## Geometry and Topology Seminar

This seminar discusses geometry and topology in a broad sense including (but not exclusively) Riemannian geometry, low dimensional topology and geometry, and geometric group theory.

### Pontryagin invariants and integral formulas for Milnor's triple linking number

by Herman Gluck | University of Pennsylvania
Time: 16:10 — 17:00  (Tuesday, Oct. 12, 2010)
Location: BA1200, Bahen Center, 40 St George St
Abstract:
To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere, while the integral itself can be viewed as the helicity of a related vector field on the 3-torus. The motivation for all this comes from the Arnol'd-Khesin search for higher-order helicities of incompressible fluid flows.

This represents joint work with Dennis DeTurck, Rafal Komendarczyk, Paul Melvin, Clayton Shonkwiler and Shea Vela-Vick.

#### Dates in this series

· Monday, Mar. 08, 2010: The fundamental group of random 2-complexes (Matthew Kahle)
· Monday, Mar. 15, 2010: Three Dimensional Manifolds All of Whose Geodesics Are Closed (John Olsen)
· Monday, Mar. 22, 2010: Curves on surfaces with large topology (Hugo Parlier)
· Friday, Mar. 26, 2010: New invariants in singularity theory (Rustam Sadykov)
· Friday, Mar. 26, 2010: The hidden smoothness of the space of stable maps (Jonathan Wise)
· Monday, Apr. 05, 2010: Moduli spaces of conics in curves (Chris Judge)
· Monday, Sep. 20, 2010: Total Q-curvature for CR manifolds (Kengo Hirachi)
· Monday, Sep. 27, 2010: Group completions of moduli spaces (Rustam Sadykov)
· Tuesday, Oct. 12, 2010: Pontryagin invariants and integral formulas for Milnor's triple linking number (Herman Gluck)
· Monday, Oct. 18, 2010: Positively curved polar manifolds (Fuquan Fang)
· Monday, Nov. 01, 2010: Rational formality and special holonomy (Manuel Amann)
· Monday, Nov. 08, 2010: Isoperimetric Inequalities -- methods and applications (Emanuel Milman)
· Monday, Nov. 15, 2010: From the sandwich to the waist (Alfredo Hubard)
· Monday, Nov. 29, 2010: TBA (Keith Burns)
· Monday, Nov. 29, 2010: Conjugate points and non-injectivity of the exponential map (Keith Burns)
· Monday, Jan. 17, 2011: Loop products and resonance for loop homology on spheres (Nancy Hingston)
· Monday, Jan. 31, 2011: A Variational Characterization of the Catenoid (Jacob Bernstein)
· Monday, Apr. 18, 2011: On the structure of conformally covariant powers of the Laplacian (Andreas Juhl)
· Monday, May. 09, 2011: Causality, Low conjecture and globally hyperbolic spacetimes (Vladimir Chernov)
· Monday, Sep. 26, 2011: Mapping degrees of self-maps of simply-connected manifolds (Manuel Amann)
· Monday, Oct. 17, 2011: Formality of Positive Quaternion Kähler manifolds (Vitali Kapovitch)
· Friday, Oct. 21, 2011: Kahler-Einstein metrics singular along a divisor (Yanir Rubinstein)
· Monday, Oct. 24, 2011: Diastolic inequality on a 2-sphere (Yevgeny Liokumovich)
· Monday, Nov. 14, 2011: TBA (Ian Hambleton)
· Monday, Nov. 14, 2011: Coarse geometry and group actions (Ian Hambleton)
· Monday, Nov. 21, 2011: Fundamental groups of manifolds with lower Ricci curvature bounds (Vitali Kapovitch)
· Monday, Nov. 28, 2011: Fundamental groups of manifolds with lower Ricci curvature bounds II (Vitali Kapovitch)
· Tuesday, Nov. 29, 2011: Asymptotic geometry, finite generation of fundamental groups, and harmonic functions (Bruce Kleiner)
· Friday, Dec. 09, 2011: Topological Complexity and H-Spaces (Gregory Lupton)
· Tuesday, Jan. 24, 2012: A new construction of compact $G_2$ manifolds by glueing families of Eguchi-Hanson spaces (Spiro Karigiannis)
· Thursday, Jan. 26, 2012: Coarse Differentiation and the rank of Teichmüller space (Kasra Rafi)
· Thursday, Feb. 16, 2012: Manifolds and spaces with Ricci curvature bounds (Toby Colding)
· Monday, Feb. 27, 2012: Homological and homotopical filling functions (Robert Young)
· Monday, Mar. 12, 2012: Algorithmic decidability of Thurston equivalence and geometrization of branched coverings (Michael Yampolsky)
· Monday, Mar. 26, 2012: On some problems in the geometry of surfaces in normed spaces (Dmitri Burago)
· Monday, Apr. 02, 2012: CANCELLED -- Scattering rigidity with trapped geodesics; the two dimensional case (Chris Croke)
· Monday, Sep. 10, 2012: Realisation and dismantlability (Piotr Przytycki)
· Friday, Oct. 05, 2012: The metric completion of Outer Space (Yael Algom Kfir)
· Monday, Oct. 15, 2012: Complexes which require large filling distortion in Euclidean space (Dominic Dotterrer)
· Tuesday, Nov. 06, 2012: A theorem of existence for Asymptotically Conical Calabi-Yau manifolds (Ronan Conlon)
· Monday, Nov. 19, 2012: Ricci Solitons of Cohomogeneity One (McKenzie Wang)
· Monday, Nov. 26, 2012: Complete affine 3-manifolds, Lorentzian and hyperbolic geometry (Bill Goldman)
· Monday, Dec. 03, 2012: Hyperbolic Group Boundaries and Cannon-Thurston Maps (Owen Baker)