21 research outputs found
Heaps and unpointed stable homotopy theory
In this paper, we show how certain ``stability phenomena'' in unpointed model
categories provide the sets of homotopy classes with the structure of abelian
heaps, i.e. abelian groups without a choice of a zero. In contrast with the
classical situation of stable (pointed) model categories, these sets can be
empty
Geometric Embeddability of Complexes Is ??-Complete
We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in ?^d is complete for the Existential Theory of the Reals for all d ? 3 and k ? {d-1,d}. Consequently, the problem is polynomial time equivalent to determining whether a polynomial equation system has a real solution and other important problems from various fields related to packing, Nash equilibria, minimum convex covers, the Art Gallery Problem, continuous constraint satisfaction problems, and training neural networks. Moreover, this implies NP-hardness and constitutes the first hardness result for the algorithmic problem of geometric embedding (abstract simplicial) complexes. This complements recent breakthroughs for the computational complexity of piece-wise linear embeddability
Algorithmic aspects of immersibility and embeddability
We analyze an algorithmic question about immersion theory: for which ,
, and or is the question of whether an
-dimensional -manifold is immersible in decidable? As a
corollary, we show that the smooth embeddability of an -manifold with
boundary in is undecidable when is even and .Comment: 20 pages, 1 figure. Revised in response to comments by several
referees, no major changes in mathematical conten
Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the -Metastable Range
Motivated by Tverberg-type problems in topological combinatorics and by
classical results about embeddings (maps without double points), we study the
question whether a finite simplicial complex K can be mapped into R^d without
higher-multiplicity intersections. We focus on conditions for the existence of
almost r-embeddings, i.e., maps from K to R^d without r-intersection points
among any set of r pairwise disjoint simplices of K.
Generalizing the classical Haefliger-Weber embeddability criterion, we show
that a well-known necessary deleted product condition for the existence of
almost r-embeddings is sufficient in a suitable r-metastable range of
dimensions (r d > (r+1) dim K +2). This significantly extends one of the main
results of our previous paper (which treated the special case where d=rk and
dim K=(r-1)k, for some k> 3).Comment: 35 pages, 10 figures (v2: reference for the algorithmic aspects
updated & appendix on Block Bundles added