4,453 research outputs found

    Weighted Modulo Orientations of Graphs

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    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

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    An additive cellular automaton is a linear map on the set of infinite multidimensional arrays of elements in a finite cyclic group Z/mZ\mathbb{Z}/m\mathbb{Z}. 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 Z/mZ\mathbb{Z}/m\mathbb{Z} with the same multiplicity. For any additive cellular automaton of dimension 11 or higher, the existence of infinitely many balanced simplices of Z/mZ\mathbb{Z}/m\mathbb{Z} appearing in such orbits is shown, and this, for an infinite number of values mm. 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

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    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

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    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

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    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

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    Let XX be a 2n2n-manifold with a locally standard action of a compact torus TnT^n. If the free part of action is trivial and proper faces of the orbit space QQ are acyclic, then there are three types of homology classes in XX: (1) classes of face submanifolds; (2) kk-dimensional classes of QQ swept by actions of subtori of dimensions <k<k; (3) relative kk-classes of QQ modulo ∂Q\partial Q swept by actions of subtori of dimensions ⩾k\geqslant k. The submodule of H∗(X)H_*(X) spanned by face classes is an ideal in H∗(X)H_*(X) with respect to the intersection product. It is isomorphic to (Z[SQ]/Θ)/W(\mathbb{Z}[S_Q]/\Theta)/W, where Z[SQ]\mathbb{Z}[S_Q] is the face ring of the Buchsbaum simplicial poset SQS_Q dual to QQ; Θ\Theta is the linear system of parameters determined by the characteristic function; and WW is a certain submodule, lying in the socle of Z[SQ]/Θ\mathbb{Z}[S_Q]/\Theta. Intersections of homology classes different from face submanifolds are described in terms of intersections on QQ and TnT^n.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

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    In math.DG/9903083 (henceforth referred to as EA) we defined an integer invariant h(Y)h(Y) for oriented integral homology 3-spheres YY which only depends on the rational homology cobordism class of YY and is additive under connected sums. In this paper we establish lower bounds for h(Y)h(Y) when YY is the boundary of a smooth, compact, oriented 4-manifold with b2+=1b_2^+=1. As applications, we give an upper bound for how much hh changes under -1 surgery on knots in terms of the slice genus of the knot, and compute hh 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

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    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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