4,453 research outputs found
Weighted Modulo Orientations of Graphs
This dissertation focuses on the subject of nowhere-zero flow problems on graphs. Tutte\u27s 5-Flow Conjecture (1954) states that every bridgeless graph admits a nowhere-zero 5-flow, and Tutte\u27s 3-Flow Conjecture (1972) states that every 4-edge-connected graph admits a nowhere-zero 3-flow. Extending Tutte\u27s flows conjectures, Jaeger\u27s Circular Flow Conjecture (1981) says every 4k-edge-connected graph admits a modulo (2k+1)-orientation, that is, an orientation such that the indegree is congruent to outdegree modulo (2k+1) at every vertex. Note that the k=1 case of Circular Flow Conjecture coincides with the 3-Flow Conjecture, and the case of k=2 implies the 5-Flow Conjecture. This work is devoted to providing some partial results on these problems.
In Chapter 2, we study the problem of modulo 5-orientation for given multigraphic degree sequences. We prove that a multigraphic degree sequence d=(d1,..., dn) has a realization G with a modulo 5-orientation if and only if di≤1,3 for each i. In addition, we show that every multigraphic sequence d=(d1,..., dn) with min{1≤i≤n}di≥9 has a 9-edge-connected realization that admits a modulo 5-orientation for every possible boundary function. Jaeger conjectured that every 9-edge-connected multigraph admits a modulo 5-orientation, whose truth would imply Tutte\u27s 5-Flow Conjecture. Consequently, this supports the conjecture of Jaeger.
In Chapter 3, we show that there are essentially finite many exceptions for graphs with bounded matching numbers not admitting any modulo (2k+1)-orientations for any positive integers t. We additionally characterize all infinite many graphs with bounded matching numbers but without a nowhere-zero 3-flow. This partially supports Jaeger\u27s Circular Flow Conjecture and Tutte\u27s 3-Flow Conjecture.
In 2018, Esperet, De Verclos, Le and Thomass introduced the problem of weighted modulo orientations of graphs and indicated that this problem is closely related to modulo orientations of graphs, including Tutte\u27s 3-Flow Conjecture. In Chapter 4 and Chapter 5, utilizing properties of additive bases and contractible configurations, we reduced the Esperet et al\u27s edge-connectivity lower bound for some (signed) graphs families including planar graphs, complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families.
In Chapter 6, we show that the assertion of Jaeger\u27s Circular Flow Conjecture with k=2 holds asymptotically almost surely for random 9-regular graphs
Balanced simplices
An additive cellular automaton is a linear map on the set of infinite
multidimensional arrays of elements in a finite cyclic group
. In this paper, we consider simplices appearing in the
orbits generated from arithmetic arrays by additive cellular automata. We prove
that they are a source of balanced simplices, that are simplices containing all
the elements of with the same multiplicity. For any
additive cellular automaton of dimension or higher, the existence of
infinitely many balanced simplices of appearing in
such orbits is shown, and this, for an infinite number of values . The
special case of the Pascal cellular automata, the cellular automata generating
the Pascal simplices, that are a generalization of the Pascal triangle into
arbitrary dimension, is studied in detail.Comment: 33 pages ; 11 figures ; 1 tabl
Whitney tower concordance of classical links
This paper computes Whitney tower filtrations of classical links. Whitney
towers consist of iterated stages of Whitney disks and allow a tree-valued
intersection theory, showing that the associated graded quotients of the
filtration are finitely generated abelian groups. Twisted Whitney towers are
studied and a new quadratic refinement of the intersection theory is
introduced, measuring Whitney disk framing obstructions. It is shown that the
filtrations are completely classified by Milnor invariants together with new
higher-order Sato-Levine and higher-order Arf invariants, which are
obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link
in the 3-sphere. Applications include computation of the grope filtration, and
new geometric characterizations of Milnor's link invariants.Comment: Only change is the addition of this comment: This paper subsumes the
entire preprint "Geometric Filtrations of Classical Link Concordance"
(arXiv:1101.3477v2 [math.GT]) and the first six sections of the preprint
"Universal Quadratic Forms and Untwisting Whitney Towers" (arXiv:1101.3480v2
[math.GT]
For Complex Orientations Preserving Power Operations, p-typicality is Atypical
We show, for primes p less than or equal to 13, that a number of well-known
MU_(p)-rings do not admit the structure of commutative MU_(p)-algebras. These
spectra have complex orientations that factor through the Brown-Peterson
spectrum and correspond to p-typical formal group laws. We provide computations
showing that such a factorization is incompatible with the power operations on
complex cobordism. This implies, for example, that if E is a Landweber exact
MU_(p)-ring whose associated formal group law is p-typical of positive height,
then the canonical map MU_(p) --> E is not a map of H_\infty ring spectra. It
immediately follows that the standard p-typical orientations on BP, E(n), and
E_n do not rigidify to maps of E_\infty ring spectra. We conjecture that
similar results hold for all primes.Comment: Minor revisions, results extended up to the prime 13. Accepted for
publication. 22 page
Instantons and odd Khovanov homology
We construct a spectral sequence from the reduced odd Khovanov homology of a
link converging to the framed instanton homology of the double cover branched
over the link, with orientation reversed. Framed instanton homology counts
certain instantons on the cylinder of a 3-manifold connect-summed with a
3-torus. En route, we provide a new proof of Floer's surgery exact triangle for
instanton homology using metric stretching maps, and generalize the exact
triangle to a link surgeries spectral sequence. Finally, we relate framed
instanton homology to Floer's instanton homology for admissible bundles.Comment: 64 pages, 19 figure
Homology cycles in manifolds with locally standard torus actions
Let be a -manifold with a locally standard action of a compact torus
. If the free part of action is trivial and proper faces of the orbit
space are acyclic, then there are three types of homology classes in :
(1) classes of face submanifolds; (2) -dimensional classes of swept by
actions of subtori of dimensions ; (3) relative -classes of modulo
swept by actions of subtori of dimensions . The
submodule of spanned by face classes is an ideal in with
respect to the intersection product. It is isomorphic to
, where is the face ring of the
Buchsbaum simplicial poset dual to ; is the linear system of
parameters determined by the characteristic function; and is a certain
submodule, lying in the socle of . Intersections of
homology classes different from face submanifolds are described in terms of
intersections on and .Comment: 25 pages, 3 figures. Minor correction in Lemma 3.3 and a calculations
of Subsection 7.
An inequality for the h-invariant in instanton Floer theory
In math.DG/9903083 (henceforth referred to as EA) we defined an integer
invariant for oriented integral homology 3-spheres which only
depends on the rational homology cobordism class of and is additive under
connected sums. In this paper we establish lower bounds for when is
the boundary of a smooth, compact, oriented 4-manifold with . As
applications, we give an upper bound for how much changes under -1 surgery
on knots in terms of the slice genus of the knot, and compute for a family
of Brieskorn spheres.
This paper contains, in revised form, most of the material from v1 of EA that
was left out in the final version of that paper. In particular, Theorem 1 of
the present paper is virtually the same as Theorem 1 of v1 of EA. The proof is
also essentially the same, but the exposition has been improved, with more
details.Comment: 35 pages. One section has been added outlining the proof of the main
theorem, and one appendix has been added. To appear in Topolog
The Hodge ring of Kaehler manifolds
We determine the structure of the Hodge ring, a natural object encoding the
Hodge numbers of all compact Kaehler manifolds. As a consequence of this
structure, there are no unexpected relations among the Hodge numbers, and no
essential differences between the Hodge numbers of smooth complex projective
varieties and those of arbitrary Kaehler manifolds. The consideration of
certain natural ideals in the Hodge ring allows us to determine exactly which
linear combinations of Hodge numbers are birationally invariant, and which are
topological invariants. Combining the Hodge and unitary bordism rings, we are
also able to treat linear combinations of Hodge and Chern numbers. In
particular, this leads to a complete solution of a classical problem of
Hirzebruch's.Comment: Dedicated to the memory of F. Hirzebruch. To appear in Compositio
Mat
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