34 results on '"Diarmuid Crowley"'
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2. Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification
- Author
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Arkadiy Skopenkov and Diarmuid Crowley
- Subjects
Group (mathematics) ,General Mathematics ,010102 general mathematics ,Geometric Topology (math.GT) ,57R52, 57R67, 55R15 ,Mathematics::Geometric Topology ,01 natural sciences ,Connected sum ,010101 applied mathematics ,Combinatorics ,Mathematics - Geometric Topology ,Simply connected space ,FOS: Mathematics ,Torsion (algebra) ,Isotopy ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,0101 mathematics ,Invariant (mathematics) ,Signature (topology) ,Quotient ,Mathematics - Abstract
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 indeterminancy. 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 $H_1\ne0$, with an indeterminancy. 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 $\mathbb Z\oplus\mathbb Z\oplus\mathbb Z_{12}$., Comment: 19 pages, exposition improved
- Published
- 2021
3. Correction to: Connected sum decompositions of high-dimensional manifolds
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Imre Bokor, Diarmuid Crowley, Stefan Friedl, Fabian Hebestreit, Daniel Kasprowski, Markus Land, and Johnny Nicholson
- Published
- 2022
4. Simply-connected manifolds with large homotopy stable classes
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ANTHONY CONWAY, DIARMUID CROWLEY, MARK POWELL, and JOERG SIXT
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Mathematics - Geometric Topology ,General Mathematics ,57R65, 57R67 ,FOS: Mathematics ,Geometric Topology (math.GT) ,QA - Abstract
For every $k \geq 2$ and $n \geq 2$ we construct $n$ pairwise homotopically inequivalent simply-connected, closed $4k$-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension $4$, we exhibit an analogous phenomenon for spin$^{c}$ structures on $S^2 \times S^2$. For $m\geq 1$, we also provide similar $(4m{-}1)$-connected $8m$-dimensional examples, where the number of homotopy types in a stable diffeomorphism class is related to the order of the image of the stable $J$-homomorphism $\pi_{4m-1}(SO) \to \pi^s_{4m-1}$., Comment: Minor revision: corrected a mistake in the proof of Theorem 4.11 and improved exposition in Section 4. 27 pages
- Published
- 2022
5. Exotic $$G_2$$-manifolds
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Diarmuid Crowley and Johannes Nordström
- Subjects
Mathematics - Differential Geometry ,Computer Science::Machine Learning ,Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,Fano plane ,Rank (differential topology) ,Computer Science::Digital Libraries ,01 natural sciences ,Connected sum ,Mathematics - Algebraic Geometry ,Mathematics - Geometric Topology ,Statistics::Machine Learning ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Invariant (mathematics) ,Mathematics::Symplectic Geometry ,Algebraic Geometry (math.AG) ,Mathematics ,010102 general mathematics ,Holonomy ,Geometric Topology (math.GT) ,14J28, 57R55 (Primary), 53C25 (Secondary) ,Differential Geometry (math.DG) ,Computer Science::Mathematical Software ,Mathematics::Differential Geometry ,010307 mathematical physics ,Diffeomorphism - Abstract
We exhibit the first examples of closed 7-dimensional Riemannian manifolds with holonomy G_2 that are homeomorphic but not diffeomorphic. These are also the first examples of closed Ricci-flat manifolds that are homeomorphic but not diffeomorphic. The examples are generated by applying the twisted connected sum construction to Fano 3-folds of Picard rank 1 and 2. The smooth structures are distinguished by the generalised Eells-Kuiper invariant introduced by the authors in arXiv:1406.2226., Comment: v4: minor corrections; 28 pages; to appear in Math. Ann
- Published
- 2020
6. The classification of 2‐connected 7‐manifolds
- Author
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Diarmuid Crowley and Johannes Nordström
- Subjects
Mathematics - Differential Geometry ,Topological manifold ,Pure mathematics ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Geometric Topology (math.GT) ,01 natural sciences ,57R55, 57R50 ,Cohomology ,Characteristic class ,Mathematics - Geometric Topology ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,Torsion (algebra) ,Classification theorem ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Spin-½ - Abstract
We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is an extension of the Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether the spin characteristic class p_M in the fourth integral cohomology of M is torsion. In addition we determine the inertia group of 2-connected M - equivalently the number of oriented smooth structures on the underlying topological manifold - in terms of p_M and the torsion linking form., Comment: Corrected the definition of pseudo-isotopy of almost diffeomorphisms, 56 pages. To appear in Proc. Lond. Math. Soc
- Published
- 2018
7. Harmonic spinors and metrics of positive curvature via the Gromoll filtration and Toda brackets
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Thomas Schick, Wolfgang Steimle, and Diarmuid Crowley
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Homotopy group ,Spinor ,Homotopy ,010102 general mathematics ,Order (ring theory) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Filtration (mathematics) ,Geometry and Topology ,Sectional curvature ,0101 mathematics ,Ricci curvature ,Scalar curvature ,Mathematics - Abstract
We construct non-trivial elements of order 2 in the homotopy groups $\pi_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Morlet's homotopy equivalence between $Diff(D^6,\partial)$ and $\Omega^7(PL_6/O_6)$, and Toda brackets in $PL_6/O_6$. We also construct non-trivial elements of order 2 in $\pi_* PL_m$ for every m greater or equal to 6 and * congruent to 1 or 2 modulo 8, which are detected by the alpha-invariant. As consequences, we (a) obtain non-trivial elements of order 2 in $\pi_* Diff(D^m,\partial)$ for m greater or equal to 6, and * + m congruent 0 or 1 modulo 8; (b) these elements remain non-trivial in $\pi_* Diff(M)$ where M is a closed spin manifold of the same dimension m and * > 0; (c) they act non-trivially on the corresponding homotopy group of the space of metrics of positive scalar curvature of such an M; in particular these homotopy groups are all non-trivial. The same applies to all other diffeomorphism invariant metrics of positive curvature, like the space of metrics of positive sectional curvature, or the space of metrics of positive Ricci curvature, provided they are non-empty. Further consequences are: (d) any closed spin manifold of dimension m greater or equal to 6 admits a metric with harmonic spinors; (e) there is no analogue of the odd-primary splitting of $(PL/O)_{(p)}$ for the prime 2; (f) for any $bP_{8j+4}$-sphere (where j > 0) of order which divides 4, the corresponding element in $\pi_0 Diff(D^{8j+2},\partial)$ lifts to $\pi_{8j-4} Diff(D^6,\partial)$, i.e., lies correspondingly deep down in the Gromoll filtration.
- Published
- 2018
8. Open Problems in the Topology of Manifolds
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James F. Davis, Stefan Friedl, Jonathan Bowden, Stephan Tillmann, Carmen Rovi, and Diarmuid Crowley
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Topology ,Topology (chemistry) ,Mathematics - Published
- 2021
9. Connected Sum Decompositions of High-Dimensional Manifolds
- Author
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Imre Bokor, Markus Land, Fabian Hebestreit, Daniel Kasprowski, Johnny Nicholson, Diarmuid Crowley, and Stefan Friedl
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Pure mathematics ,Degree (graph theory) ,Decomposition (computer science) ,Mathematics::Metric Geometry ,Uniqueness ,High dimensional ,Mathematics::Geometric Topology ,Connected sum ,Manifold ,Mathematics - Abstract
The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fails in higher dimensions.
- Published
- 2021
10. The rational homotopy type of (n-1)-connected manifolds of dimension up to 5n-3
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Diarmuid Crowley and Johannes Nordström
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Pure mathematics ,Topological space ,Type (model theory) ,01 natural sciences ,Mathematics::Algebraic Topology ,General Relativity and Quantum Cosmology ,Mathematics - Geometric Topology ,Tensor (intrinsic definition) ,0103 physical sciences ,55P62, 57N65 ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,0101 mathematics ,Mathematics ,Homotopy ,010102 general mathematics ,Geometric Topology (math.GT) ,Cohomology ,Manifold ,Linear map ,55P62 ,57R19 (primary) ,010307 mathematical physics ,Mathematics::Differential Geometry ,Geometry and Topology ,Subquotient - Abstract
We define the Bianchi-Massey tensor of a topological space X to be a linear map from a subquotient of the fourth tensor power of H*(X). We then prove that if M is a closed (n-1)-connected manifold of dimension at most 5n-3 (and n > 1) then its rational homotopy type is determined by its cohomology algebra and Bianchi-Massey tensor, and that M is formal if and only if the Bianchi-Massey tensor vanishes. We use the Bianchi-Massey tensor to show that there are many (n-1)-connected (4n-1)-manifolds that are not formal but have no non-zero Massey products, and to present a classification of simply-connected 7-manifolds up to finite ambiguity., 31 pages. v3: Corrected statement of Theorem 1.5. To appear in the Journal of Topology
- Published
- 2020
11. Distinguishing $$G_2$$-Manifolds
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Sebastian Goette, Diarmuid Crowley, and Johannes Nordström
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Pure mathematics - Published
- 2020
12. Differential Geometry in the Large
- Author
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Owen Dearricott, Wilderich Tuschmann, Yuri Nikolayevsky, Thomas Leistner, Diarmuid Crowley, Owen Dearricott, Wilderich Tuschmann, Yuri Nikolayevsky, Thomas Leistner, and Diarmuid Crowley
- Subjects
- Mathematical recreations, Geometry, Differential, Topology, Geometry
- Abstract
The 2019'Australian-German Workshop on Differential Geometry in the Large'represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures on manifolds, non-negative curvature and Alexandrov geometry, and topics in differential topology. A joy to the expert and novice alike, this proceedings volume touches on topics as diverse as Ricci and mean curvature flow, geometric invariant theory, Alexandrov spaces, almost formality, prescribed Ricci curvature, and Kähler and Sasaki geometry.
- Published
- 2020
13. Finite group actions on Kervaire manifolds
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Diarmuid Crowley and Ian Hambleton
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55Q10, 57N65, 57S17 ,Finite group ,Mathematics::Commutative Algebra ,Kervaire invariant ,General Mathematics ,Geometric Topology (math.GT) ,Surgery theory ,Homology (mathematics) ,16. Peace & justice ,Mathematics::Algebraic Topology ,Mathematics::Geometric Topology ,Manifold ,Combinatorics ,Mathematics - Geometric Topology ,Kervaire manifold ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics::Differential Geometry ,Mathematics - Algebraic Topology ,Mathematics::Symplectic Geometry ,Smooth structure ,Mathematics - Abstract
The (4k+2)-dimensional Kervaire manifold is a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product of two (2k+1)-dimensional spheres. We show that a finite group of odd order acts freely on a Kervaire manifold if and only if it acts freely on the corresponding product of spheres. If the Kervaire manifold M is smoothable, then each smooth structure on M admits a free smooth involution. If k + 1 is not a 2-power, then the Kervaire manifold in dimension 4k+2 does not admit any free TOP involutions. Free "exotic" (PL) involutions are constructed on the Kervaire manifolds of dimensions 30, 62, and 126. Each smooth structure on the 30-dimensional Kervaire manifold admits a free Z/2 x Z/2 action., 36 pages. Final version: Advances in Mathematics (to appear)
- Published
- 2015
14. The Topological Period-Index Conjecture for spin$^c$ 6-manifolds
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Diarmuid Crowley and Mark Grant
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Conjecture ,Index (economics) ,twisted $K$-theory ,14F22 ,Function (mathematics) ,Assessment and Diagnosis ,Topology ,Space (mathematics) ,Twisted K-theory ,19L50 ,FOS: Mathematics ,period-index problems ,Algebraic Topology (math.AT) ,Geometry and Topology ,Mathematics - Algebraic Topology ,57R19 ,Analysis ,Brauer group ,Period (music) ,Mathematics ,Brauer groups - Abstract
The Topological Period-Index Conjecture is an hypothesis which relates the period and index of elements of the cohomological Brauer group of a space. It was identified by Antieau and Williams as a topological analogue of the Period-Index Conjecture for function fields. In this paper we show that the Topological Period-Index Conjecture holds and is in general sharp for spin$^c$ 6-manifolds. We also show that it fails in general for 6-manifolds., v2, 14 pages. Significant improvements were made in the exposition, following comments from an anonymous referee. The main results and the essence of their proofs remain unchanged. v3: Corrected one reference
- Published
- 2018
15. Diagnosis and management of femoroacetabular impingement: a review of the literature
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Diarmuid Crowley, Claire Crowley, and Aidan O’Shea
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musculoskeletal diseases ,030222 orthopedics ,medicine.medical_specialty ,Femoroacetabular ,Hip ,Cam ,business.industry ,Rehabilitation ,Physical Therapy, Sports Therapy and Rehabilitation ,030229 sport sciences ,medicine.disease ,03 medical and health sciences ,0302 clinical medicine ,Pincer ,Occupational Therapy ,Impingement ,Physical therapy ,medicine ,business ,Femoroacetabular impingement - Abstract
Femoroacetabular impingement (FAI) is being increasingly diagnosed as a cause of hip pain among young and middle-aged adults, and is now recognised as a likely cause of early osteoarthritis (OA) of the hip. There are two main forms, “cam” impingement and “pincer” impingement, although the vast majority of cases have a combination of both forms known as “mixed” impingement. Over time, repetitive abnormal contact between the femoral head and acetabulum can result in chondral, labral, and eventually osseous pathology. A detailed history and physical examination in conjunction with radiological imaging help in the diagnosis of FAI. Diagnosis is made difficult by the fact that many asymptomatic patients have the characteristic features of FAI on imaging. Most patients should have a trial of conservative management prior to consideration for surgery. Arthroscopic and open surgical treatments are available, with neither showing superior outcomes. In general, early outcomes of surgical treatment appear to be favourable, however these outcomes are limited to a hip joint with little or no evidence of OA. Further high quality research is required to investigate the optimal approach to diagnosing and managing this complex condition.
- Published
- 2018
16. Functorial seminorms on singular homology and (in)flexible manifolds
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Clara Löh and Diarmuid Crowley
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Pure mathematics ,Homotopy ,mapping degrees ,Homology (mathematics) ,Mathematics::Geometric Topology ,Mathematics::Algebraic Topology ,57N65 ,55P62 ,Mathematics::K-Theory and Homology ,55N35 ,functorial seminorms on homology ,55N10 ,Simply connected space ,simply connected manifolds ,Geometry and Topology ,Mathematics::Symplectic Geometry ,Mathematics ,Singular homology - Abstract
A functorial seminorm on singular homology is a collection of seminorms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial seminorms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. ¶ In this paper, we use information about the degrees of maps between manifolds to construct new functorial seminorms with interesting properties. In particular, we answer a question of Gromov by providing a functorial seminorm that takes finite positive values on homology classes of certain simply connected spaces. Our construction relies on the existence of simply connected manifolds that are inflexible in the sense that all their self-maps have degree [math] , [math] or [math] . The existence of such manifolds was first established by Arkowitz and Lupton; we extend their methods to produce a wide variety of such manifolds.
- Published
- 2015
17. On the cardinality of the manifold set
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Diarmuid Crowley and Tibor Macko
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Mathematics::General Topology ,Algebraic geometry ,01 natural sciences ,Combinatorics ,Set (abstract data type) ,Mathematics - Geometric Topology ,0103 physical sciences ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Cardinality (SQL statements) ,Mathematics - Algebraic Topology ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Homotopy ,010102 general mathematics ,Geometric Topology (math.GT) ,Surgery theory ,Mathematics::Geometric Topology ,Manifold ,Homeomorphism ,Mathematics::Logic ,Differential geometry ,57R65, 57R67 ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry - Abstract
We study the cardinality of the set of manifolds homotopy equivalent to a given manifold M and compare it to the cardinality of the structure set of M., Comment: 23 pages. Added smooth versions of the main theorems. Improved exposition. Corrected the mistake from the first version in definition of r_k in section 3, current display (3.2), where we omitted factor t
- Published
- 2017
- Full Text
- View/download PDF
18. The Poincar\'e-Hopf Theorem for line fields revisited
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Diarmuid Crowley and Mark Grant
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Mathematics - Differential Geometry ,Pure mathematics ,Fundamental theorem ,Picard–Lindelöf theorem ,010102 general mathematics ,General Physics and Astronomy ,01 natural sciences ,Differential geometry ,57R22 (Primary), 57R25, 55M25, 53C80, 76A15 (Secondary) ,0103 physical sciences ,No-go theorem ,Line (geometry) ,Gravitational singularity ,Geometry and Topology ,Mathematics - Algebraic Topology ,0101 mathematics ,Poincaré–Hopf theorem ,010306 general physics ,Brouwer fixed-point theorem ,Mathematical Physics ,Mathematics - Abstract
A Poincar\'e-Hopf Theorem for line fields with point singularities on orientable surfaces can be found Hopf's 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus' statement only holds in even dimensions $2k \geq 4$. In 1984 J\"{a}nich presented a Poincar\'{e}-Hopf theorem for line fields with more complicated singularities and focussed on the complexities arising in the generalised setting. In this expository note we review the Poincar\'e-Hopf Theorem for line fields with point singularities, presenting a careful proof which is valid in all dimensions., Comment: 13 pages, 3 figures. v2: Final version, to appear in Journal of Geometry and Physics
- Published
- 2016
19. Contact structures on $$M \times S^2$$ M × S 2
- Author
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Jonathan Bowden, Diarmuid Crowley, and András I. Stipsicz
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Pure mathematics ,Boundary component ,General Mathematics ,Solid torus ,Structure (category theory) ,Geometry ,Surgery theory ,Mathematics::Geometric Topology ,Mathematics::Symplectic Geometry ,Manifold ,Mathematics - Abstract
We show that if a manifold \(M\) admits a contact structure, then so does \(M \times S^2\). Our proof relies on surgery theory, a theorem of Eliashberg on contact surgery and a theorem of Bourgeois showing that if \(M\) admits a contact structure then so does \(M \times T^2\).
- Published
- 2013
20. Embeddings of non-simply-connected 4-manifolds in 7-space. I. Classification modulo knots
- Author
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Diarmuid Crowley and Arkadiy Skopenkov
- Subjects
General Mathematics ,57R40, 57R52 ,Geometric Topology (math.GT) ,Bilinear form ,Mathematics::Geometric Topology ,Connected sum ,Combinatorics ,Mathematics - Geometric Topology ,Simply connected space ,Isotopy ,Torsion (algebra) ,FOS: Mathematics ,Equivalence relation ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Invariant (mathematics) ,Quotient ,Mathematics - Abstract
We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q:=H_q(N;Z)$. Our main result is a complete readily calculable classification of embeddings $N\to R^7$, up to the equivalence relation generated by isotopy and embedded connected sum with embeddings $S^4\to R^7$. Such a classification was already known only for $H_1=0$ by the work of Bo\'echat, Haefliger and Hudson from 1970. Our classification involves the Bo\'echat-Haefliger invariant $\varkappa(f)\in H_2$, Seifert bilinear form $\lambda(f):H_3\times H_3\to Z$ and $\beta$-invariant $\beta(f)$ which assumes values in a quotient of $H_1$ defined by values of $\varkappa(f)$ and $\lambda(f)$. In particular, for $N=S^1\times S^3$ we give a geometrically defined 1-1 correspondence between the set of equivalence classes of embeddings and an explicit quotient of the set $Z\oplus Z$. Our proof is based on development of Kreck modified surgery approach, involving some simpler reformulations, and also uses parametric connected sum., Comment: 52 pages, exposition improved
- Published
- 2016
- Full Text
- View/download PDF
21. A CLASSIFICATION OF SMOOTH EMBEDDINGS OF FOUR-MANIFOLDS IN SEVEN-SPACE, II
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Diarmuid Crowley and Arkadiy Skopenkov
- Subjects
Combinatorics ,Discrete mathematics ,General Mathematics ,Isotopy ,Invariant (mathematics) ,Injective function ,Connected sum ,Mathematics - Abstract
Let N be a closed connected smooth four-manifold with H1(N; ℤ) = 0. Our main result is the following classification of the set E7(N) of smooth embeddings N → ℝ7 up to smooth isotopy. Haefliger proved that E7(S4) together with the connected sum operation is a group isomorphic to ℤ12. This group acts on E7(N) by an embedded connected sum. Boéchat and Haefliger constructed an invariant ℵ: E7(N) → H2(N;ℤ) which is injective on the orbit space of this action; they also described im (ℵ). We determine the orbits of the action: for u ∈ im (ℵ) the number of elements in ℵ-1(u) is GCD (u/2, 12) if u is divisible by 2, or is GCD(u, 3) if u is not divisible by 2. The proof is based on Kreck's modified formulation of surgery.
- Published
- 2011
22. On the mapping class groups of # r (S p × S p ) for p = 3, 7
- Author
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Diarmuid Crowley
- Subjects
Combinatorics ,Orientation (vector space) ,Discrete mathematics ,Class (set theory) ,Group (mathematics) ,General Mathematics ,Homotopy ,Isotopy ,Mathematics - Abstract
For \({M_r := \sharp_r(S^p \times S^p),\,p=3, 7}\), we calculate \({\pi_0{\rm Diff}(M_r)/\Theta_{2p+1}}\) and \({\mathcal{E}(M_r)}\), respectively the group of isotopy classes of orientation preserving diffeomorphisms of Mr modulo isotopy classes with representatives which are the identity outside a 2p-disc and the group of homotopy classes of orientation preserving homotopy equivalences of Mr.
- Published
- 2010
23. The smooth structure set of S p × S q
- Author
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Diarmuid Crowley
- Subjects
Combinatorics ,Group structure ,Discrete mathematics ,Exact sequence ,Surgery exact sequence ,Hyperbolic geometry ,Image (category theory) ,Geometry and Topology ,Algebraic geometry ,Smooth structure ,Mathematics - Abstract
We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of Sp × Sq, for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.
- Published
- 2010
24. Positive Ricci curvature on highly connected manifolds
- Author
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David J. Wraith and Diarmuid Crowley
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Geometric Topology (math.GT) ,53C20, 57R65 ,01 natural sciences ,Mathematics::Geometric Topology ,Manifold ,Mathematics - Geometric Topology ,Homotopy sphere ,Differential Geometry (math.DG) ,0103 physical sciences ,Metric (mathematics) ,FOS: Mathematics ,Algebraic Topology (math.AT) ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry ,Mathematics - Algebraic Topology ,0101 mathematics ,Mathematics::Symplectic Geometry ,Analysis ,Ricci curvature ,Mathematics - Abstract
For $k \ge 2,$ let $M^{4k-1}$ be a $(2k{-}2)$-connected closed manifold. If $k \equiv 1$ mod $4$ assume further that $M$ is $(2k{-}1)$-parallelisable. Then there is a homotopy sphere $\Sigma^{4k-1}$ such that $M \sharp \Sigma$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings., Comment: Corrected some minor typos and changed document class to amsart. The new document class added 10 pages, so the paper is now now 46 pages
- Published
- 2014
25. Contact structures on $$M \\times S^2$$ M \xd7 S 2
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Jonathan Bowden, Diarmuid Crowley, and Andrxe1s I. Stipsicz
- Published
- 2014
26. The rational classification of links of codimension > 2
- Author
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Steven C. Ferry, Mikhail Skopenkov, and Diarmuid Crowley
- Subjects
Discrete mathematics ,Homotopy group ,Exact sequence ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Codimension ,Rank (differential topology) ,Mathematics::Geometric Topology ,Combinatorics ,Mathematics Subject Classification ,Lie algebra ,Isotopy ,Mathematics - Abstract
Let m and p1;:::;pr 2,E m . F rD1S p k /, is the set of smooth isotopy classes of smooth embeddings Fr kD1 S p k ! S m . Haefliger showed thatE m . Fr kD1 S p k/ is a finitely generated abelian group with respect to embedded connected summation and computed its rank in the case of knots, i.e. r D 1. For r > 1 and for restrictions on p1;:::;pr the rank of this group can be computed using results of Haefliger or Nezhinsky. Our main result determines the rank of the group E m . F rD1S pk/ in general. In particular we determine precisely when E m . FrD1 S pk/ is finite. We also accomplish these tasks for framed links. Our proofs are based on the Haefliger exact sequence for groups of links and the theory of Lie algebras. Keywords. Smooth manifold, embedding, isotopy, link, homotopy group, Lie algebra. 2010 Mathematics Subject Classification. Primary 57R52, 57Q45; secondary 55P62, 17B01.
- Published
- 2014
27. The topology of Stein fillable manifolds in high dimensions I
- Author
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András I. Stipsicz, Jonathan Bowden, and Diarmuid Crowley
- Subjects
Connection (fibred manifold) ,Mathematics - Differential Geometry ,General Mathematics ,Structure (category theory) ,Cobordism ,Geometric Topology (math.GT) ,Surgery theory ,Mathematics::Geometric Topology ,Connected sum ,Manifold ,Combinatorics ,Mathematics - Geometric Topology ,Differential Geometry (math.DG) ,Product (mathematics) ,Symplectic filling ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashberg's h-principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability., 39 pages, more explanation added. To appear in Proc. London Math. Soc
- Published
- 2014
28. New invariants of G_2-structures
- Author
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Diarmuid Crowley and Johannes Nordström
- Subjects
$G_2$–structures ,Mathematics - Differential Geometry ,Tangent bundle ,Pure mathematics ,Divisor (algebraic geometry) ,53C25 ,Connected sum ,exceptional holonomy ,Mathematics - Geometric Topology ,symbols.namesake ,$h$–principle ,Euler characteristic ,FOS: Mathematics ,57R15 ,Invariant (mathematics) ,Mathematics::Symplectic Geometry ,Mathematics ,diffeomorphisms ,spin geometry ,Homotopy ,Holonomy ,Geometric Topology (math.GT) ,53C10 ,53C10, 57R15 (Primary) 53C25, 53C27, 57R50, 57R90 (Secondary) ,53C27 ,Differential Geometry (math.DG) ,symbols ,Geometry and Topology ,Mathematics::Differential Geometry ,Signature (topology) - Abstract
We define a Z/48-valued homotopy invariant nu of a G_2-structure on the tangent bundle of a closed 7-manifold in terms of the signature and Euler characteristic of a coboundary with a Spin(7)-structure. For manifolds of holonomy G_2 obtained by the twisted connected sum construction, the associated torsion-free G_2-structure always has nu = 24. Some holonomy G_2 examples constructed by Joyce by desingularising orbifolds have odd nu. We define a further homotopy invariant xi of G_2-structures such that if M is 2-connected then the pair (nu, xi) determines a G_2-structure up to homotopy and diffeomorphism. The class of a G_2-structure is determined by nu on its own when the greatest divisor of p_1(M) modulo torsion divides 224; this sufficient condition holds for many twisted connected sum G_2-manifolds. We also prove that the parametric h-principle holds for coclosed G_2-structures., Comment: 26 pages, 1 figure. v3: Defined further invariant, strengthened classification results, changed title
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- 2012
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29. The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature
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Diarmuid Crowley and Thomas Schick
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Mathematics - Differential Geometry ,Pure mathematics ,positive scalar curvature ,53C21 ,01 natural sciences ,Mathematics - Geometric Topology ,0103 physical sciences ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Conjecture ,010102 general mathematics ,Gromoll filtration ,58B20 ,Geometric Topology (math.GT) ,Exotic sphere ,Characteristic class ,53C27 ,57R60 ,Differential Geometry (math.DG) ,$\alpha$–invariant ,Homomorphism ,010307 mathematical physics ,Geometry and Topology ,exotic sphere ,Scalar curvature - Abstract
Let X be a closed m-dimensional spin manifold which admits a metric of positive scalar curvature and let Pos(X) be the space of all such metrics. For any g in Pos(X), Hitchin used the KO-valued alpha-invariant to define a homomorphism A_{n-1} from \pi_{n-1}(Pos(X) to KO_{m+n}. He then showed that A_0 is not 0 if m = 8k or 8k+1 and that A_1 is not 0 if m = 8k-1 or 8$. In this paper we use Hitchin's methods and extend these results by proving that A_{8j+1-m} is not 0 whenever m>6 and 8j - m >= 0. The new input are elements with non-trivial alpha-invariant deep down in the Gromoll filtration of the group \Gamma^{n+1} = \pi_0(\Diff(D^n, \del)). We show that \alpha(\Gamma^{8j+2}_{8j-5}) is not 0 for j>0. This information about elements existing deep in the Gromoll filtration is the second main new result of this note., Comment: 14 pages, ams-latex. v2: corrections. Based on a referee's report we added Lemma 2.5 v2 and we gave more details in the proof of Lemma 2.14 v2. We also removed Corollary 1.2 v1 since we found a gap in the proof. v3: typo (wrong index) in Lemma 2.14 corrected
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- 2012
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30. Stably diffeomorphic manifolds and l 2q+1(ℤ[π])
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Jörg Sixt and Diarmuid Crowley
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Set (abstract data type) ,Combinatorics ,Simple (abstract algebra) ,Applied Mathematics ,General Mathematics ,Diffeomorphism ,Directed graph ,Mathematics::Algebraic Topology ,Mathematics::Symplectic Geometry ,Mathematics::Geometric Topology ,Surgery obstruction ,Mathematics - Abstract
The monoids l2q+1(Z[�]) detect s-cobordisms amongst certain bordisms between stably diffeomorphic 2q-dimensional manifolds and generalise the Wall simple surgery obstruction groups, L sq+1(Z[�]) � l2q+1(Z[�]). In this paper we identify l2q+1(Z[�]) as the edge set of a directed graph
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- 2011
31. The additivity of the $\rho$-invariant and periodicity in topological surgery
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Tibor Macko and Diarmuid Crowley
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Topological manifold ,medicine.medical_specialty ,Finite group ,Classifying space ,Algebraic structure ,Topology ,Surgery ,surgery ,$\rho$–invariant ,Mathematics - Geometric Topology ,57R65 ,medicine ,57S25 ,57R65, 57S25 ,Homomorphism ,Geometry and Topology ,Mathematics - Algebraic Topology ,Abelian group ,Invariant (mathematics) ,Subquotient ,Mathematics - Abstract
For a closed topological manifold M with dim (M) >= 5 the topological structure set S(M) admits an abelian group structure which may be identified with the algebraic structure group of M as defined by Ranicki. If dim (M) = 2d-1, M is oriented and M is equipped with a map to the classifying space of a finite group G, then the reduced rho-invariant defines a function, \wrho : S(M) \to \QQ R_{hat G}^{(-1)^d}, to a certain sub-quotient of the complex representation ring of G. We show that the function \wrho is a homomorphism when 2d-1 >= 5. Along the way we give a detailed proof that a geometrically defined map due to Cappell and Weinberger realises the 8-fold Siebenmann periodicity map in topological surgery., Comment: Contains minor corrections suggested by the referee and also more details in the proofs of Proposition 2.9 and Lemma 8.1 and an explanation in Remark 2.10
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- 2010
32. Kreck-Stolz invariants for quaternionic line bundles
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Sebastian Goette and Diarmuid Crowley
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Pure mathematics ,Chern class ,Applied Mathematics ,General Mathematics ,Geometric Topology (math.GT) ,Homology (mathematics) ,Manifold ,Algebra ,Mathematics - Geometric Topology ,Line bundle ,58J28, 57R55, 57R20 ,Quaternionic representation ,FOS: Mathematics ,Invariant (mathematics) ,Mathematics::Symplectic Geometry ,Mathematics ,Splitting principle - Abstract
We generalise the Kreck-Stolz invariants s_2 and s_3 by defining a new invariant, the t-invariant, for quaternionic line bundles E over closed spin-manifolds M of dimension 4k-1 with H^3(M; \Q) = 0 such that c_2(E)\in H^4(M) is torsion. The t-invariant classifies closed smooth oriented 2-connected rational homology 7-spheres up to almost-diffeomorphism, that is, diffeomorphism up to connected sum with an exotic sphere. It also detects exotic homeomorphisms between such manifolds. The t-invariant also gives information about quaternionic line bundles over a fixed manifold and we use it to give a new proof of a theorem of Feder and Gitler about the values of the second Chern classes of quaternionic line bundles over HP^k. The t-invariant for S^{4k-1} is closely related to the Adams e-invariant on the (4k-5)-stem., Comment: We extended the scope of the definition of our central invariant and significantly improved the presentation: to appear in the Transactions of the AMS. 35 pages
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- 2010
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33. Principal Bundles and the Dixmier Douady Class
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Diarmuid Crowley, Michael K. Murray, and Alan L. Carey
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Physics ,High Energy Physics - Theory ,Pure mathematics ,Class (set theory) ,Reduction (recursion theory) ,Group (mathematics) ,Principal (computer security) ,Structure (category theory) ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Principal bundle ,String (physics) ,High Energy Physics - Theory (hep-th) ,Variety (universal algebra) ,Mathematical Physics - Abstract
A systematic consideration of the problem of the reduction and extension of the structure group of a principal bundle is made and a variety of techniques in each case are explored and related to one another. We apply these to the study of the Dixmier-Douady class in various contexts including string structures, U-res bundles and other examples motivated by considerations from quantum field theory., 28 pages, latex, no figures, uses amsmath, amsthm, amsfonts. Revised version - only change a lot of irritating typos removed
- Published
- 1997
34. A classification of S3-bundles over S4
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Christine Escher and Diarmuid Crowley
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,FOS: Mathematics ,55R15 ,55R40 ,57T35 ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Sectional curvature ,Sphere bundles over spheres ,μ-invariant ,Equivalence (measure theory) ,Homeomorphism ,Mathematics ,Fiber (mathematics) ,Homotopy ,Orientation (vector space) ,Algebra ,Homotopy sphere ,Computational Theory and Mathematics ,Differential Geometry (math.DG) ,Geometry and Topology ,Diffeomorphism ,Mathematics::Differential Geometry ,Analysis ,Diffeomorphism classification - Abstract
We classify the total spaces of bundles over the four sphere with fiber a three sphere up to orientation preserving and reversing homotopy equivalence, homeomorphism and diffeomorphism. These total spaces have been of interest to both topologists and geometers. It has recently been shown by Grove and Ziller that each of these total spaces admits metrics with nonnegative sectional curvature., Comment: 19 pages
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