-rigidity of Lagrangian submanifolds and punctured holomorphic disks in the cotangent bundle

Our main result is the $\mathbb {\mathcal {C}}^{0}$ -rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic disks in cotangent bundles with boundary on the zero section, whose boundaries represent any integral homology class. We discuss further applications of these punctured disks in symplectic geometry.

Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár

In their paper from 2012, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer–Hopf conjecture in the complex projective setting.

Slopes of links and signature formulas

We present a new invariant, called slope, of a colored link in an integral homology sphere and use this invariant to complete the signature formula for the splice of two links. We develop a number of ways of computing the slope and a few vanishing results. Besides, we discuss the concordance invariance of the slope and establish its close relation to the Conway polynomials, on the one hand, and to the Kojima–Yamasaki η \eta -function (in the univariate case) and Cochran invariants, on the other hand.

Rational cobordisms and integral homology

We consider the question of when a rational homology $3$-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology group injects in the first homology group of any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite-rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of $2$-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.

Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations

We compute the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu’s conjecture that $\unicode[STIX]{x1D6FD}=-\bar{\unicode[STIX]{x1D707}}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\unicode[STIX]{x1D6F4}(a_{1},\ldots ,a_{n})$ are not homology cobordant to any $-\unicode[STIX]{x1D6F4}(b_{1},\ldots ,b_{n})$. We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer spectrum provides homology cobordism obstructions distinct from $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$. In particular, we identify an $\mathbb{F}[U]$-module called connected Seiberg–Witten Floer homology, whose isomorphism class is a homology cobordism invariant.

Homology of Borel Subgroup of SL(2,\mathbb{F}_p)

In this paper we compute the integral homology of the Borel subgroup $B$ of the special linear group $SL(2,\mathbb{F}_p), p$ is a prime number. Arcoding to Adem \cite{AJM} these are periodic groups. In order to compute the integral homology of $B,$ we decompose it into $\ell-$ primary parts. We compute the first summand based on Invariant Theory and compute the rest summand based on Lyndon-Hochschild-Serre spectral sequence. We assume that $p$ is an odd prime and larger than $3.$

Cohomology of Homotopy 2-types

This chapter introduces some of the basic ingredients in the classification of homotopy 2-types and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: the fundamental crossed modules of a CW-complex, cat-1-groups, simplicial groups, Moore complexes, the Dold-Kan correspondence, integral homology of simplicial groups, homological perturbation theory. A manual classification of homotopy classes of maps from a surface to the projective plane is also included.

Sasaki-Einstein 7-Manifolds, Orlik Polynomials and Homology

In this article, we give ten examples of 2-connected seven dimensional Sasaki-Einstein manifolds for which the third homology group is completely determined. Using the Boyer-Galicki construction of links over particular Kähler-Einstein orbifolds, we apply a valid case of Orlik’s conjecture to the links so that one is able to explicitly determine the entire third integral homology group. We give ten such new examples, all of which have the third Betti number satisfy 10 ≤ b 3 ( L f ) ≤ 20 .

Symmetric Powers of Motives

This chapter develops the basic theory of symmetric powers of smooth varieties. The constructions in this chapter are based on an analogy with the corresponding symmetric power constructions in topology. If 𝐾 is a set (or even a topological space) then the symmetric power 𝑆𝑚𝐾 is defined to be the orbit space 𝐾𝑚/Σ‎𝑚, where Σ‎𝑚 is the symmetric group. If 𝐾 is pointed, there is an inclusion 𝑆𝑚𝐾 ⊂ 𝑆𝑚+1𝐾 and 𝑆∞𝐾 = ∪𝑆𝑚𝐾 is the free abelian monoid on 𝐾 − {*}. When 𝐾 is a connected topological space, the Dold–Thom theorem says that ̃𝐻*(𝐾, ℤ) agrees with the homotopy groups π‎ *(𝑆∞𝐾). In particular, the spaces 𝑆∞(𝑆 𝑛) have only one homotopy group (𝑛 ≥ 1) and hence are the Eilenberg–Mac Lane spaces 𝐾(ℤ, 𝑛) which classify integral homology.