacuum itself is not obvious due to the qu...