Intersections and embeddings of polyhedra

Harris, Leona

doi:10.1016/0040-9383(69)90026-3

Cited by 15 publications

(12 citation statements)

References 7 publications

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“…In many cases, mixed-valence compounds tend to undergo charge-order transitions at low temperatures. In lacunar spinels it is believed that the metal ions within the cubane M4X4 clusters form M4 units with a unique molecular electron density [19]. It seems interesting to check if these electronic molecular structures are stable up to elevated temperatures.…”

Section: Introductionmentioning

confidence: 99%

On the multiferroic skyrmion-host GaV₄S₈

Widmann

Ruff

Günther

et al. 2016

Philosophical Magazine

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Broadband microwave spectroscopy has been performed on single-crystalline GaV4S8, which exhibits a complex magnetic phase diagram including cycloidal, Néel-type skyrmion lattice, as well as field-polarized ferromagnetic phases below 13 K. At zero and small magnetic fields two collective modes are found at 5 and 15 GHz, which are characteristic of the cycloidal state in this easy-axis magnet. In finite fields, entering the skyrmion lattice phase, the spectrum transforms into a multi-mode pattern with absorption peaks near 4, 8, and 15 GHz. The spin excitation spectra in GaV4S8 and their field dependencies are found to be in close relation to those observed in materials with Bloch-type skyrmions. Distinct differences arise from the strong uniaxial magnetic anisotropy of GaV4S8 not present in so-far known skyrmion hosts. PACS numbers: 76.50.+g, 12.39.Dc, 75.30.Gw The occurence of nontrivial topology in the spin pattern of magnets has gained considerable interest in condensed matter physics. Recent research focuses on magnetic skyrmions which are thermodynamically stabilized in compounds with noncentrosymmetric crystal structures , in a limited region of the magnetic field versus temperature phase diagram [1-3]. Skyrmions are whirl-like objects of spins which can crystallize in skyrmion lattices (SkLs) with typical lattice constants from ten to hundred nanometers and give rise to emergent electrody-namics, like the topological Hall effect [4-6] or magnetic monopoles [7]. Individual skyrmions have been proposed as building blocks for novel nanomagnetic storage devices [8, 9]. The SkL has raised high interest for microwave-technology applications after collective spin excitations predicted in the GHz range [10] were evidenced in the insulating chiral magnet Cu 2 OSeO 3 [11-16]. Later it was found that different metallic, semiconducting, and insulating chiral magnets support the same set of characteristic excitations, i.e., three SkL modes characterized as clockwise (CW), counterclockwise (CCW) and breathing (BR) modes, that all follow a universal behavior [17]. Besides the Bloch-type skyrmions reported in the aforementioned works, a Néel-type SkL has recently been discovered in GaV 4 S 8 [18], where the spins rotate radially towards the vortex core. In this semiconductor characterized by V 4 S 4 clusters with spin S = 1 2 [19], a structural Jahn-Teller transition [20] at 44 K is followed by the onset of magnetic order at the Curie temperature T C = 13 K. At the structural transition the lattice is stretched along one of the four body diagonals, resulting in a strongly anisotropic easy-axis magnet. The magnetic multi-domain structure strongly depends on the orientation and strength of the applied magnetic field and gives rise to complex magnetic phase diagrams [see Figures 1(a), 1(c) and 2(a)] including cycloidal (Cyc), SkL, and ferromagnetic (FM) regions. Specifically, the skyrmions do not follow the external magnetic field but are confined to the magnetic easy axes. The phases have been interpreted in terms of a comp...

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Section: Introductionmentioning

confidence: 99%

On the multiferroic skyrmion-host GaV₄S₈

Widmann

Ruff

Günther

et al. 2016

Philosophical Magazine

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“…These all are controlled versions of Harris' criteria [Harr], however we use some ideas, additional (proof of 7.2) and alternative (proof of 7.4) to that of [Harr]. In fact, our proof of 7.4 is a new geometric proof of [Harr,Th. 2] (roughly a half of 1.7a-) and a good candidate for generalization for k-tuples of points.…”

Section: Figurementioning

confidence: 99%

“…The equivariant homotopy H p,q+1 is defined analogously to as F ′ in [Harr,proof of Proposition 9], and it follows from the construction of F ′ in [Harr]…”

Section: Proof Of Criterion 17mentioning

confidence: 99%

On maps with unstable singularities

Melikhov

2002

Topology and its Applications

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If a continuous map f : X → Q is approximable arbitrary closely by embeddings X ֒→ Q, can some embedding be taken onto f by a pseudo-isotopy? This question, called Isotopic Realization Problem, was raised byŠčepin and Akhmet'ev. We consider the case where X is a compact n-polyhedron, Q a PL m-manifold and show that the answer is 'generally no' for (n, m) = (3, 6); (1, 3), and 'yes' when:and ∆(f ) = {(x, y) | f (x) = f (y)} has an equivariant (with respect to the factor exchanging involution) mapping cylinder neighborhood in X × X;3) m > n + 2 and f is the composition of a PL map and a TOP embedding. In doing this, we answer affirmatively (with a minor preservation) a question of Kirby: does small smooth isotopy imply small smooth ambient isotopy in the metastable range, verify a conjecture of Kearton-Lickorish: small PL concordance implies small PL ambient isotopy in codimension ≥ 3, and a conjecture set of Repovs-Skopenkov.

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“…The method of [Web67,Har69] was actually designed for the proof of Theorem 1.1α and thus is not optimal: the result of [Web67,Har69] concerning the existence of an immersion has the dimension restriction 2m ≥ 3n + 3 instead of 2m ≥ 3n + 2 as in Theorem 1.3β. The proofs [Web67,Har69] can be perhaps modified to work for 2m ≥ 3n + 2 (at least to obtain some immersion h, i.e. without the property h eq Φ on SN ).…”

Section: Construction Of An Immersionmentioning

confidence: 99%

On the Haefliger-Hirsch-Wu invariants for embeddings and immersions

Skopenkov¹

2002

Comment. Math. Helv.

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Abstract. We prove beyond the metastable dimension the PL cases of the classical theorems due to Haefliger, Harris, Hirsch and Weber) on the deleted product criteria for embeddings and immersions. The isotopy and regular homotopy versions of the above theorems are also improved. We show by examples that they cannot be improved further. These results have many interesting corollaries, e.g. 1) Any closed homologically 2-connected smooth 7-manifold smoothly embeds in R 11 .2) If p ≤ q and m ≥ 3q 2 Mathematics Subject Classification (2000). Primary: 57Q35, 57Q37; Secondary: 54C25, 55S15, 57Q30, 57Q60, 57Q65, 57R40.

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Intersections and embeddings of polyhedra

Cited by 15 publications

References 7 publications

On the multiferroic skyrmion-host GaV₄S₈

On the multiferroic skyrmion-host GaV₄S₈

On maps with unstable singularities

On the Haefliger-Hirsch-Wu invariants for embeddings and immersions

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Intersections and embeddings of polyhedra

Cited by 15 publications

References 7 publications

On the multiferroic skyrmion-host GaV4S8

On the multiferroic skyrmion-host GaV4S8

On maps with unstable singularities

On the Haefliger-Hirsch-Wu invariants for embeddings and immersions

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Product

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On the multiferroic skyrmion-host GaV₄S₈

On the multiferroic skyrmion-host GaV₄S₈