Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups

Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $$\mathbf {G}$$ G be a semisimple algebraic $${\mathbb {R}}$$ R -group such that $$G=\mathbf {G}({\mathbb {R}})^{\circ }$$ G = G ( R ) ∘ is of Hermitian type. If $$\Gamma \le L$$ Γ ≤ L is a torsion-free lattice of a finite connected covering of $$\mathrm{PU}(1,1)$$ PU ( 1 , 1 ) , given a standard Borel probability $$\Gamma $$ Γ -space $$(\Omega ,\mu _\Omega )$$ ( Ω , μ Ω ) , we introduce the notion of Toledo invariant for a measurable cocycle $$\sigma :\Gamma \times \Omega \rightarrow G$$ σ : Γ × Ω → G . The Toledo invariant remains unchanged along G-cohomology classes and its absolute value is bounded by the rank of G. This allows to define maximal measurable cocycles. We show that the algebraic hull $$\mathbf {H}$$ H of a maximal cocycle $$\sigma $$ σ is reductive and the centralizer of $$H=\mathbf {H}({\mathbb {R}})^{\circ }$$ H = H ( R ) ∘ is compact. If additionally $$\sigma $$ σ admits a boundary map, then H is of tube type and $$\sigma $$ σ is cohomologous to a cocycle stabilizing a unique maximal tube type subdomain. This result is analogous to the one obtained for representations. In the particular case $$G=\mathrm{PU}(n,1)$$ G = PU ( n , 1 ) maximality is sufficient to prove that $$\sigma $$ σ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles.

Sifat-Sifat Morfisma di dalam Kategori Ruang Penutup Ruang Topologis yang Terhubung Lintasan (On the Morphisms of the Category of Path Connected Covering Spaces )

Untuk sebarang ruang topologis $X$ dapat dibentuk $Cov_X$ yaitu kategori \linebreak ruang penutup $X$ yang terhubung lintasan. Pada tulisan ini akan dibahas syarat perlu dan cukup eksistensi morfisma antara dua ruang penutup yang terhubung lintasan lokal. Untuk sebarang $x_0 \in X$ dan grup fundamental $G=\pi_1(X,x_0)$, dapat dibentuk kategori $SetG$, yaitu kategori semua himpunan yang dilengkapi aksi kanan oleh $G$. Selanjutnya dibentuk fungtor $F$ dari $Cov_X$ ke $SetG$. Dalam tulisan dibuktikan bahwa $F$ bersifat \textit{fully faithful} jika $X$ terhubung lintasan dan terhubung lintasan lokal. Akibatnya untuk mengidentifikasi morfisma-morfisma antara dua obyek $A$ dan $B$ di $Cov_X$ dapat dilakukan dengan cara melihat sifat morfisma-morfisma antara $F(A)$ dan $F(B)$. (For any topological space $X$, we can construct the category of path \linebreak connected covering spaces of $X$, denoted by $Cov_X$. In this paper we study a sufficient and necesarry condition for the existence of morphism between two locally path \linebreak connected covering spaces. For every $x_0 \in X$ and fundamental group $G=\pi_1(X,x_0)$, we can construct the category of sets with right action of $G$, denoted by $SetG$. \linebreak Furthermore, we can define a functor $F$ from $Cov_X$ to $SetG$. We proof that the functor $F$ is fully faithul if $X$ is path connected and locally path connected. From this result, we can identify morphisms between $A$ and $B$ in $Cov_X$ by using the properties of morphisms between $F(A)$ and $F(B)$. )

On Steenrod 𝕃-homology, generalized manifolds, and surgery

The aim of this paper is to show the importance of the Steenrod construction of homology theories for the disassembly process in surgery on a generalized n-manifold Xn, in order to produce an element of generalized homology theory, which is basic for calculations. In particular, we show how to construct an element of the nth Steenrod homology group $H^{st}_{n} (X^{n}, \mathbb {L}^+)$, where 𝕃+ is the connected covering spectrum of the periodic surgery spectrum 𝕃, avoiding the use of the geometric splitting procedure, the use of which is standard in surgery on topological manifolds.

GLOBAL ACTIONS AND VECTOR -THEORY

Purely algebraic objects like abstract groups, coset spaces, and G-modules do not have a notion of hole as do analytical and topological objects. However, equipping an algebraic object with a global action reveals holes in it and thanks to the homotopy theory of global actions, the holes can be described and quantified much as they are in the homotopy theory of topological spaces. Part I of this article, due to the first author, starts by recalling the notion of a global action and describes in detail the global actions attached to the general linear, elementary, and Steinberg groups. With these examples in mind, we describe the elementary homotopy theory of arbitrary global actions, construct their homotopy groups, and revisit their covering theory. We then equip the set $Um_{n}(R)$ of all unimodular row vectors of length $n$ over a ring $R$ with a global action. Its homotopy groups $\unicode[STIX]{x1D70B}_{i}(Um_{n}(R)),i\geqslant 0$ are christened the vector $K$ -theory groups $K_{i+1}(Um_{n}(R)),i\geqslant 0$ of $Um_{n}(R)$ . It is known that the homotopy groups $\unicode[STIX]{x1D70B}_{i}(\text{GL}_{n}(R))$ of the general linear group $\text{GL}_{n}(R)$ viewed as a global action are the Volodin $K$ -theory groups $K_{i+1,n}(R)$ . The main result of Part I is an algebraic construction of the simply connected covering map $\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R)$ where $\mathit{EUm}_{n}(R)$ is the path connected component of the vector $(1,0,\ldots ,0)\in Um_{n}(R)$ . The result constructs the map as a specific quotient of the simply connected covering map $St_{n}(R)\rightarrow E_{n}(R)$ of the elementary global action $E_{n}(R)$ by the Steinberg global action $St_{n}(R)$ . As expected, $K_{2}(Um_{n}(R))$ is identified with $\text{Ker}(\mathit{StUm}_{n}(R)\rightarrow \mathit{EUm}_{n}(R))$ . Part II of the paper provides an exact sequence relating stability for the Volodin $K$ -theory groups $K_{1,n}(R)$ and $K_{2,n}(R)$ to vector $K$ -theory groups.