ANR divisors and absolute neighborhood contractibility

A1-contractibility and topological contractibility

These notes are an extended transcript of two lectures given at the University of Ottawa during the workshop “Group actions, generalized cohomology theories, and affine algebraic geometry." The first lecture was a general introduction to A1-homotopy theory focusing on motivating the choice of definitions in the construction, together with a definition of A1-contractible spaces. The second lectu...

A Characterization of Absolute Neighborhood Retracts

By an absolute neighborhood retract (ANR) I mean a separable metrizable space which is a neighborhood retract of every separable metrizable space which contains it and in which it is closed. This generalization of Borsuk's original definition was given by Kuratowski for the purpose of enlarging the class of absolute neighborhood retracts to include certain spaces which are not compact. The spac...

Sparse Days and ANR SOLSTICE

Contractibility and NP-completeness

For a fixed graph H, let H-CON denote the problem of determining whether a given graph is contractible to H. The complexity of H-CON is studied for H belonging to certain classes of graphs, together covering all connected graphs of order at most 4. In particular, H-CON is NP-complete if H is a connected triangle-free graph other than a star. For each connected graph H of order a t most 4 other ...

Contractibility and the Hadwiger Conjecture

Consider the following relaxation of the Hadwiger Conjecture: For each t there exists Nt such that every graph with no Kt minor admits a vertex partition into dαt + βe parts, such that each component of the subgraph induced by each part has at most Nt vertices. The Hadwiger Conjecture corresponds to the case α = 1, β = −1 and Nt = 1. Kawarabayashi and Mohar [K. Kawarabayashi, B. Mohar, A relaxe...