scholarly journals Lescop's invariant and gauge theory

2015 ◽  
Vol 24 (09) ◽  
pp. 1550050 ◽  
Author(s):  
Prayat Poudel
Keyword(s):  
Gauge Theory ◽  
Euler Characteristic ◽  
Betti Number ◽  
Floer Homology ◽  
Integral Homology ◽  
Similar Relationship

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and its instanton Floer homology. The proof uses surgery techniques.

scholarly journals INSTANTON FLOER HOMOLOGY FOR TWO-COMPONENT LINKS

2012 ◽  
Vol 21 (05) ◽  
pp. 1250054 ◽  
Author(s):  
ERIC HARPER ◽  
NIKOLAI SAVELIEV
Keyword(s):  
Euler Characteristic ◽  
Floer Homology ◽  
Integral Homology ◽  
Oriented Link ◽  
Linking Number ◽  
Two Component

For any oriented link of two components in an integral homology 3-sphere, we define an instanton Floer homology whose Euler characteristic is twice the linking number between the components of the link. We show that, for two-component links in the 3-sphere, this Floer homology does not vanish unless the link is split. We also relate our Floer homology to the Kronheimer–Mrowka instanton Floer homology for links.

Casson-Lin's Invariant and Floer Homology

1997 ◽  
Vol 06 (06) ◽  
pp. 851-877 ◽  
Author(s):  
Weiping Li
Keyword(s):  
Euler Number ◽  
Floer Homology ◽  
Natural Generalization ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Representation Space ◽  
Integral Homology

Casson defined an invariant which can be thought of as the number of conjugacy classes of irreducible representations of π1(Y) into SU(2) counted with signs, where Y is an oriented integral homology 3-sphere. Lin defined a similar invariant (the signature of a knot) for a braid representative of a knot in S3. In this paper, we give a natural generalization of Casson-Lin's invariant. Our invariant is the symplectic Floer homology for the representation space of π1(S3 \ K) into SU(2) with trace-zero along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its Euler number is the negative of Casson-Lin's invariant.

scholarly journals On the iterated Hamiltonian Floer homology

2020 ◽  
pp. 2050026
Author(s):  
Erman Çı̇nelı̇ ◽  
Viktor L. Ginzburg
Keyword(s):  
Fixed Point ◽  
Euler Characteristic ◽  
Group Action ◽  
Floer Homology ◽  
Local Homology ◽  
Lefschetz Index ◽  
Cyclic Group Action ◽  
Symplectically Aspherical Manifold ◽  
Aspherical Manifold ◽  
Symplectically Aspherical

The focus of the paper is the behavior under iterations of the filtered and local Floer homology of a Hamiltonian on a symplectically aspherical manifold. The Floer homology of an iterated Hamiltonian comes with a natural cyclic group action. In the filtered case, we show that the supertrace of a generator of this action is equal to the Euler characteristic of the homology of the un-iterated Hamiltonian. For the local homology the supertrace is the Lefschetz index of the fixed point. We also prove an analog of the classical Smith inequality for the iterated local homology and the equivariant versions of these results.

scholarly journals On the signature and Euler characteristic of certain four-manifolds

1993 ◽  
Vol 114 (3) ◽  
pp. 431-437 ◽  
Author(s):  
F. E. A. Johnson ◽  
D. Kotschick†
Keyword(s):  
Euler Characteristic ◽  
Betti Number ◽  
Four Manifolds ◽  
Image Position

Let M be a smooth closed connected oriented 4-manifold; we shall say that M satisfies Winkelnkemper's inequality when its signature, σ(M), and Euler characteristic, X(M), are related byThis inequality is trivially true for manifolds M with first Betti number b1(M) ≤ 1.

Transfer maps for fibrations

1996 ◽  
Vol 120 (2) ◽  
pp. 221-235 ◽  
Author(s):  
W. G. Dwyer
Keyword(s):  
Euler Characteristic ◽  
Homotopy Equivalent ◽  
Integral Homology ◽  
Finite Complex ◽  
Connected Space ◽  
Transfer Maps ◽  
Transfer Map ◽  
Image Position

Let f: E → B be a fibration with fibre F over a connected space B. If F is homotopy equivalent to a finite complex, Becker and Gottlieb [2, 3] and others have constructed a transfer mapwhere for simplicity X+ denotes the suspension spectrum of the space obtained from X adding a disjoint basepoint. One key property of τ(f) is the fact that the composite map f+. τ(f): B+ → B+ induces a map on integral homology which is multiplication the Euler characteristic X(F).

scholarly journals Link Floer Homology Categorifies the Conway Function

2016 ◽  
Vol 59 (4) ◽  
pp. 813-836
Author(s):  
Mounir Benheddi ◽  
David Cimasoni
Keyword(s):  
Euler Characteristic ◽  
Floer Homology ◽  
Alexander Polynomial ◽  
Oriented Link

AbstractGiven an oriented link in the 3-sphere, the Euler characteristic of its link Floer homology is known to coincide with its multi-variable Alexander polynomial, an invariant only defined up to a sign and powers of the variables. In this paper we remove this ambiguity by proving that this Euler characteristic is equal to the so-called Conway function, the representative of the multi-variable Alexander polynomial introduced by Conway in 1970 and explicitly constructed by Hartley in 1983. This is achieved by creating a model of the Conway function adapted to rectangular diagrams, which is then compared to the Euler characteristic of the combinatorial version of link Floer homology.

scholarly journals The knot Floer cube of resolutions and the composition product

2020 ◽  
Vol 29 (03) ◽  
pp. 2050006
Author(s):  
Nathan Dowlin
Keyword(s):  
Euler Characteristic ◽  
Floer Homology ◽  
Product Formula ◽  
Knot Floer Homology ◽  
Direct Sum ◽  
Heegaard Diagram ◽  
Composition Product ◽  
The Relationship

We examine the relationship between the oriented cube of resolutions for knot Floer homology and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot [Formula: see text], we see that the filtered complex decomposes as a direct sum of HOMFLY-PT complexes of various subdiagrams. Applying Jaeger’s composition product formula for knot polynomials, we deduce that the graded Euler characteristic of this direct sum is the HOMFLY-PT polynomial of [Formula: see text].

scholarly journals TOPOLOGICAL GAUGE THEORIES FROM SUPERSYMMETRIC QUANTUM MECHANICS ON SPACES OF CONNECTIONS

1993 ◽  
Vol 08 (03) ◽  
pp. 573-585 ◽  
Author(s):  
MATTHIAS BLAU ◽  
GEORGE THOMPSON
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theory ◽  
Euler Characteristic ◽  
Moduli Spaces ◽  
Riemannian Geometry ◽  
Gauge Theories ◽  
Supersymmetric Quantum Mechanics ◽  
Three Dimensions ◽  
Flat Connections ◽  
Infinite Dimensional

We rederive the recently introduced N=2 topological gauge theories, representing the Euler characteristic of moduli spaces ℳ of connections, from supersymmetric quantum mechanics on the infinite-dimensional spaces [Formula: see text] of gauge orbits. To that end we discuss variants of ordinary supersymmetric quantum mechanics which have meaningful extensions to infinite-dimensional target spaces, and introduce supersymmetric quantum mechanics actions modeling the Riemannian geometry of submersions and embeddings, relevant to the projections [Formula: see text] and inclusions [Formula: see text] respectively. We explain the relation between Donaldson theory and the gauge theory of flat connections in three dimensions and illustrate the general construction by other two- and four-dimensional examples.

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