Asaeda-Przytycki-Sikora, Manturov, and Gabrovšek extended Khovanov homology to links in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R double-struck upper P cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> …
Asaeda-Przytycki-Sikora, Manturov, and Gabrovšek extended Khovanov homology to links in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R double-struck upper P cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {RP}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We construct a Lee-type deformation of their theory, and use it to define an analogue of Rasmussen’s <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant in this setting. We show that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant gives constraints on the genera of link cobordisms in the cylinder <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I times double-struck upper R double-struck upper P cubed"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">I \times \mathbb {RP}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As an application, we give examples of freely <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-periodic knots in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S cubed"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that are concordant but not standardly equivariantly concordant.
Asaeda-Przytycki-Sikora, Manturov, and Gabrov\v{s}ek extended Khovanov homology to links in $\mathbb{RP}^3$. We construct a Lee-type deformation of their theory, and use it to define an analogue of Rasmussen's s-invariant in …
Asaeda-Przytycki-Sikora, Manturov, and Gabrov\v{s}ek extended Khovanov homology to links in $\mathbb{RP}^3$. We construct a Lee-type deformation of their theory, and use it to define an analogue of Rasmussen's s-invariant in this setting. We show that the s-invariant gives constraints on the genera of link cobordisms in the cylinder $I \times \mathbb{RP}^3$. As an application, we give examples of freely 2-periodic knots in $S^3$ that are concordant but not standardly equivariantly concordant.
Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product …
Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered Floer homology invariants of 3-manifolds with codimension-2 corners, and prove that the bordered Floer homology of a 3-manifold with boundary, split into two pieces with corners, can be recovered as a tensor product of the cornered invariants of the pieces.
We show that monopole Floer homology (as defined by Kronheimer and Mrowka) is isomorphic to the S^1-equivariant homology of the Seiberg-Witten Floer spectrum constructed by the second author.
We show that monopole Floer homology (as defined by Kronheimer and Mrowka) is isomorphic to the S^1-equivariant homology of the Seiberg-Witten Floer spectrum constructed by the second author.
The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with an emphasis on the …
The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with an emphasis on the local equivalence methods coming from Pin(2)- equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homology.
The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with an emphasis on the …
The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with an emphasis on the local equivalence methods coming from Pin(2)- equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homology.
We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive …
We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.
We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive …
We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.
We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pin(2) symmetry of the Seiberg-Witten equations. We …
We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pin(2) symmetry of the Seiberg-Witten equations. We also explore a related construction, of an involutive version of Heegaard Floer homology.
We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a …
We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer / Heegaard Floer correspondence, we deduce that if a 3-manifold Y admits a p^n-sheeted regular cover that is a Z/pZ-L-space (for p prime), then Y is a Z/pZ-L-space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots.
We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of …
We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with $\underline{d}(Y) \neq d(Y) \neq \overline{d}(Y)$. We also construct a homomorphism from the three-dimensional homology cobordism group to an algebraically defined Abelian group, consisting of certain complexes (equipped with a homotopy involution) modulo a notion of local equivalence.
We show that monopole Floer homology (as defined by Kronheimer and Mrowka) is isomorphic to the S^1-equivariant homology of the Seiberg-Witten Floer spectrum constructed by the second author.
We show that monopole Floer homology (as defined by Kronheimer and Mrowka) is isomorphic to the S^1-equivariant homology of the Seiberg-Witten Floer spectrum constructed by the second author.
We define <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P i n left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Pin</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Pin}(2)</mml:annotation> </mml:semantics> </mml:math> …
We define <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P i n left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Pin</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Pin}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivariant Seiberg-Witten Floer homology for rational homology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-spheres equipped with a spin structure. The analogue of Frøyshov’s correction term in this setting is an integer-valued invariant of homology cobordism whose mod <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> reduction is the Rokhlin invariant. As an application, we show that there are no homology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-spheres <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Rokhlin invariant one such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y number-sign upper Y"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mi mathvariant="normal">#<!-- # --></mml:mi> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Y \#Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounds an acyclic smooth <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.
We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. …
We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then describe Floer stable homotopy types, the related Pin(2)-equivariant Seiberg-Witten Floer homology, and its application to the triangulation conjecture.
Ozsváth and Szabó gave a combinatorial description of knot Floer homology based on a cube of resolutions, which uses maps with twisted coefficients. We study the t=1 specialization of their …
Ozsváth and Szabó gave a combinatorial description of knot Floer homology based on a cube of resolutions, which uses maps with twisted coefficients. We study the t=1 specialization of their construction. The associated spectral sequence converges to knot Floer homology, and we conjecture that its E_1 page is isomorphic to the HOMFLY-PT chain complex of Khovanov and Rozansky. At the level of each E_1 summand, this conjecture can be stated in terms of an isomorphism between certain Tor groups. As evidence for the conjecture, we prove that such an isomorphism exists in degree zero.
This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid diagrams, and a …
This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid diagrams, and a combinatorial description in terms of the cube of resolutions. We discuss the geometric information carried by knot Floer homology, and the connection to three- and four-dimensional topology via surgery formulas. We also describe some conjectural relations to Khovanov-Rozansky homology.
Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we …
Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nil-Coxeter sequential 2-algebra and to a surface with connected boundary an algebra-module over this 2-algebra, such that a natural gluing property is satisfied. Moreover, with a view towards the structure of a potential Floer homology theory of 3-manifolds with codimension-2 corners, we present a decomposition theorem for the Floer complex of a planar grid diagram, with respect to vertical and horizontal slicing.
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology …
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there are no homology 3-spheres Y of Rokhlin invariant one such that Y # Y bounds an acyclic smooth 4-manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.
Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product …
Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered Floer homology invariants of 3-manifolds with codimension-2 corners, and prove that the bordered Floer homology of a 3-manifold with boundary, split into two pieces with corners, can be recovered as a tensor product of the cornered invariants of the pieces.
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology …
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there are no homology 3-spheres Y of Rokhlin invariant one such that Y # Y bounds an acyclic smooth 4-manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.
We prove Furuta-type bounds for the intersection forms of spin cobordisms between homology 3-spheres. The bounds are in terms of a new numerical invariant of homology spheres, obtained from Pin(2)-equivariant …
We prove Furuta-type bounds for the intersection forms of spin cobordisms between homology 3-spheres. The bounds are in terms of a new numerical invariant of homology spheres, obtained from Pin(2)-equivariant Seiberg-Witten Floer K-theory. In the process we introduce the notion of a Floer K_G-split homology sphere; this concept may be useful in an approach to the 11/8 conjecture.
We review the use of grid diagrams in the development of Heegaard Floer theory. We describe the construction of the combinatorial link Floer complex, and the resulting algorithm for unknot …
We review the use of grid diagrams in the development of Heegaard Floer theory. We describe the construction of the combinatorial link Floer complex, and the resulting algorithm for unknot detection. We also explain how grid diagrams can be used to show that the Heegaard Floer invariants of 3-manifolds and 4-manifolds are algorithmically computable (mod 2).
Ozsvath and Szabo gave a combinatorial description of knot Floer homology based on a cube of resolutions, which uses maps with twisted coefficients. We study the t=1 specialization of their …
Ozsvath and Szabo gave a combinatorial description of knot Floer homology based on a cube of resolutions, which uses maps with twisted coefficients. We study the t=1 specialization of their construction. The associated spectral sequence converges to knot Floer homology, and we conjecture that its E_1 page is isomorphic to the HOMFLY-PT chain complex of Khovanov and Rozansky. At the level of each E_1 summand, this conjecture can be stated in terms of an isomorphism between certain Tor groups. As evidence for the conjecture, we prove that such an isomorphism exists in degree zero.
Let L be a link in an integral homology three-sphere. We give a description of the Heegaard Floer homology of integral surgeries on L in terms of some data associated …
Let L be a link in an integral homology three-sphere. We give a description of the Heegaard Floer homology of integral surgeries on L in terms of some data associated to L, which we call a complete system of hyperboxes for L. Roughly, a complete systems of hyperboxes consists of chain complexes for (some versions of) the link Floer homology of L and all its sublinks, together with several chain maps between these complexes. Further, we introduce a way of presenting closed four-manifolds with b_2^+ > 1 by four-colored framed links in the three-sphere. Given a link presentation of this kind for a four-manifold X, we then describe the Ozsvath-Szabo mixed invariants of X in terms of a complete system of hyperboxes for the link. Finally, we explain how a grid diagram produces a particular complete system of hyperboxes for the corresponding link.
We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2). The descriptions are based on presenting the three-manifold as an integer surgery on …
We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 Ozsvath-Szabo mixed invariants of closed four-manifolds, in terms of grid diagrams.
Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant, in the form of a relatively Z/8-graded abelian group. Our motivation is …
Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant, in the form of a relatively Z/8-graded abelian group. Our motivation is to have a well-defined symplectic side of the Atiyah-Floer Conjecture, for arbitrary three-manifolds. The symplectic manifold used in the construction is the extended moduli space of flat SU(2)-connections on the Heegaard surface. An open subset of this moduli space carries a symplectic form, and each of the two handlebodies in the decomposition gives rise to a Lagrangian inside the open set. In order to define their Floer homology, we compactify the open subset by symplectic cutting; the resulting manifold is only semipositive, but we show that one can still develop a version of Floer homology in this setting.
Journal Article A Concordance Invariant from the Floer Homology of Double Branched Covers Get access Ciprian Manolescu, Ciprian Manolescu 1 Department of Mathematics, Columbia University New York, NY 10027, USA …
Journal Article A Concordance Invariant from the Floer Homology of Double Branched Covers Get access Ciprian Manolescu, Ciprian Manolescu 1 Department of Mathematics, Columbia University New York, NY 10027, USA Correspondence to be sent to: Ciprian Manolescu, Department of Mathematics, Columbia University New York, NY 10027, USA. e-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Brendan Owens Brendan Owens 2 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2007, 2007, rnm077, https://doi.org/10.1093/imrn/rnm077 Published: 01 January 2007 Article history Received: 27 September 2006 Revision received: 27 September 2006 Published: 01 January 2007 Accepted: 30 July 2007
In a previous paper we have constructed an invariant of four-dimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum …
In a previous paper we have constructed an invariant of four-dimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum of the boundary. Here we prove that when one glues two four-manifolds along their boundaries, the Bauer-Furuta invariant of the resulting manifold is obtained by applying a natural pairing to the invariants of the pieces. As an application, we show that the connected sum of three copies of the K3 surface contains no exotic nuclei. In the process we also compute the Floer spectrum for several Seifert fibrations.
For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We …
For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conjecture them to be equal to those of Khovanov and Rozansky after a collapsation of the bigrading. Our work is a generalization of that of Seidel and Smith, who treated the case n=2.
Given a crossing in a planar diagram of a link in the three-sphere, we show that the knot Floer homologies of the link and its two resolutions at that crossing …
Given a crossing in a planar diagram of a link in the three-sphere, we show that the knot Floer homologies of the link and its two resolutions at that crossing are related by an exact triangle. As a consequence, we deduce that for any quasi-alternating knot, the total rank of its knot Floer homology is equal to the determinant of the knot.
Ozsvath and Szabo defined an analog of the Froyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover …
Ozsvath and Szabo defined an analog of the Froyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover of the 3-sphere branched over a knot K, we obtain an invariant delta of knot concordance. We show that delta is determined by the signature for alternating knots and knots with up to nine crossings, and conjecture a similar relation for all H-thin knots. We also use delta to prove that for all knots K with tau(K)>0, the positive untwisted double of K is not smoothly slice.
Ozsvath and Szabo defined an analog of the Froyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover …
Ozsvath and Szabo defined an analog of the Froyshov invariant in the form of a correction term for the grading in Heegaard Floer homology. Applying this to the double cover of the 3-sphere branched over a knot K, we obtain an invariant delta of knot concordance. We show that delta is determined by the signature for alternating knots and knots with up to nine crossings, and conjecture a similar relation for all H-thin knots. We also use delta to prove that for all knots K with tau(K)>0, the positive untwisted double of K is not smoothly slice.
In [27], we introduced Floer homology theories HF -(Y, s), HF ∞ (Y, s), HF + (Y, t), HF (Y, s),and HF red (Y, s) associated to closed, oriented three-manifolds …
In [27], we introduced Floer homology theories HF -(Y, s), HF ∞ (Y, s), HF + (Y, t), HF (Y, s),and HF red (Y, s) associated to closed, oriented three-manifolds Y equipped with a Spin c structures s ∈ Spin c (Y ).In the present paper, we give calculations and study the properties of these invariants.The calculations suggest a conjectured relationship with Seiberg-Witten theory.The properties include a relationship between the Euler characteristics of HF ± and Turaev's torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences.We also include some applications of these techniques to three-manifold topology.
The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y , equipped with a Spin c structure.Given a Heegaard splitting of Y = U …
The aim of this article is to introduce certain topological invariants for closed, oriented three-manifolds Y , equipped with a Spin c structure.Given a Heegaard splitting of Y = U 0 ∪ Σ U 1 , these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of Σ relative to certain totally real subspaces associated to U 0 and U 1 .
Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. …
Using Furuta's idea of finite dimensional approximation in Seiberg-Witten theory, we refine Seiberg-Witten Floer homology to obtain an invariant of homology 3-spheres which lives in the S^1-equivariant graded suspension category. In particular, this gives a construction of Seiberg-Witten Floer homology that avoids the delicate transversality problems in the standard approach. We also define a relative invariant of four-manifolds with boundary which generalizes the Bauer-Furuta stable homotopy invariant of closed four-manifolds.
We calculate the Heegaard Floer homologies for three-manifolds obtained by plumbings of spheres specified by certain graphs.Our class of graphs is sufficiently large to describe, for example, all Seifert fibered …
We calculate the Heegaard Floer homologies for three-manifolds obtained by plumbings of spheres specified by certain graphs.Our class of graphs is sufficiently large to describe, for example, all Seifert fibered rational homology spheres.These calculations can be used to determine also these groups for other three-manifolds, including the product of a circle with a genus two surface.
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call …
We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call CF_r. It carries information about the Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on Floer homology induced by certain surgery cobordisms. We define a certain class of \em{perfect} knots in S^3 for which CF_r has a particularly simple form. For these knots, formal properties of the Ozsvath-Szabo theory enable us to make a complete calculation of the Floer homology. This is the author's thesis; many of the results have been independently discovered by Ozsvath and Szabo in math.GT/0209056.
We define <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P i n left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Pin</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Pin}(2)</mml:annotation> </mml:semantics> </mml:math> …
We define <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P i n left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Pin</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Pin}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivariant Seiberg-Witten Floer homology for rational homology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-spheres equipped with a spin structure. The analogue of Frøyshov’s correction term in this setting is an integer-valued invariant of homology cobordism whose mod <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> reduction is the Rokhlin invariant. As an application, we show that there are no homology <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-spheres <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Rokhlin invariant one such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y number-sign upper Y"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mi mathvariant="normal">#<!-- # --></mml:mi> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Y \#Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounds an acyclic smooth <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.
Using the conjugation symmetry on Heegaard Floer complexes, we define a 3-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z4-equivariant Seiberg–Witten Floer homology. Further, we …
Using the conjugation symmetry on Heegaard Floer complexes, we define a 3-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z4-equivariant Seiberg–Witten Floer homology. Further, we obtain two new invariants of homology cobordism, d̲ and d¯, and two invariants of smooth knot concordance, V̲0 and V¯0. We also develop a formula for the involutive Heegaard Floer homology of large integral surgeries on knots. We give explicit calculations in the case of L-space knots and thin knots. In particular, we show that V̲0 detects the nonsliceness of the figure-eight knot. Other applications include constraints on which large surgeries on alternating knots can be homology-cobordant to other large surgeries on alternating knots.
To my teachers and friends 0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's.Of the mysteries still remaining after that period of great success the …
To my teachers and friends 0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's.Of the mysteries still remaining after that period of great success the most compelling seemed to lie in dimensions three and four.Although experience suggested that manifold theory at these dimensions has a distinct character, the dream remained since my graduate school days 1 that some key principle from the high dimensional theory would extend, at least to dimension four, and bring with it the beautiful adherence of topology to algebra familiar in dimensions greater than or equal to five.There is such a principle.It is a homotopy theoretic criterion for imbedding (relatively) a topological 2-handle in a smooth four-dimensional manifold with boundary.The main impact, as outlined in §1, is to the classification of 1-connected 4-manifolds and topological end recognition.However, certain applications to nonsimply connected problems such as knot concordance are also obtained.The discovery of this principle was made in three stages.From 1973 to 1975 Andrew Casson developed his theory of "flexible handles" 2 .These are certain pairs having the proper homotopy type of the common place open 2-handle H = (D 2 X D 2 , dD 2 X D 2 ) but "flexible" in the sense that finding imbeddings is rather easy; in fact imbedding is implied by a homotopy theoretic criterion.It was clear to Casson 3 that: (1) no known invariant-link theoretic
Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a nontrivial knot in the three-sphere.To obtain this result, we use a …
Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a nontrivial knot in the three-sphere.To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a nonvanishing theorem, which shows that monopole Floer homology detects the unknot.In addition, we apply these techniques to give information about knots which admit lens space surgeries, and to exhibit families of three-manifolds which do not admit taut foliations.
Let Σ(a 1 , a 2 , . . ., a n ) be a Seifert fibered homology 3-sphere with a 1 even.We show that if µ(Σ(a 1 , a …
Let Σ(a 1 , a 2 , . . ., a n ) be a Seifert fibered homology 3-sphere with a 1 even.We show that if µ(Σ(a 1 , a 2 , . . ., a n )) = 1 mod 2, then the class of Σ(a 1 , a 2 , . . ., a n ) has infinite order in the homology cobordism group of homology 3-spheres.In the proof we use Seiberg-Witten's monopole equation on fourdimensional V-manifolds.
In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y. …
In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y. In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (generalizing a result of Rasmussen for two-bridge knots). Applications include new restrictions on the Alexander polynomial of alternating knots.
Soit X une u-variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee Q est definie positive. Alors cette forme est equivalente, sur les entiers, a la …
Soit X une u-variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee Q est definie positive. Alors cette forme est equivalente, sur les entiers, a la forme diagonale standard, soit dans une base: Q(u 1 ,u 2 ,...u 2 )=u 2 1 +u 2 2 +...+u 2 2
An early result by Donaldson says that if Z is closed and JZ is negative definite then JZ is isomorphic to some diagonal form 〈−1〉 ⊕ · · · ⊕ …
An early result by Donaldson says that if Z is closed and JZ is negative definite then JZ is isomorphic to some diagonal form 〈−1〉 ⊕ · · · ⊕ 〈−1〉. More generally one may ask which negative definite forms can occur if Z is allowed to have some fixed oriented rational homology sphere Y as boundary. The main purpose of the present paper is to apply the equations recently introduced by Seiberg and Witten [W] to prove a finiteness result about the definite forms associated to an arbitrary Y . It is useful to consider the more general situation where the boundary of Z is a disjoint union of rational homology spheres: ∂Z = Y1 ∪ · · · ∪ Yl. (Of course, ∪jYj and #jYj bound the same intersection forms, since the standard cobordism connecting them has no rational homology in dimension 2.) Let JZ = m〈−1〉 ⊕ JZ , where JZ has no elements of square −1. Note that
We give a new construction of monopole Floer homology for spin c rational homology 3-spheres.As applications we define two invariants of certain 4-manifolds with b 1 = 1 and b …
We give a new construction of monopole Floer homology for spin c rational homology 3-spheres.As applications we define two invariants of certain 4-manifolds with b 1 = 1 and b + = 0. 11 Reduced Floer groups 46 12 The h-invariant 48 13 Proof of Theorem 8 53 14 A finite-dimensional analogue 55
We give inequalities for the Manolescu invariants α , β , γ under the connected sum operation. We compute the Manolescu invariants of connected sums of some Seifert fiber spaces. …
We give inequalities for the Manolescu invariants α , β , γ under the connected sum operation. We compute the Manolescu invariants of connected sums of some Seifert fiber spaces. Using these same invariants, we provide a proof of Furuta's Theorem, the existence of a Z ∞ subgroup of the homology cobordism group. To our knowledge, this is the first proof of Furuta's Theorem using monopoles. We also provide information about Manolescu invariants of the connected sum of n copies of a three-manifold Y , for large n.
We compute the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu's conjecture that …
We compute the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu's conjecture that $β=-\barμ$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $α, β,$ and $γ$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $Σ(a_1,...,a_n)$ are not homology cobordant to any $-Σ(b_1,...,b_n)$. We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the $\mathrm{Pin}(2)$-equivariant Seiberg-Witten Floer spectrum provides homology cobordism obstructions distinct from $α,β,$ and $γ$. In particular, we identify an $\mathbb{F}[U]$-module called connected Seiberg-Witten Floer homology, whose isomorphism class is a homology cobordism invariant.
Journal Article A Concordance Invariant from the Floer Homology of Double Branched Covers Get access Ciprian Manolescu, Ciprian Manolescu 1 Department of Mathematics, Columbia University New York, NY 10027, USA …
Journal Article A Concordance Invariant from the Floer Homology of Double Branched Covers Get access Ciprian Manolescu, Ciprian Manolescu 1 Department of Mathematics, Columbia University New York, NY 10027, USA Correspondence to be sent to: Ciprian Manolescu, Department of Mathematics, Columbia University New York, NY 10027, USA. e-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Brendan Owens Brendan Owens 2 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2007, 2007, rnm077, https://doi.org/10.1093/imrn/rnm077 Published: 01 January 2007 Article history Received: 27 September 2006 Revision received: 27 September 2006 Published: 01 January 2007 Accepted: 30 July 2007
In this paper, we give an algorithm to compute the hat version of Heegaard Floer homology of a closed oriented three-manifold.This method also allows us to compute the filtration coming …
In this paper, we give an algorithm to compute the hat version of Heegaard Floer homology of a closed oriented three-manifold.This method also allows us to compute the filtration coming from a null-homologous link in a three-manifold.
On stable properties of the solution set of an ordinary differential equation Elementary properties of flows The Morse index Continuation Bibliography.
On stable properties of the solution set of an ordinary differential equation Elementary properties of flows The Morse index Continuation Bibliography.
In a previous paper we have constructed an invariant of four-dimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum …
In a previous paper we have constructed an invariant of four-dimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum of the boundary. Here we prove that when one glues two four-manifolds along their boundaries, the Bauer-Furuta invariant of the resulting manifold is obtained by applying a natural pairing to the invariants of the pieces. As an application, we show that the connected sum of three copies of the K3 surface contains no exotic nuclei. In the process we also compute the Floer spectrum for several Seifert fibrations.
Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus …
Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares.Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.
In this paper, we investigate the Seiberg-Witten gauge theory for Seifert fibered spaces. The monopoles over these three-manifolds, for a particular choice of metric and perturbation, are completely described. Gradient …
In this paper, we investigate the Seiberg-Witten gauge theory for Seifert fibered spaces. The monopoles over these three-manifolds, for a particular choice of metric and perturbation, are completely described. Gradient flow lines between monopoles are identified with holomorphic data on an associated ruled surface, and a dimension formula for such flows is calculated.
This article is one of three highly influential articles on the topology of manifolds written by Robert D. Edwards in the 1970's but never published. It presents the initial solutions …
This article is one of three highly influential articles on the topology of manifolds written by Robert D. Edwards in the 1970's but never published. It presents the initial solutions of the fabled Double Suspension Conjecture. (The other two articles are: 'Approximating certain cell-like maps by homeomorphisms' and 'Topological regular neighborhoods') The manuscripts of these three articles have circulated privately since their creation. The organizers of the Workshops in Geometric Topology (http://www.math.oregonstate.edu/~topology/workshop.htm) with the support of the National Science Foundation have facilitated the preparation of electronic versions of these articles to make them publicly available. The second and third articles are still in preparation. The current article contains four major theorems: I. The double suspension of Mazur's homology 3-sphere is a 5-sphere, II. The double suspension of any homology n-sphere that bounds a contractible (n+1)-manifold is an (n+2)-sphere, III. The double suspension of any homology 3-sphere is the cell-like image of a 5-sphere. IV. The triple suspension of any homology 3-sphere is a 6-sphere. Edwards' proof of I. was the first evidence that the suspension process could transform a non-simply connected manifold into a sphere, thereby answering a question that had puzzled topologists since the mid-1950's if not earlier. Results II, III and IV represent significant advances toward resolving the general double suspension conjecture: the double suspension of every homology n-sphere is an (n+2)-sphere. [That conjecture was subsequently proved by J. W. Cannon (Annals of Math. 110 (1979), 83-112).]
We develop monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of A. Juhász. "Holomorphic discs and sutured manifolds." Algebr. Geom. Topol., 6:1429–1457 (electronic), 2006. Applications …
We develop monopole and instanton Floer homology groups for balanced sutured manifolds, in the spirit of A. Juhász. "Holomorphic discs and sutured manifolds." Algebr. Geom. Topol., 6:1429–1457 (electronic), 2006. Applications include a new proof of Property P for knots.
Every Brieskorn homology sphere S(p, q, r) is a double cover of the 3-sphere ramified over a Montesinos knot k(p,q,r).We express the Floer homology of E(p, q,r) in terms of …
Every Brieskorn homology sphere S(p, q, r) is a double cover of the 3-sphere ramified over a Montesinos knot k(p,q,r).We express the Floer homology of E(p, q,r) in terms of certain invariants of the knot k(p,q,r), among which are the knot signature and the Jones polynomial.We also define an integer valued invariant of integral homology 3-spheres which agrees with the fiinvariant of W. Neumann and L. Siebenmann for Seifert fibered homology spheres, and investigate its behavior with respect to homology 4-cobordism.
By equivariantly pasting together exteriors of links in S 3 that are invariant under several different involutions of S 3 , we construct closed orientable 3-manifolds that are two-fold branched …
By equivariantly pasting together exteriors of links in S 3 that are invariant under several different involutions of S 3 , we construct closed orientable 3-manifolds that are two-fold branched covering spaces of S 3 in distinct ways, that is, with different branch sets.Sufficient conditions are given to guarantee when the constructed manifold M admits an induced involution, h, and when M/h = S 3 .Using the theory of characteristic submanifolds for Haken manifolds with incompressible boundary components, we also prove that doubles, D(K,ρ), of prime knots that are not strongly invertible are characterized by their two-fold branched covering spaces, when p Φ 0. If, however, K is strongly invertible, then the manifold branch covers distinct knots.Finally, the authors characterize the type of a prime knot by the double covers of the doubled knots, D(K; p, η) and D(K*; p, η), of K and its mirror image K* when p and η are fixed, with p Φ 0 and η e { -2,2}.With each two-fold branched covering map, p: M 3 -» iV 3 , there is associated a PL involution, h: M -> M, that induces p.There can, however, be other PL involutions on M that are not equivalent to Λ, but nevertheless are covering involutions for two-fold branched covering maps of M (cf.[BGM]).Our purpose, in this paper, is to introduce ways of detecting such involutions and controlling their number.We begin with compact 3-manifolds with several obvious PL involutions.An oriented link L in M is 2-symmetric, if N 3 s S 3 and if h(L) = L.In §1, we give examples of knots and links in S 3 that are 2-symmetric in two or more ways; for example, a trefoil knot is both strongly invertible and periodic (definitions in §1).In §2, we paste the exteriors, E(L) and E{L'), of 2-symmetric links, L and L', together along a torus-boundary component of each exterior; Proposition 2.1 gives the pasting instructions / that must be followed in order for the involutions, h and h\ of E(L) and E(L') to extend to an involution h f of E(L) U f E{L').Theorems 2.2 and 2.3 allow us to conclude, under fairly relaxed conditions, that the orbit space of h f is S 3 .
We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot …
We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, tau gives sharp bounds on the four-ball genera of torus knots. As another illustration, we calculate the invariant for several ten-crossing knots.
Flow-type suspension and homotopy suspension agree for attractor–repellor homotopy data. The connection maps associated in Conley index theory to an attractor–repellor decomposition with respect to the direct flow and its …
Flow-type suspension and homotopy suspension agree for attractor–repellor homotopy data. The connection maps associated in Conley index theory to an attractor–repellor decomposition with respect to the direct flow and its inverse are Spanier–Whitehead duals in the stably parallelizable context and are duals modulo a certain Thom construction in general.
In the present work we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on …
In the present work we generalize the construction of monopole Floer homology due to Kronheimer and Mrowka to the case of a gradient flow with Morse-Bott singularities. Focusing then on the special case of a three-manifold equipped with a spin$^c$ structure isomorphic to its conjugate, we define the counterpart in this context of Manolescu's recent $\mathrm{Pin}(2)$-equivariant Seiberg-Witten-Floer homology. In particular, we provide an alternative approach to his disproof of the celebrated Triangulation conjecture.
Euclidean n-space En, n > 5, has the following simple DISJOINT DISK PROPERTY: singular 2-dimensional disks in En may be adjusted slightly so as to be disjoint. We show that …
Euclidean n-space En, n > 5, has the following simple DISJOINT DISK PROPERTY: singular 2-dimensional disks in En may be adjusted slightly so as to be disjoint. We show that for a large class of cell-like decompositions of manifolds this property in the decomposition space is sufficient in order that the decomposition space be a manifold. As a consequence we deduce the DOUBLE SUSPENSION THEOREM proved in a large number of cases by R. D. Edwards: The double suspension of any homology sphere is a topological sphere. We also obtain a sweeping generalization of Edwards' MANIFOLD FACTOR THEOREM; Edwards' theorem states that, if X is a single cell-like set in Euclidean n-dimensional space En, then (En/X) x E = En+l.