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

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.

ABELIAN CYCLES IN THE HOMOLOGY OF THE TORELLI GROUP

In the early 1980s, Johnson defined a homomorphism $\mathcal {I}_{g}^1\to \bigwedge ^3 H_1\left (S_{g},\mathbb {Z}\right )$ , where $\mathcal {I}_{g}^1$ is the Torelli group of a closed, connected, and oriented surface of genus g with a boundary component and $S_g$ is the corresponding surface without a boundary component. This is known as the Johnson homomorphism. We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding-pair maps, in order to compute a large quotient of $H_n\left (\mathcal {I}_{g}^1,\mathbb {Q}\right )$ in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group $\mathcal {I}_{g,1}$ of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of $H_n\left (\mathcal {I}_{g,1}\right )$ for $n\ge 2$ and g large enough.

Complexity of virtual multistrings

A virtual[Formula: see text]-string [Formula: see text] consists of a closed, oriented surface [Formula: see text] and a collection [Formula: see text] of [Formula: see text] oriented, closed curves immersed in [Formula: see text]. We consider virtual [Formula: see text]-strings up to virtual homotopy, i.e. stabilizations, destabilizations, stable homeomorphism, and homotopy. Recently, Cahn proved that any virtual 1-string can be virtually homotoped to a minimally filling and crossing-minimal representative by monotonically decreasing both genus and the number of self-intersections. We generalize her result to the case of non-parallel virtual [Formula: see text]-strings. Cahn also proved that any two crossing-irreducible representatives of a virtual 1-string are related by isotopy, Type 3 moves, stabilizations, destabilizations, and stable homeomorphism. Kadokami claimed that this held for virtual [Formula: see text]-strings in general, but Gibson found a counterexample for 5-strings. We show that Kadokami’s statement holds for non-parallel [Formula: see text]-strings and exhibit a counterexample for general virtual 3-strings.

Solar Radiation on a Parabolic Concave Surface

Curved structures are used in buildings and may be integrated with photovoltaic modules. Self-shading occurs on non-flat (curved) surface collectors resulting in a non-uniform distribution of the direct beam and the diffuse incident solar radiation along the curvature the surface. The present study uses analytical expressions for calculating and analyzing the incident solar radiation on a general parabolic concave surface. Concave surfaces facing north, south and east/west are considered, and numerical values for the annual incident irradiations (in kWh) are demonstrated for two locations: 32° N (Tel Aviv, Israel) and 52.2° N (Lindenberg, Germany). The numerical results show that the difference in the incident global irradiation for the different surface orientations is not very wide. At 32° N, the irradiation difference between the south and north-oriented surface is about 15 percent, and between the south and east surface orientation it is about 9.6 percent. For latitude 52.2° N, the global irradiation difference between the south and north-oriented surface is about 16 percent, and between the south and east orientation it is about 3 percent.

LOCAL COORDINATES FOR COMPLEX AND QUATERNIONIC HYPERBOLIC PAIRS

Let $G(n)={\textrm {Sp}}(n,1)$ or ${\textrm {SU}}(n,1)$ . We classify conjugation orbits of generic pairs of loxodromic elements in $G(n)$ . Such pairs, called ‘nonsingular’, were introduced by Gongopadhyay and Parsad for ${\textrm {SU}}(3,1)$ . We extend this notion and classify $G(n)$ -conjugation orbits of such elements in arbitrary dimension. For $n=3$ , they give a subspace that can be parametrized using a set of coordinates whose local dimension equals the dimension of the underlying group. We further construct twist-bend parameters to glue such representations and obtain local parametrization for generic representations of the fundamental group of a closed (genus $g \geq 2$ ) oriented surface into $G(3)$ .