SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS

Heegaard Floer ◽

We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to$S^{2}$but do not admit a spine (that is, a piecewise linear embedding of$S^{2}$that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer$d$invariants.

TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

Glasgow Mathematical Journal ◽

10.1017/s0017089521000215 ◽

2021 ◽

pp. 1-8

Author(s):

DANIEL KASPROWSKI ◽

MARKUS LAND

Keyword(s):

Fundamental Group ◽

Homotopy Equivalent ◽

Poincaré Duality ◽

Poincare Duality ◽

Canonical Map ◽

Duality Space ◽

Simply Connected ◽

Poincaré Duality Space

Abstract Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

On open 3-manifolds proper homotopy equivalent to geometrically simply-connected polyhedra☆☆This preprint is available electronically at http://www-fourier.ujf-grenoble.fr/~funar.

Topology and its Applications ◽

10.1016/s0166-8641(99)00154-6 ◽

2001 ◽

Vol 109 (2) ◽

pp. 191-200 ◽

Cited By ~ 2

Author(s):

L. Funar ◽

T.L. Thickstun

Keyword(s):

Homotopy Equivalent ◽

Proper Homotopy ◽

Classification of closed oriented 4-manifolds modulo connected sum with simply connected manifolds

Hiroshima Mathematical Journal ◽

10.32917/hmj/1206126759 ◽

1998 ◽

Vol 28 (2) ◽

pp. 207-212 ◽

Cited By ~ 1

Author(s):

Ichiji Kurazono ◽

Takao Matumoto

Keyword(s):

Connected Sum ◽

Translations of Mathematical Monographs - Functions on Manifolds: Algebraic and Topological Aspects ◽

Minimal Morse functions of non-simply-connected manifolds

10.1090/mmono/131/07 ◽

1993 ◽

pp. 155-172

Keyword(s):

Morse Functions ◽

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

Open Mathematics ◽

10.2478/s11533-012-0122-7 ◽

2012 ◽

Vol 10 (6) ◽

Author(s):

Grzegorz Graff ◽

Agnieszka Kaczkowska

Keyword(s):

Natural Number ◽

Homotopy Class ◽

Periodic Points ◽

Topological Invariant ◽

Periodic Sequence ◽

Connected Manifold ◽

Lefschetz Numbers ◽

AbstractLet f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f.In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH 2(M; ℚ) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.

Concerning Lp resolvent estimates for simply connected manifolds of constant curvature

Journal of Functional Analysis ◽

10.1016/j.jfa.2014.08.016 ◽

2014 ◽

Vol 267 (12) ◽

pp. 4635-4666 ◽

Cited By ~ 7

Author(s):

Shanlin Huang ◽

Christopher D. Sogge

Keyword(s):

Constant Curvature ◽

Resolvent Estimates ◽

The group of homotopy self-equivalences of non-simply-connected spaces using Postnikov decompositions II

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500021004 ◽

1992 ◽

Vol 122 (1-2) ◽

pp. 127-135 ◽

Cited By ~ 1

Author(s):

John W. Rutter

Keyword(s):

Group Extension ◽

Equivalence Classes ◽

Homotopy Groups ◽

Geometric Proof ◽

Homotopy Equivalence ◽

Connected Space ◽

Abelian Kernel ◽

Simply Connected ◽

Extension Sequence ◽

Connected Spaces

SynopsisWe give here an abelian kernel (central) group extension sequence for calculating, for a non-simply-connected space X, the group of pointed self-homotopy-equivalence classes . This group extension sequence gives in terms of , where Xn is the nth stage of a Postnikov decomposition, and, in particular, determines up to extension for non-simplyconnected spaces X having at most two non-trivial homotopy groups in dimensions 1 and n. We give a simple geometric proof that the sequence splits in the case where is the generalised Eilenberg–McLane space corresponding to the action ϕ: π1 → aut πn, and give some information about the class of the extension in the general case.