3,941 research outputs found
On retracts, absolute retracts, and folds in cographs
Let G and H be two cographs. We show that the problem to determine whether H
is a retract of G is NP-complete. We show that this problem is fixed-parameter
tractable when parameterized by the size of H. When restricted to the class of
threshold graphs or to the class of trivially perfect graphs, the problem
becomes tractable in polynomial time. The problem is also soluble when one
cograph is given as an induced subgraph of the other. We characterize absolute
retracts of cographs.Comment: 15 page
Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds
The basic result of Oka theory, due to Gromov, states that every continuous
map from a Stein manifold to an elliptic manifold can be deformed
to a holomorphic map. It is natural to ask whether this can be done for all
at once, in a way that depends continuously on and leaves fixed if it
is holomorphic to begin with. In other words, is \scrO(S,X) a deformation
retract of \scrC(S,X)? We prove that it is if has a strictly
plurisubharmonic Morse exhaustion with finitely many critical points; in
particular, if is affine algebraic. The only property of used in the
proof is the parametric Oka property with approximation with respect to finite
polyhedra, so our theorem holds under the weaker assumption that is an Oka
manifold. Our main tool, apart from Oka theory itself, is the theory of
absolute neighbourhood retracts. We also make use of the mixed model structure
on the category of topological spaces.Comment: Version 2: A few very minor improvements to the exposition. Version
3: Another few very minor improvements to the exposition. To appear in
Proceedings AM
On Murayama's theorem on extensor properties of G-spaces of given orbit types
We develop a method of extending actions of compact transformation groups
which is then applied to the problem of preservation of equivariant extensor
property by passing to a subspace of given orbit types
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