In this article, we present a number of results of the following type: A given subgraph of an embedded graph either is embedded in a disc or it contains a face chain that has a non-contractible curve through it. One application is that any two faces of a 4representative embedding are simultaneously contained in a disc bounded by a polygon. Such results are applied to giving proofs of the existe...

Exact squares in Cat are not necessarily absolute (i.e. preserved by any 2-functor Cat→ Cat), or even preserved by any 2-functor given by exponentiation (−)I : Cat → Cat: if a square is preserved by exponentiation it will be called a contractible exact square. We will characterize diagrammatically these contractible squares, and among them the contractible categories, and the so called fibering...

The following theorem is surprising because it says that if A is a suspension, then for any map A→ X, the homotopy fiber of the cofiber Fib(X → X/A) and A have unstably quite a lot in common (see corollary 8.9): Theorem. Let A→ X → X/A be a cofibration sequence of pointed and connected spaces and Y be a pointed space. If A is weakly equivalent to a suspension of a connected space, then the foll...

This paper analyzes Bayesian normal form games in which players write contracts that condition their actions on the contracts of other players. These contracts are required to be representable in a formal language. This is accomplished by constructing contracts which are de…nable functions of the Godel code of every other player’s contract. We provide a complete characterization of the set of a...

Let G be a 3-connected graph on more than 4 vertices. We show that every depth-first-search tree of G contains a contractible edge. Moreover, we show that if G is 3-regular or does not have two disjoint pairs of adjacent degree-3 vertices, every spanning tree of G contains a contractible edge.

Contraction of an edge merges its end points into a new vertex which is adjacent to each neighbor of the end points of the edge. An edge in a k-connected graph is contractible if its contraction does not result in a graph of lower connectivity. We characterize contractible edges in chordal graphs using properties of tree decompositions with respect to minimal vertex separators.

The non-Hausdorff suspension of the one-sphere S1 of complex numbers fails to model the group’s continuous multiplication. Moreover, finite connected H-spaces are contractible, and therefore cannot model infinite connected non-contractible H-spaces. For an H-space and a finite model of the topology, the multiplication can be realized on the finite model after barycentric subdivision.

We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A-homotopy theory; when k = C, the smooth manifolds X(C) are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are that of Spec k. One might hope that furthermore, and in analogy with the classification of topological vector bundles on m...

In this paper it is shovm that every maximum matching in a 3-connectccl graph, other than ](4, contains at least one contractible edge. In the case of a. perfect ma.tching, those graphs in \vhich there exists a perfect matching containing precisely one contractible edge arc characterized.