A Steenrod Square on Khovanov Homology
... Z (L) to each link L ⊂ S [Kho00]. In [LSa] we gave a space-level version of Khovanov homology. That is, to each link L we associated stable spaces (finite suspenj sion spectra) XKh (L), well-defined up to stable homotopy equivalence, so that the reduced j i e cohomology H (XKh (L)) of these spaces i ...
... Z (L) to each link L ⊂ S [Kho00]. In [LSa] we gave a space-level version of Khovanov homology. That is, to each link L we associated stable spaces (finite suspenj sion spectra) XKh (L), well-defined up to stable homotopy equivalence, so that the reduced j i e cohomology H (XKh (L)) of these spaces i ...
Computations and structures in sl(n)-link homology
... actions on some category, etc. One approach to constructing Khovanov homology begins with Kauffman’s solid-state model for the Jones polynomial. This is gotten by resolving each crossing of a link diagram in either the oriented or un-oriented resolution and assigning to each the polynomial (q+q −1 ) ...
... actions on some category, etc. One approach to constructing Khovanov homology begins with Kauffman’s solid-state model for the Jones polynomial. This is gotten by resolving each crossing of a link diagram in either the oriented or un-oriented resolution and assigning to each the polynomial (q+q −1 ) ...
sl(3) link homology 1 Introduction
... When g = sl(n) and each components of L is labelled either by the defining representation V or its dual, the invariant is determined by the skein relation in Figure 1. If we introduce a second variable p = q n , the skein relation gives rise to the HOMFLY polynomial, a 2-variable polynomial invarian ...
... When g = sl(n) and each components of L is labelled either by the defining representation V or its dual, the invariant is determined by the skein relation in Figure 1. If we introduce a second variable p = q n , the skein relation gives rise to the HOMFLY polynomial, a 2-variable polynomial invarian ...
Topology
... twentieth century it has also 1. provided notions and concepts that are of core importance for all of mathematics, such as the notion of “compactness”, 2. contributed a great variety of important methods and tools for the solution of mathematical problems in other areas — for example, there is a lar ...
... twentieth century it has also 1. provided notions and concepts that are of core importance for all of mathematics, such as the notion of “compactness”, 2. contributed a great variety of important methods and tools for the solution of mathematical problems in other areas — for example, there is a lar ...
T A G An invariant of link cobordisms
... Remark One may regard the signs + and − as linear generators of a commutative Frobenius algebra A, with multiplication and comultiplication defined by the calculus in the previous remark. It is a theorem (see e.g. [1]) that commutative Frobenius algebras are in one-to-one correspondence with (1 + 1) ...
... Remark One may regard the signs + and − as linear generators of a commutative Frobenius algebra A, with multiplication and comultiplication defined by the calculus in the previous remark. It is a theorem (see e.g. [1]) that commutative Frobenius algebras are in one-to-one correspondence with (1 + 1) ...
Mirror symmetry and T -duality in the complement of
... carries a nonvanishing holomorphic n-form Ω = σ −1 . By analogy with the Calabi-Yau situation, for a given φ ∈ R we make the following definition: Definition 2.1. A Lagrangian submanifold L ⊂ X \ D is special Lagrangian with phase φ if Im (e−iφ Ω)|L = 0. Multiplying Ω by e−iφ if necessary, in the res ...
... carries a nonvanishing holomorphic n-form Ω = σ −1 . By analogy with the Calabi-Yau situation, for a given φ ∈ R we make the following definition: Definition 2.1. A Lagrangian submanifold L ⊂ X \ D is special Lagrangian with phase φ if Im (e−iφ Ω)|L = 0. Multiplying Ω by e−iφ if necessary, in the res ...
Slides - Biomedical Informatics
... greater are extended in both directions in an attempt to find a locally optimal ungapped alignment or HSP (high scoring pair) with a score of at least S or an E value lower than the specified threshold. HSPs that meet these criteria will be reported by BLAST, provided they do not exceed the cutoff v ...
... greater are extended in both directions in an attempt to find a locally optimal ungapped alignment or HSP (high scoring pair) with a score of at least S or an E value lower than the specified threshold. HSPs that meet these criteria will be reported by BLAST, provided they do not exceed the cutoff v ...
2 - Ohio State Department of Mathematics
... homology 3–sphere, H 3 . The signature of Q(E8 ) is 8. X 4 is defined to be the union of this plumbing with c(H 3 ) (the cone on H 3 ). It is a polyhedral homology 4–manifold with one non-manifold point. By Freedman [8] we can topologically “resolve the singularity” of X 4 by replacing c(H 3 ) with ...
... homology 3–sphere, H 3 . The signature of Q(E8 ) is 8. X 4 is defined to be the union of this plumbing with c(H 3 ) (the cone on H 3 ). It is a polyhedral homology 4–manifold with one non-manifold point. By Freedman [8] we can topologically “resolve the singularity” of X 4 by replacing c(H 3 ) with ...
Geometric homology versus group homology - Math-UMN
... where the left-hand side is singular cohomology of a topological space Γ\X obtained as a quotient of a homeomorphic copy of an open ball X by the action of a discrete group Γ, and the right-hand side is group cohomology of Γ. A striking aspect of this is that the essence of the argument concerns hom ...
... where the left-hand side is singular cohomology of a topological space Γ\X obtained as a quotient of a homeomorphic copy of an open ball X by the action of a discrete group Γ, and the right-hand side is group cohomology of Γ. A striking aspect of this is that the essence of the argument concerns hom ...
A Glimpse into Symplectic Geometry
... can find an ω-compatible J on M that restricts to an almost complex structure on Q, i.e. J(T Q) = T Q. In particular, if Q = C is 2-dimensional it will be a complex submanifold with respect to any J such that J(T C) = T C. Such C are called J-holomorphic curves. It was the great insight of Gromov [G ...
... can find an ω-compatible J on M that restricts to an almost complex structure on Q, i.e. J(T Q) = T Q. In particular, if Q = C is 2-dimensional it will be a complex submanifold with respect to any J such that J(T C) = T C. Such C are called J-holomorphic curves. It was the great insight of Gromov [G ...
Homology stratifications and intersection homology Geometry & Topology Monographs Colin Rourke Brian Sanderson
... and a similar remark applies to homologies. Thus we get exactly the same groups if dπi,j is replaced by min(j , i + j − n + pn−j ). We now explain how to find a (unique) permutation π for which dπi,j has this value. Define a permutation π ∈ Σn+1 to be V –shaped if π|[0, u] is monotone decreasing and ...
... and a similar remark applies to homologies. Thus we get exactly the same groups if dπi,j is replaced by min(j , i + j − n + pn−j ). We now explain how to find a (unique) permutation π for which dπi,j has this value. Define a permutation π ∈ Σn+1 to be V –shaped if π|[0, u] is monotone decreasing and ...
PDF version - University of Warwick
... and a similar remark applies to homologies. Thus we get exactly the same groups if dπi,j is replaced by min(j , i + j − n + pn−j ). We now explain how to find a (unique) permutation π for which dπi,j has this value. Define a permutation π ∈ Σn+1 to be V –shaped if π|[0, u] is monotone decreasing and ...
... and a similar remark applies to homologies. Thus we get exactly the same groups if dπi,j is replaced by min(j , i + j − n + pn−j ). We now explain how to find a (unique) permutation π for which dπi,j has this value. Define a permutation π ∈ Σn+1 to be V –shaped if π|[0, u] is monotone decreasing and ...
A Prelude to Obstruction Theory - WVU Math Department
... Φ 2 : e2 → S 2 that contracts the boundary of e2 to e0 via a straight-line homotopy. Example. The torus in Figure 1.3 grants Φ 2 : e2 → T 2 defined by the identification of edges and, subsequently, Φ11 and Φ12 : e1i → S 1 for i = 1, 2, by identifying endpoints of each edge. Both of the these example ...
... Φ 2 : e2 → S 2 that contracts the boundary of e2 to e0 via a straight-line homotopy. Example. The torus in Figure 1.3 grants Φ 2 : e2 → T 2 defined by the identification of edges and, subsequently, Φ11 and Φ12 : e1i → S 1 for i = 1, 2, by identifying endpoints of each edge. Both of the these example ...
Homotopy type of symplectomorphism groups of × S Geometry & Topology
... homotopy equivalent to its subgroup of standard isometries SO(3) × SO(3). He also showed that this would no longer hold when one sphere is larger than the other, and in [9] McDuff constructed explicitly an element of infinite order in H1 (Gλ ), λ > 0. The main tool in their proofs is to look at the ...
... homotopy equivalent to its subgroup of standard isometries SO(3) × SO(3). He also showed that this would no longer hold when one sphere is larger than the other, and in [9] McDuff constructed explicitly an element of infinite order in H1 (Gλ ), λ > 0. The main tool in their proofs is to look at the ...
“GROUP-COMPLETION”, LOCAL COEFFICIENT SYSTEMS, AND
... Corollary 1.2. — Let M , m1 , m2 , . . . satisfy the assumptions of Theorem 1.1. Then there is a weak homotopy equivalence ...
... Corollary 1.2. — Let M , m1 , m2 , . . . satisfy the assumptions of Theorem 1.1. Then there is a weak homotopy equivalence ...
Pseudoholomorphic Curves and Mirror Symmetry
... Recall that an almost complex structure on a smooth manifold M is a smooth (1, 1)tensor J such that the vector bundle isomorphism J : T M → T M satisfies J 2 = − Id. In other words, J gives a smooth family of complex structures on the tangent spaces of M . A smooth manifold with an almost complex st ...
... Recall that an almost complex structure on a smooth manifold M is a smooth (1, 1)tensor J such that the vector bundle isomorphism J : T M → T M satisfies J 2 = − Id. In other words, J gives a smooth family of complex structures on the tangent spaces of M . A smooth manifold with an almost complex st ...
Homology Theory - Section de mathématiques
... french mathematician Henri Poincaré who gave a somewhat fuzzy definition of what “homology” should be in 1895. Thirty years later, it was realised by Emmy Noether that abelian groups were the right context to study homology and not the then known and extensively used Betti numbers. In the decades af ...
... french mathematician Henri Poincaré who gave a somewhat fuzzy definition of what “homology” should be in 1895. Thirty years later, it was realised by Emmy Noether that abelian groups were the right context to study homology and not the then known and extensively used Betti numbers. In the decades af ...
1. Introduction 1 2. Simplicial and Singular Intersection Homology 2
... Definition 2.5. A singular i-simplex σ : ∆i → X is said to be p-allowable, if σ −1 (Xm−k ) ⊆ (i − k + p(k)) − skeleton of∆i . Similarly, a Borel-Moore singular i-chain is p-allowable if each simplex that appears with nonzero coefficient is allowable. Define I p,BM Si (X) to be the vector space over ...
... Definition 2.5. A singular i-simplex σ : ∆i → X is said to be p-allowable, if σ −1 (Xm−k ) ⊆ (i − k + p(k)) − skeleton of∆i . Similarly, a Borel-Moore singular i-chain is p-allowable if each simplex that appears with nonzero coefficient is allowable. Define I p,BM Si (X) to be the vector space over ...
Aspherical manifolds that cannot be triangulated
... p. 356]). So, if K is a polyhedral homology manifold, then so is h(K). (e) If K is a polyhedral homology manifold, then f : h(K) → K pulls back the Stiefel-Whitney classes of K to those of h(K). In [4] the above version of hyperbolization is used to define a “relative hyperbolization procedure” (an ...
... p. 356]). So, if K is a polyhedral homology manifold, then so is h(K). (e) If K is a polyhedral homology manifold, then f : h(K) → K pulls back the Stiefel-Whitney classes of K to those of h(K). In [4] the above version of hyperbolization is used to define a “relative hyperbolization procedure” (an ...
![arXiv:math/0302340v2 [math.AG] 7 Sep 2003](http://s1.studyres.com/store/data/014807023_1-69877329430c783f1dba3c51b4839685-300x300.png)
arXiv:math/0302340v2 [math.AG] 7 Sep 2003
... properties of cohomology of complex algebraic varieties. One of the ingredients of mixed Hodge structure is the weight filtration. We will focus on the dual filtration in homology. We will find a relation between intersection homology defined by M. Goresky–R. MacPherson and the weight filtration. Th ...
... properties of cohomology of complex algebraic varieties. One of the ingredients of mixed Hodge structure is the weight filtration. We will focus on the dual filtration in homology. We will find a relation between intersection homology defined by M. Goresky–R. MacPherson and the weight filtration. Th ...
Partial Groups and Homology
... mani-fold” to “Γ- structure on a topological space”. Each Γ has a classifying space BΓ, and questions about foliations, in particular those involving different degrees of differentiability, can often be formulated in terms of their algebraic invariants. For example,the non-existence of codimension-o ...
... mani-fold” to “Γ- structure on a topological space”. Each Γ has a classifying space BΓ, and questions about foliations, in particular those involving different degrees of differentiability, can often be formulated in terms of their algebraic invariants. For example,the non-existence of codimension-o ...
A pairing between super Lie-Rinehart and periodic cyclic
... pairing cyclic cohomology with nontrivial coefficients in the sense of [9]. It is defined for a Hopf algebra H, an H-module algebra A, an H-comodule algebra B, an H-module coalgebra C acting on A in a suitable sense and any stable anti-Yetter-Drinfeld (SAYD) module M over H. For C = H = U (g), q = 0 ...
... pairing cyclic cohomology with nontrivial coefficients in the sense of [9]. It is defined for a Hopf algebra H, an H-module algebra A, an H-comodule algebra B, an H-module coalgebra C acting on A in a suitable sense and any stable anti-Yetter-Drinfeld (SAYD) module M over H. For C = H = U (g), q = 0 ...
Simplicial Sets - Stanford Computer Graphics
... • Linear-time algorithm to reduce the size of a complex • Can use Gaussian Elimination to compute Homology of simplified complex ...
... • Linear-time algorithm to reduce the size of a complex • Can use Gaussian Elimination to compute Homology of simplified complex ...