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From Higher Invariants


3-MANIFOLDS AND FLOER THEORIES -- Abstracts

Guillem Cazassus

  • Title: Naturality and maps from cobordisms in symplectic instanton homology.
  • Abstract: Manolescu and Woodward defined homology groups associated to a closed connected oriented 3-manifold, called symplectic instanton homology, using Lagrangian Floer homology inside a moduli space of flat SU(2)-connexions associated to a punctured Heegaard surface. Using Wehrheim and Woodward's "Floer field theory" and pseudo-holomorphic quilts, I will show that these groups only depend on the choice of a basepoint, and will define maps associated to a smooth 4-dimensional cobordism equipped with a path.

Aliakbar Daemi

  • Title: Sutures and Higher Rank Bundles
  • Abstract: Donaldson's polynomials are strong invariants of smooth closed 4-manifolds which are defined using certain moduli spaces associated to complex vector bundles of rank 2. Later on, these invariants were generalized to higher rank bundles, firstly in physics and then in math. However, the role of these 4-manifold invariants in low dimensional topology are largely mysterious. The physicists expect that these invariants do no contain any new information about 4-manifolds.
In this talk, I'll explain how to confirm the predictions from physics about some families of 4-manifolds including elliptic surfaces. Nevertheless, these computations can be used to gain new information about manifolds of lower dimensions. In particular, I'll discuss how one can define an invariant of sutured 3-manifolds and obtain structural results about the quantum cohomology of moduli spaces of stable bundles on a Riemann surface. The sutured invariant is a potential tool to study rank three unitary representations of knot groups. This talk is based on an ongoing project, joint with Yi Xie.

Kim Frøyshov

  • Title: Mod 2 instanton Floer homology
  • Abstract: I will discuss ongoing work on a new kind of mod 2 Floer theory of oriented homology 3-spheres. This theory shows some formal similarities with monopole and Heegaard Floer theory.

Paolo Ghiggini

  • Title: An application of L^2 homology to symplectic topology.
  • Abstract: Let \Lambda be a simply connected Lagrangian submanifold of R^{2n+1} endowed with the standard contact structure. I will prove that, if \Lambda is "unobstructed" (e.g. is the boundary of a Spin exact Lagrangian submanifold of the symplectisation of R^{2n+1}), every exact symplectic cobordism from \Lambda to itself is simply connected. This is a joint work with Baptiste Chantraine, Georgios Dimitroglu Rizell and Roman Golovko.

Sherry Gong

  • Title: Khovanov homologies, instanton spectral sequences, and binary dihedral representations for alternating links with singular bundle data.
  • Abstract: Consider a link $L$ in a three dimensional manifold $Y$ with singular bundle data given by 1 dimensional manifold $\omega \subset Y$ with boundary in $L$, thought of as the dual of $w_2(P)$ for singular vector bundle $P$. We study the instanton spectral sequence associated to $(Y,L)$, and in particular relate it to the Khovanov homology. We will also explore the binary dihedral representations of alternating links with such singular bundle data.

Joshua Greene

  • Title: Alternating links and definite surfaces
  • Abstract: I will describe a characterization of the class of alternating links in terms intrinsic to the link exterior and use it to derive some properties of these links, including topological proofs of some of Tait's conjectures.

Andras Juhasz

  • Title: "Concordance maps in knot Floer homology"
  • Abstract: We show that a decorated knot concordance C from K to K′ induces a homomorphism FC on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to HFˆ(S3)≅Z2 that agrees with FC on the E1 page and is the identity on the E∞ page. It follows that FC is non-vanishing on HFKˆ0(K,τ(K)). We also obtain an invariant of slice disks in homology 4-balls bounding S3. If C is invertible, then FC is injective, hence dimHFKˆj(K,i)≤dimHFKˆj(K′,i) for every i, j∈Z. This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot K to K′, then g(K)≤g(K′), where g denotes the Seifert genus. Furthermore, if g(K)=g(K′) and K′ is fibred, then so is K. This is joint work with Marco Marengon.

Christine Lescop

  • Title: Functorial behaviour of 3-manifold invariants that count graph configurations.
  • Abstract: Following Witten's study of Chern-Simons theory, Kontsevich proposed ways of counting graph configurations in rational homology 3-spheres in order to produce universal finite type invariants of these 3-manifolds, and invariants of their links. We will describe an extension of these invariants to tangles in rational homology cylinders and discuss functorial properties of this extension.

Francesco Lin

  • Title: Some properties of Pin(2)-monopole Floer homology
  • Abstract: Pin(2)-monopole Floer homology is the Morse theoretic analogue of Manolescu's Pin(2)-Seiberg-Witten Floer homology. We discuss some formal properties of it and show some applications.

Paolo Lisca

  • Title: On 3-braid knots of finite concordance order.
  • Abstract: I will sketch how to prove that if K is a 3-braid knot of finite smooth concordance order then one the following holds:
(1) K is ribbon and quasipositive,
(2) K is a symmetric union, or
(3) K is amphichiral.

Andrew Lobb

  • Title: The Khovanov space and instanton knot Floer homology.
  • Abstract: Five years ago Lipshitz and Sarkar introduced a space-level refinement of Khovanov cohomology. Khovanov cohomology is a bigraded abelian group which is an invariant of links - the gradings are the quantum degree and the cohomological degree. Fixing one of the quantum degrees, Lipshitz-Sarkar showed how to produce a space (in fact a stable homotopy type) which is a link invariant and whose singular cohomology gives Khovanov cohomology in that quantum degree. We shall give an overview of similar /spacifications/ of quantum knot cohomologies. We shall also discuss recent work (with Orson and Schuetz) suggesting a possible connection with a representation space that appears in the construction of instanton knot Floer homology.

Daniel Ruberman

  • Title: Invariants of 4-manifolds with the homology of a circle cross the 3-sphere.
  • Abstract: Gauge theoretic invariants of a closed 4-manifold X usually are defined only in the setting where the intersection form has non-zero positive part; this condition rules out reducible solutions. Many interesting questions remain, however, in the setting where there is no second homology at all. We relate several invariants that have been previously defined in this context. The first is the invariant LSW(X) defined by Mrowka-Ruberman-Saveliev; it is a count of the solutions to the Seiberg-Witten equations plus an index-theoretic correction term. The invariant h(Y) for a rational homology 3-sphere was introduced by Froyshov; if Y is embedded in X, then h(Y) is actually an invariant of X alone.
Theorem: Let X be an integral homology circle cross the 3-sphereand assume that the Poincare dual of the homology orientation is realized by a rational homology sphere Y in X. Then
LSW+ h(Y) = Lef (W*: HMred(Y) -> HMred(Y)),
where W is the cobordism from Y to itself obtained by cutting X open along Y, and the right hand side of the above formula is the Lefschetz number of the map induced by W in the reduced monopole Floer homology of Y.
This is joint work with Jianfeng Lin and Nikolai Saveliev.

Gilberto Spano

  • Title: Twisted Gromov-Taubes invariants
  • Abstract: We define a twisted version of the Gromov-Taubes series for symplectic 4-manifolds, where the twisted coefficients are induced by the choice of a surface bundle over the 4-manifold. We will sketch the proof of the symplectic invariance of our twisted Gromov-Taubes series and relate it to the twisted Reidemeister torsions of 3-manifolds.

Matthew Stoffregen

  • Title: Two-fold Quasi-Alternating Links
  • Abstract: This is joint work with Chris Scaduto. We introduce a class of links, called "Two-fold Quasi-Alternating Links", strictly containing quasi-alternating links, and for which mod 2 reduced Khovanov homology is always thin. We also discuss evidence for a spectral sequence from a twisted variant of Khovanov homology to the framed instanton homology of the double branched cover, which collapses for our generalized class of links.

Saso Strle

  • Title: On the Thom conjecture in CP^3
  • Abstract: The original Thom conjecture states that holomorphic curves are minimal genus representatives of 2-dimensional homology classes in CP^2. It has been known for a long time that the analogous claim for codimension 2 homology classes in CP^n does not hold; Freedman showed that for n even any such class is represented by a submanifold which has smaller middle homology than a complex hypersurface representing this class and which on the level of homotopy behaves as a complex hypersurface. We consider the case of 4-manifolds in CP^3 and show that the rank of the 2nd homology in any given class can be significantly reduced. This is joint work with D. Ruberman and M. Slapar.

Ian Zemke

  • Title: Graph cobordism maps in Heegaard Floer homology
  • Abstract: We will discuss cobordism maps in Heegaard Floer homology for cobordisms with embedded graphs. We will discuss maps appearing in the construction, as well as the structure of the TQFT. We will describe applications of the theory to computations of basepoint moving maps, as well as applications to the involutive theory.

Speakers

  • Guillem Cazassus (University of Toulouse)
  • Aliakbar Daemi (Simons Center, Stony Brook)
  • Kim Frøyshov (University of Oslo)
  • Paolo Ghiggini (University of Nantes)
  • Sherry Gong (Massachusetts Institute of Technology)
  • Joshua Greene (Boston College)
  • Andras Juhasz (University of Oxford)
  • Christine Lescop (University Joseph Fourier Grenoble)
  • Francesco Lin (Massachusetts Institute of Technology)
  • Paolo Lisca (University of Pisa)
  • Andrew Lobb (Durham University)
  • Daniel Ruberman (Brandeis University)
  • Gilberto Spano (Renyi Institute of Mathematics)
  • Matthew Stoffregen (University of California Los Angeles)
  • Saso Strle (University of Liubliana)
  • Ian Zemke (University of California Los Angeles)