scholarly journals Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification

2021 ◽  
pp. 1-19
Author(s):  
D. Crowley ◽  
A. Skopenkov
Keyword(s):  
Connected Sum ◽  
Smooth Category ◽  
Simply Connected ◽  
Torsion Free ◽  
Smooth Classification

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$ , where $H_q := H_q(N; {\mathbb Z} )$ . Our main result is a readily calculable classification of embeddings $N \to {\mathbb R}^7$ up to isotopy, with an indeterminacy. Such a classification was only known before for $H_1=0$ by our earlier work from 2008. Our classification is complete when $H_2=0$ or when the signature of $N$ is divisible neither by 64 nor by 9. The group of knots $S^4\to {\mathbb R}^7$ acts on the set of embeddings $N\to {\mathbb R}^7$ up to isotopy by embedded connected sum. In Part I we classified the quotient of this action. The main novelty of this paper is the description of this action for $H_1 \ne 0$ , with an indeterminacy. Besides the invariants of Part I, detecting the action of knots involves a refinement of the Kreck invariant from our work of 2008. For $N=S^1\times S^3$ we give a geometrically defined 1–1 correspondence between the set of isotopy classes of embeddings and a certain explicitly defined quotient of the set ${\mathbb Z} \oplus {\mathbb Z} \oplus {\mathbb Z} _{12}$ .

scholarly journals On the K-theory of the loop space of a Lie group

1974 ◽  
Vol 76 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Francis Clarke
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Simple Structure ◽  
Homology Theory ◽  
Compact Lie Group ◽  
Classifying Space ◽  
K Theory ◽  
Simply Connected ◽  
Torsion Free ◽  
Multiplicative Structures

Let G be a simply connected, semi-simple, compact Lie group, let K* denote Z/2-graded, representable K-theory, and K* the corresponding homology theory. The K-theory of G and of its classifying space BG are well known, (8),(1). In contrast with ordinary cohomology, K*(G) and K*(BG) are torsion-free and have simple multiplicative structures. If ΩG denotes the space of loops on G, it seems natural to conjecture that K*(ΩG) should have, in some sense, a more simple structure than H*(ΩG).

scholarly journals The even Clifford structure of the fourth Severi variety

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Maurizio Parton ◽  
Paolo Piccinni
Keyword(s):  
Vector Bundle ◽  
Riemannian Manifolds ◽  
Severi Variety ◽  
Simply Connected

AbstractTheHermitian symmetric spaceM = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ : Cl

Surgery Theory and the Classification of Closed, Simply Connected 4-manifolds

2021 ◽  
pp. 331-352
Author(s):  
Patrick Orson ◽  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Embedding Theorem ◽  
Surgery Theory ◽  
Simply Connected

Surgery theory and the classification of simply connected 4-manifolds comprise two key consequences of the disc embedding theorem. The chapter begins with an introduction to surgery theory from the perspective of 4-manifolds. In particular, the terms and maps in the surgery sequence are defined, and an explanation is given as to how the sphere embedding theorem, with the added ingredient of topological transversality, can be used to define the maps in the surgery sequence and show that it is exact. The surgery sequence is applied to classify simply connected closed 4-manifolds up to homeomorphism. The chapter closes with a survey of related classification results.

The Development of Topological 4-manifold Theory

2021 ◽  
pp. 295-330
Author(s):  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Smooth Structures ◽  
Normal Bundles ◽  
Exotic Smooth Structures ◽  
Poincare Conjecture ◽  
Manifold Theory ◽  
Simply Connected

The development of topological 4-manifold theory is described in the form of a flowchart showing the interdependence among many key statements in the theory. In particular, the flowchart demonstrates how the theory crucially relies on the constructions in this book, what goes into the work of Quinn on smoothing, normal bundles, and transversality, and what is needed to deduce the famous consequences, such as the classification of closed, simply connected, topological 4-manifolds, the category preserving Poincaré conjecture, and the existence of exotic smooth structures on 4-dimensional Euclidean space. Precise statements of the results, brief indications of some proofs, and extensive references are provided.

scholarly journals Unitary spherical representations of Drinfeld doubles

2018 ◽  
Vol 2018 (742) ◽  
pp. 157-186 ◽  
Author(s):  
Yuki Arano
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
Complete Classification ◽  
Unitary Representations ◽  
Compact Lie Group ◽  
Drinfeld Double ◽  
Quantum Analogue ◽  
Property T ◽  
Simply Connected

Abstract We study irreducible spherical unitary representations of the Drinfeld double of the q-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of \mathrm{SU}_{q}(3) , we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group \mathrm{SU}_{q}(2n+1) has property (T), which also implies central property (T) of the dual of \mathrm{SU}_{q}(2n+1) .

scholarly journals Classification of stable homotopy types with torsion-free homology

Topology ◽  
2001 ◽  
Vol 40 (4) ◽  
pp. 789-821 ◽  
Author(s):  
Hans-Joachim Baues ◽  
Yuri Drozd
Keyword(s):  
Stable Homotopy ◽  
Homotopy Types ◽  
Torsion Free

scholarly journals A BARTH–LEFSCHETZ THEOREM FOR SUBMANIFOLDS OF A PRODUCT OF PROJECTIVE SPACES

2009 ◽  
Vol 20 (01) ◽  
pp. 77-96
Author(s):  
LUCIAN BĂDESCU ◽  
FLAVIA REPETTO
Keyword(s):  
Normal Bundle ◽  
Projective Spaces ◽  
Restriction Map ◽  
Vanishing Theorem ◽  
Type Result ◽  
Picard Groups ◽  
Lefschetz Type ◽  
Simply Connected ◽  
Join Construction ◽  
Torsion Free

Let X be a complex submanifold of dimension d of ℙm × ℙn (m ≥ n ≥ 2) and denote by α: Pic(ℙm × ℙn) → Pic(X) the restriction map of Picard groups, by NX|ℙm × ℙn the normal bundle of X in ℙm × ℙn. Set t := max{dim π1(X), dim π2(X)}, where π1 and π2 are the two projections of ℙm × ℙn. We prove a Barth–Lefschetz type result as follows: Theorem. If [Formula: see text] then X is algebraically simply connected, the map α is injective and Coker(α) is torsion-free. Moreover α is an isomorphism if [Formula: see text], or if [Formula: see text] and NX|ℙm×ℙn is decomposable. These bounds are optimal. The main technical ingredients in the proof are: the Kodaira–Le Potier vanishing theorem in the generalized form of Sommese ([18, 19]), the join construction and an algebraization result of Faltings concerning small codimensional subvarieties in ℙN (see [9]).

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