scholarly journals Khovanov–Lipshitz–Sarkar homotopy type for links in thickened higher genus surfaces

2021 ◽  
Author(s):  
Louis H. Kauffman ◽  
Igor Mikhailovich Nikonov ◽  
Eiji Ogasa
Keyword(s):  
Homotopy Type ◽  
Khovanov Homology ◽  
Stable Homotopy ◽  
Product Manifold ◽  
Oriented Surface ◽  
Steenrod Square ◽  
Higher Genus ◽  
Genus Formula ◽  
Thickened Surface ◽  
Closed Oriented Surface

We discuss links in thickened surfaces. We define the Khovanov–Lipshitz–Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov–Lipshitz–Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus [Formula: see text] are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. It is the first meaningful Khovanov–Lipshitz–Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.

scholarly journals Generating the mapping class groups by torsions

2017 ◽  
Vol 26 (07) ◽  
pp. 1750037
Author(s):  
Xiaoming Du
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Oriented Surface ◽  
Genus Formula ◽  
Closed Oriented Surface

Let [Formula: see text] be a closed oriented surface of genus [Formula: see text] and let [Formula: see text] be the mapping class group. When the genus is at least 3, [Formula: see text] can be generated by torsion elements. We prove the following results: For [Formula: see text], [Formula: see text] can be generated by four torsion elements. Three generators are involutions and the fourth one is an order three element. [Formula: see text] can be generated by five torsion elements. Four generators are involutions and the fifth one is an order three element.

scholarly journals The extended mapping class group can be generated by two torsions

2017 ◽  
Vol 26 (11) ◽  
pp. 1750061
Author(s):  
Xiaoming Du
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
The Other ◽  
Mapping Class ◽  
Oriented Surface ◽  
Genus Formula ◽  
Extended Mapping ◽  
Closed Oriented Surface

Let [Formula: see text] be the closed-oriented surface of genus [Formula: see text] and let [Formula: see text] be the extended mapping class group of [Formula: see text]. When the genus is at least 5, we prove that [Formula: see text] can be generated by two torsion elements. One of these generators is of order [Formula: see text], and the other one is of order [Formula: see text].

scholarly journals A Steenrod square on Khovanov homology

2014 ◽  
Vol 7 (3) ◽  
pp. 817-848 ◽  
Author(s):  
Robert Lipshitz ◽  
Sucharit Sarkar
Keyword(s):  
Khovanov Homology ◽  
Steenrod Square

Cuspidality in higher genus

2016 ◽  
Vol 12 (08) ◽  
pp. 2043-2060
Author(s):  
Dania Zantout
Keyword(s):  
Linear Operator ◽  
Half Space ◽  
Modular Forms ◽  
Cusp Forms ◽  
General Notion ◽  
Global Operator ◽  
Higher Genus ◽  
Genus Formula ◽  
Holomorphic Modular Forms

We define a global linear operator that projects holomorphic modular forms defined on the Siegel upper half space of genus [Formula: see text] to all the rational boundaries of lower degrees. This global operator reduces to Siegel's [Formula: see text] operator when considering only the maximal standard cusps of degree [Formula: see text]. One advantage of this generalization is that it allows us to give a general notion of cusp forms in genus [Formula: see text] and to bridge this new notion with the classical one found in the literature.

Stable summands of U(n)

1997 ◽  
Vol 127 (5) ◽  
pp. 963-973 ◽  
Author(s):  
M. C. Crabb ◽  
J. R. Hubbuck ◽  
J. A. W. McCall
Keyword(s):  
Homotopy Type ◽  
Unitary Group ◽  
Stable Homotopy ◽  
Finite Complex ◽  
Special Unitary Group ◽  
Prime 2

SynopsisThe special unitary group SU(n) has the stable homotopy type of a wedge of n − 1 finite complexes. The ‘first’ of these complexes is ΣℂPn–1, which is well known to be indecomposable at the prime 2 whether n is finite or infinite. We show that the ‘second’ finite complex is again indecomposable at the prime 2 when n is finite, but splits into a wedge of two pieces when n is infinite.

On the Stable Homotopy Type of Thom Complexes

1973 ◽  
Vol 25 (6) ◽  
pp. 1285-1294 ◽  
Author(s):  
R. P. Held ◽  
D. Sjerve
Keyword(s):  
Vector Bundle ◽  
Homotopy Type ◽  
Stable Homotopy ◽  
Homotopy Equivalence ◽  
Real Vector ◽  
Fiber Homotopy

Let α be a real vector bundle over a finite CW complex X and let T(α;X) be its associated Thorn complex. We propose to study the S-type (stable homotopy type) of Thorn complexes in the framework of the Atiyah-Adams J-Theory. Therefore we focus our attention on the group JR(X) which is defined to be the group of orthogonal sphere bundles over X modulo stable fiber homotopy equivalence.

An n-dimensional Klein bottle

2019 ◽  
Vol 149 (5) ◽  
pp. 1207-1221
Author(s):  
Donald M. Davis
Keyword(s):  
Euclidean Space ◽  
Homotopy Type ◽  
Klein Bottle ◽  
Stable Homotopy ◽  
Cohomology Algebra ◽  
Topological Complexity ◽  
Integral Cohomology ◽  
Planar Polygon

AbstractAn n-dimensional analogue of the Klein bottle arose in our study of topological complexity of planar polygon spaces. We determine its integral cohomology algebra and stable homotopy type, and give an explicit immersion and embedding in Euclidean space.

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