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Homotopical intersection theory
, 2009
"... Abstract. We give a new approach to intersection theory. Our “cycles ” are closed manifolds mapping into compact manifolds and our “intersections ” are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjuncti ..."
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Abstract. We give a new approach to intersection theory. Our “cycles ” are closed manifolds mapping into compact manifolds and our “intersections ” are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn [HQ], but our proofs are fundamentally different. Contents
GRADED POISSON ALGEBRAS ON BORDISM GROUPS OF GARLANDS AND THEIR APPLICATIONS
, 2006
"... Let M be an oriented manifold and let N be a set consisting of oriented closed manifolds of possibly different dimensions. Roughly speaking, the space GN,M of Ngarlands in M is the space of mappings into M of singular manifolds obtained by gluing manifolds from N at some marked points. In our prev ..."
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Cited by 4 (1 self)
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Let M be an oriented manifold and let N be a set consisting of oriented closed manifolds of possibly different dimensions. Roughly speaking, the space GN,M of Ngarlands in M is the space of mappings into M of singular manifolds obtained by gluing manifolds from N at some marked points. In our previous work with Rudyak we introduced a rich algebra structure on the bordism group Ω∗(GN,M). In this work we introduce the operations ⋆ and [·, ·] on the tensor product of Q and a certain bordism group ̂Ω∗(GN,M). For N consisting of odddimensional manifolds, these operations make ̂Ω∗(GN,M) ⊗ Q into a graded Poisson algebra (Gerstenhaberlike algebra). For N consisting of evendimensional manifolds, ⋆ satisfies a graded Leibniz rule with respect to [·, ·], but [·, ·] does not satisfy a graded Jacobi identity. The mod 2analogue of [·, ·] for oneelement sets N was previously constructed in our preprint with Rudyak. For N = {S 1} and a surface F 2, the subalgebra ̂ Ω0(G {S 1},F 2) ⊗ Q of our algebra is related to the GoldmanTuraev algebra of loops on a surface and to the AndersenMattesReshetikhin Poisson algebra of chorddiagrams. As an application, our Lie bracket allows one to compute the minimal number of intersection points of loops in two given homotopy classes ̂ δ1, ̂ δ2 of free loops on F 2, provided that ̂ δ1, ̂ δ2 do not contain powers of the same element of π1(F 2).
A Manifold Calculus Approach to Link Maps and the Linking Number
, 2007
"... We study the space of link maps Link(P1,... Pk;N), the space of maps P1 · · · Pk → N such that the images of the Pi are pairwise disjoint. We apply the manifold calculus of functors developed by Goodwillie and Weiss to study the difference between it and its linear and quadratic approximations. We ..."
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We study the space of link maps Link(P1,... Pk;N), the space of maps P1 · · · Pk → N such that the images of the Pi are pairwise disjoint. We apply the manifold calculus of functors developed by Goodwillie and Weiss to study the difference between it and its linear and quadratic approximations. We identify an appropriate generalization of the linking number as the geometric object which measures the difference between the space of link maps and its linear approximation. Our analysis of the difference between link maps and its quadratic approximation connects with recent work of the author, and is used to show that the Borromean rings are linked. Key words: links, calculus of functors, linking number PACS: 1
Symplectic monodromy, Leray residues and quasihomogeneous polynomials
, 2012
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We
, 2007
"... study the space of link maps Link(P1,... Pk;N), which is the space of maps P1 · · · Pk → N such that the images of the Pi are pairwise disjoint. We apply the manifold calculus of functors developed by Goodwillie and Weiss to study the difference between it and its linear and quadratic approximatio ..."
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study the space of link maps Link(P1,... Pk;N), which is the space of maps P1 · · · Pk → N such that the images of the Pi are pairwise disjoint. We apply the manifold calculus of functors developed by Goodwillie and Weiss to study the difference between it and its linear and quadratic approximations. We identify an appropriate generalization of the linking number as the geometric object which measures the difference between the space of link maps and its linear approximation. Our analysis of the difference between link maps and its quadratic approximation connects with recent work of the author, and is used to show that the Borromean rings are linked. Key words: links, calculus of functors, linking number PACS: 1
CONFORMAL INVARIANTS FROM NODAL SETS. I. NEGATIVE EIGENVALUES AND CURVATURE PRESCRIPTION
"... In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a ..."
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In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant’s Nodal Domain Theorem. We also show that on any manifold of dimension n ≥ 3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n ≥ 3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. We describe the invariants arising from the Yamabe and Paneitz operators associated to leftinvariant metrics on Heisenberg manifolds. Finally, in the appendix, the 2nd named author and Andrea Malchiodi study the Qcurvature prescription problems for noncritical Qcurvatures.