scholarly journals On the homotopy classification of proper Fredholm maps into a Hilbert space

2020 ◽  
Vol 2020 (759) ◽  
pp. 161-200 ◽  
Author(s):  
Alberto Abbondandolo ◽  
Thomas O. Rot
Keyword(s):  
Model Space ◽  
Complete Classification ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Infinite Dimensional ◽  
Ambient Manifold ◽  
Alternative Approach ◽  
Framed Cobordism ◽  
Fredholm Maps

AbstractWe classify the homotopy classes of proper Fredholm maps from an infinite-dimensional Hilbert manifold into its model space in terms of a suitable version of framed cobordism. Our construction is an alternative approach to the classification introduced by Elworthy and Tromba in 1970 and does not make use of further structures on the ambient manifold, such as Fredholm structures. In the special case of index zero, we obtain a complete classification involving the Caccioppoli–Smale mod 2 degree and the absolute value of the oriented degree.

scholarly journals Golod–Shafarevich-Type Theorems and Potential Algebras

2018 ◽  
Vol 2019 (15) ◽  
pp. 4822-4844 ◽  
Author(s):  
Natalia Iyudu ◽  
Agata Smoktunowicz
Keyword(s):  
Linear Growth ◽  
Complete Classification ◽  
Koszul Complex ◽  
Model Program ◽  
Minimal Model Program ◽  
Infinite Dimensional ◽  
Homogeneous Potential ◽  
Finite Dimensional ◽  
The Difference

Abstract Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree $n\geqslant 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class $\mathcal {P}_{n}$ of potential algebras with homogeneous potential of degree $n+1\geqslant 4$, the minimal Hilbert series is $H_{n}=\frac {1}{1-2t+2t^{n}-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.

scholarly journals 2-Local derivations on the W-algebra W(2,2)

2020 ◽  
pp. 2150237
Author(s):  
Xiaomin Tang
Keyword(s):  
Lie Algebra ◽  
Complete Classification ◽  
Infinite Dimensional ◽  
Local Derivation ◽  
Infinite Dimensional Lie Algebra

This paper is devoted to study 2-local derivations on [Formula: see text]-algebra [Formula: see text] which is an infinite-dimensional Lie algebra with some outer derivations. We prove that all 2-local derivations on the [Formula: see text]-algebra [Formula: see text] are derivations. We also give a complete classification of the 2-local derivation on the so-called thin Lie algebra and prove that it admits many 2-local derivations which are not derivations.

scholarly journals Homotopy Classification of Mappings of a 4-Dimensional Complex into a 2-Dimensional Sphere

1953 ◽  
Vol 5 ◽  
pp. 127-144 ◽  
Author(s):  
Nobuo Shimada
Keyword(s):  
Dimensional Sphere ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Cup Product

Steenrod [1] solved the problem of enumerating the homotopy classes of maps of an (n + 1)-complex K into an n-sphere Sn utilizing the cup-i-product, the far-reaching generalization of the Alexander-Čech-Whitney cup product [7] and the Pontrjagin *-product [5].

scholarly journals Tilting modules over tame hereditary algebras

2012 ◽  
Vol 2013 (682) ◽  
pp. 1-48
Author(s):  
Lidia Angeleri Hügel ◽  
Javier Sánchez
Keyword(s):  
Complete Classification ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Hereditary Algebras ◽  
Tame Hereditary Algebra

Abstract. We give a complete classification of the infinite dimensional tilting modules over a tame hereditary algebra R. We start our investigations by considering tilting modules of the form where is a union of tubes, and denotes the universal localization of R at in the sense of Schofield and Crawley-Boevey. Here is a direct sum of the Prüfer modules corresponding to the tubes in . Over the Kronecker algebra, large tilting modules are of this form in all but one case, the exception being the Lukas tilting module L whose tilting class consists of all modules without indecomposable preprojective summands. Over an arbitrary tame hereditary algebra, T can have finite dimensional summands, but the infinite dimensional part of T is still built up from universal localizations, Prüfer modules and (localizations of) the Lukas tilting module. We also recover the classification of the infinite dimensional cotilting R-modules due to Buan and Krause.

Homotopy Classification of Projections in the Corona Algebra of a Non-simple C*-algebra

2012 ◽  
Vol 64 (4) ◽  
pp. 755-777 ◽  
Author(s):  
Lawrence G. Brown ◽  
Hyun Ho Lee
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Homotopy Classification ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

AbstractWe study projections in the corona algebra of C(X) ⊗ K, where K is the C*-algebra of compact operators on a separable infinite dimensional Hilbert space and X = [0, 1], [0,∞), (−∞,∞), or [0, 1]/﹛0, 1﹜. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in K0, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.

scholarly journals Global defect topology in nematic liquid crystals

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160265 ◽  
Author(s):  
Thomas Machon ◽  
Gareth P. Alexander
Keyword(s):  
Homology Group ◽  
Double Cover ◽  
Explicit Computation ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
General Domain ◽  
Global Classification ◽  
Local Characterization ◽  
Global Defect

We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in R 3 , showing that the distinct homotopy classes have a 1–1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group has representatives that are planar and characterize the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterization. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of τ lines in cholesterics.

scholarly journals Homotopy classification of filtered complexes

1973 ◽  
Vol 15 (3) ◽  
pp. 298-318 ◽  
Author(s):  
Ross Street
Keyword(s):  
Abelian Groups ◽  
Classification Theorem ◽  
Homotopy Equivalence ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Homology Groups ◽  
Chain Complexes ◽  
Kunneth Formula ◽  
Homology Functor

The homology functor from the category of free abelian chain complexes and homotopy classes of maps to that of graded abelian groups is full and replete (surjective on objects up to isomorphism) and reflects isomorphisms. Thus such a complex is determined to within homotopy equivalence (although not a unique homotopy equivalence) by its homology. The homotopy classes of maps between two such complexes should therefore be expressible in terms of the homology groups, and such an expression is in fact provided by the Künneth formula for Hom, sometimes called ‘the homotopy classification theorem’.

scholarly journals On a Homotopy Classification of Mappings of an (n+1) Dimensional Complex Into an Arcwise Connected Topological Space Which is Aspherical in Dimensions Less Thann (n >2)

1951 ◽  
Vol 3 ◽  
pp. 67-72 ◽  
Author(s):  
Nobuo Shimada ◽  
Hiroshi Uehara
Keyword(s):  
Topological Space ◽  
Nauk Sssr ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Continuous Mappings ◽  
Arcwise Connected ◽  
New Type

Pontrjagin classified mappings of a three dimensional sphere into anndimensional complex, where he made use of a new type of product of cocycles. By the aid of the generalized Pontrjagin’s product of cocycles Steenrod enumerated effectively all the homotopy classes of mappings of an (n+1) dimensional complex into annsphere. According to the recent issue of the Mathematical Reviews it is reported that M. M. Postnikov extended Steenrod’s case to the case where an arcwise connected topological space which is aspherical in dimensions less thann, takes place of annsphere. (Postnikov M. M., Classification of continuous mappings of an(n+1)dimensional complex into a connected topological space which is aspherical in dimensions less thann. Doklady Akad. Nauk SSSR (N.S.) 71., 1027-1028, 1950 (Russian. No. proof is given.)) But here in Japan no details are yet to hand. We intend to give a solution to this problem in case wheren>2, and also to give an application concerning the(n+ 3)-extension cocycle.

Homotopy classification and the generalized Swan homomorphism

2009 ◽  
Vol 4 (3) ◽  
pp. 491-536 ◽  
Author(s):  
F.E.A. Johnson
Keyword(s):  
Finite Group ◽  
Homotopy Theory ◽  
Group Cohomology ◽  
Projective Modules ◽  
Finitely Generated ◽  
Homotopy Classification ◽  
Homotopy Classes ◽  
Algebraic Homotopy Theory ◽  
Fundamental Paper

AbstractIn his fundamental paper on group cohomology [20] R.G. Swan defined a homomorphism for any finite group G which, in this restricted context, has since been used extensively both in the classification of projective modules and the algebraic homotopy theory of finite complexes ([3], [18], [21]). We extend the definition so that, for suitable modules J over reasonably general rings Λ, it takes the form here is the quotient of the category of Λ-homomorphisms obtained by setting ‘projective = 0’. We then employ it to give an exact classification of homotopy classes of extensions 0 → J → Fn → … → F0 → F0 → M → 0 where each Fr is finitely generated free.

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