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.

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.

Stable cohomotopy Seiberg–Witten invariants of connected sums of four-manifolds with positive first Betti number, I: Non-vanishing theorem

2015 ◽  
Vol 26 (06) ◽  
pp. 1541004 ◽  
Author(s):  
Masashi Ishida ◽  
Hirofumi Sasahira
Keyword(s):  
Betti Number ◽  
Witten Invariant ◽  
Vanishing Theorem ◽  
Connected Sums ◽  
Four Manifolds ◽  
Stable Cohomotopy ◽  
Invent Math

We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg–Witten invariant [S. Bauer and M. Furuta, Stable cohomotopy refinement of Seiberg–Witten invariants: I, Invent. Math.155 (2004) 1–19; S. Bauer, Stable cohomotopy refinement of Seiberg–Witten invariants: II, Invent. Math.155 (2004) 21–40.] of connected sums of 4-manifolds with positive first Betti number.

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).

A rational invariant for certain infinite discrete groups

1993 ◽  
Vol 113 (3) ◽  
pp. 473-478
Author(s):  
F. E. A. Johnson
Keyword(s):  
Euler Characteristic ◽  
Discrete Groups ◽  
Fundamental Groups ◽  
Intersection Form ◽  
Absolute Value ◽  
Rational Invariant ◽  
The Absolute ◽  
Image Position ◽  
Aspherical Manifolds

We introduce a rational-valued invariant which is capable of distinguishing between the commensurability classes of certain discrete groups, namely, the fundamental groups of smooth closed orientable aspherical manifolds of dimensional 4k(k ≥ 1) whose Euler characteristic χ(Λ) is non-zero. The invariant in question is the quotientwhere Sign (Λ) is the absolute value of the signature of the intersection formand [Λ] is a generator of H4k(Λ; ℝ).

NORMALIZED RICCI FLOW, SURGERY, AND SEIBERG–WITTEN INVARIANTS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450005
Author(s):  
MASASHI ISHIDA
Keyword(s):  
Existence Theorem ◽  
Euler Characteristic ◽  
Ricci Flow ◽  
Einstein Metrics ◽  
Nonsingular Solution ◽  
Ricci Flows ◽  
Normalized Ricci Flow ◽  
Four Manifolds ◽  
Special Case

We investigate the behavior of solutions of the normalized Ricci flow under surgeries of four-manifolds along circles by using Seiberg–Witten invariants. As a by-product, we prove that any pair (α, β) of integers satisfying α + β ≡ 0 (mod 2) can be realized as the Euler characteristic χ and signature τ of infinitely many closed smooth 4-manifolds with negative Perelman's [Formula: see text] invariants and on which there is no nonsingular solution to the normalized Ricci flows for any initial metric. In particular, this includes the existence theorem of non-Einstein 4-manifolds due to Sambusetti [An obstruction to the existence of Einstein metrics on 4-manifolds, Math. Ann.311 (1998) 533–547] as a special case.

On the Euler characteristic mod 2 of real projective varieties

1988 ◽  
Vol 104 (3) ◽  
pp. 479-481 ◽  
Author(s):  
Zbigniew Szafraniec
Keyword(s):  
Euler Characteristic ◽  
Homogeneous Polynomials ◽  
Geometric Invariant ◽  
Projective Varieties ◽  
Image Position

Let f1, …, fs: ℝn+1→ℝ be homogeneous polynomials, and letThe Euler characteristic χ(X) is an important topological and geometric invariant of X.

On discrete generalised triangle groups

1995 ◽  
Vol 38 (3) ◽  
pp. 397-412 ◽  
Author(s):  
M. Hagelberg ◽  
C. MacLachlan ◽  
G. Rosenberger
Keyword(s):  
Euler Characteristic ◽  
Discrete Groups ◽  
Triangle Group ◽  
Triangle Groups ◽  
Reduced Word ◽  
Discrete Representations ◽  
Image Position ◽  
The Relationship

A generalised triangle group has a presentation of the formwhere R is a cyclically reduced word involving both x and y. When R = xy, these classical triangle groups have representations as discrete groups of isometrics of S2, R2, H2 depending onIn this paper, for other words R, faithful discrete representations of these groups in Isom +H3 = PSL(2, C) are considered with particular emphasis on the case R = [x, y] and also on the relationship between the Euler characteristic χ and finite covolume representations.

scholarly journals Embedded surfaces in four-manifolds, branched covers, and SO(3)-invariants

1995 ◽  
Vol 117 (2) ◽  
pp. 275-286 ◽  
Author(s):  
D. Kotschick ◽  
G. Matić
Keyword(s):  
Algebraic Curves ◽  
Homology Class ◽  
Algebraic Surfaces ◽  
Branched Covers ◽  
Minimal Genus ◽  
Adjunction Formula ◽  
Embedded Surfaces ◽  
Four Manifolds ◽  
Image Position ◽  
Simply Connected

One of the outstanding problems in four-dimensional topology is to find the minimal genus of an oriented smoothly embedded surface representing a given homology class in a smooth four-manifold. For an arbitrary homology class in an arbitrary smooth manifold not even a conjectural lower bound is known. However, for the classes represented by smooth algebraic curves in (simply connected) algebraic surfaces, it is possible that the genus of the algebraic curve, given by the adjunction formulais the minimal genus. This is usually called the (generalized) Thom conjecture. It is mentioned in Kirby's problem list [11] as Problem 4·36.

Groups acting freely on R-trees

1991 ◽  
Vol 11 (4) ◽  
pp. 737-756 ◽  
Author(s):  
John W. Morgan ◽  
Richard K. Skora
Keyword(s):  
Euler Characteristic ◽  
Cyclic Group ◽  
Abelian Subgroup ◽  
Closed Surface ◽  
Infinite Cyclic Group ◽  
Precise Result ◽  
Finitely Presented Group ◽  
Hnn Extension ◽  
Image Position ◽  
Finitely Presented

AbstractIn this paper we study the question of which groups act freely on R-trees. The paper has two parts. The first part concerns groups which contain a non-cyclic, abelian subgroup. The following is the main result in this case.Let the finitely presented group G act freely on an R-tree. If A is a non-cyclic, abelian subgroup of G, then A is contained in an abelian subgroup A′ which is a free factor of G.The second part of the paper concerns groups whch split as an HNN-extension along an infinite cyclic group. Here is one formulation of our main result in that case.Let the finitely presented group G act freely on an R-tree. If G has an HNN-decompositionwhere (s) is infinite cyclic, then there is a subgroup H′ ⊂ H such that either(a); or(b),where S is a closed surface of non-positive Euler characteristic.A slightly different, more precise result is also given.

Structure Theory for Montgomery-Samelson Fiberings between Manifolds, II

1969 ◽  
Vol 21 ◽  
pp. 180-186 ◽  
Author(s):  
Peter L. Antonelli
Keyword(s):  
Euler Characteristic ◽  
Betti Number ◽  
Elementary Proof ◽  
Total Space ◽  
Structure Theory ◽  
Singular Set ◽  
Sphere Bundle ◽  
Simply Connected

Let f: Mn→ Npbe the projection map of an MS-fibering of manifolds β with finite non-empty singular set Aand simply connected total space (see 1). Results of Timourian (10) imply that (n, p) = (4, 3), (8, 5) or (16, 9), while a theorem of Conner (2) yields that #(A), the cardinality of the singular set, is equal to the Euler characteristic of Mn. We give an elementary proof of this fact and, in addition, prove that #(A) is actually determined by bn/2(Mn), the middle betti number of Mn, or what is the same, by bn/2(Np – f(A)). It is then shown that β is topologically the suspension of a (Hopf) sphere bundle when Np is a sphere and bn/2(Mn) = 0. It follows as a corollary that β must also be a suspension when Mn is n/4-connected with vanishing bn/2. Examples where bn/2 is not zero are constructed and we state a couple of conjectures concerning the classification of such objects.

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