4,887 research outputs found
The Directed Homotopy Hypothesis
The homotopy hypothesis was originally stated by Grothendieck: topological spaces should be "equivalent" to (weak) infinite-groupoids, which give algebraic representatives of homotopy types. Much later, several authors developed geometrizations of computational models, e.g., for rewriting, distributed systems, (homotopy) type theory etc.
But an essential feature in the work set up in concurrency theory, is that time should be considered irreversible, giving rise to the field of directed algebraic topology. Following the path proposed by Porter, we state here a directed homotopy hypothesis: Grandis\u27 directed topological spaces should be "equivalent" to a weak form of topologically enriched categories, still very close to (infinite,1)-categories. We develop, as in ordinary algebraic topology, a directed homotopy equivalence and a weak equivalence, and show invariance of a form of directed homology
On growth rate and contact homology
It is a conjecture of Colin and Honda that the number of Reeb periodic orbits
of universally tight contact structures on hyperbolic manifolds grows
exponentially with the period, and they speculate further that the growth rate
of contact homology is polynomial on non-hyperbolic geometries. Along the line
of the conjecture, for manifolds with a hyperbolic component that fibers on the
circle, we prove that there are infinitely many non-isomorphic contact
structures for which the number of Reeb periodic orbits of any non-degenerate
Reeb vector field grows exponentially. Our result hinges on the exponential
growth of contact homology which we derive as well. We also compute contact
homology in some non-hyperbolic cases that exhibit polynomial growth, namely
those of universally tight contact structures non-transverse to the fibers on a
circle bundle
Homotopy groups of ascending unions of infinite-dimensional manifolds
Let M be a topological manifold modelled on topological vector spaces, which
is the union of an ascending sequence of such manifolds M_n. We formulate a
mild condition ensuring that the k-th homotopy group of M is the direct limit
of the k-th homotopy groups of the steps M_n, for each non-negative integer k.
This result is useful for Lie theory, because many important examples of
infinite-dimensional Lie groups G can be expressed as ascending unions of
finite- or infinite-dimensional Lie groups (whose homotopy groups may be easier
to access). Information on the k-th homotopy groups of G, for k=0, k=1 and k=2,
is needed to understand the Lie group extensions of G with abelian kernels.
The above conclusion remains valid if the union of the steps M_n is merely
dense in M (under suitable hypotheses). Also, ascending unions can be replaced
by (possibly uncountable) directed unions.Comment: 44 pages, LaTeX; v2: update of reference
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