Fundamental Group Of Space Research Articles

Fundamental Groups of Spaces with Bakry–Emery Ricci Tensor Bounded Below

We first extend Cheeger–Colding’s Almost Splitting Theorem (Ann Math 144:189–237, 1996) to smooth metric measure spaces. Arguments utilizing this extension show that if a smooth metric measure space has almost nonnegative Bakry–Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abelian. Second, if the smooth metric measure space has Bakry–Emery Ricci curvature bounded from below then the number of generators of the fundamental group is uniformly bounded. These results are extensions of theorems which hold for Riemannian manifolds with Ricci curvature bounded from below. The first result extends a result of Yun (Proc Amer Math Soc 125:1517–1522, 1997), while the second extends a result of Kapovitch and Wilking (Structure of fundamental groups of manifolds with Ricci curvature bounded below, 2011).

Fundamental groups of spaces of generalized perfect splines

The fundamental groups π1(Ωn) of the spaces Ωn of generalized perfect splines are calculated. For n ≥ 2, the groups are trivial; π1(Ω1) is a free group with three generators.

Fundamental groups of spaces of holomorphic maps and group actions

For integers $d \geq 0$ and $1 \neq k \neq n$, let $\mathrm{Hol}_{d}(\mathbb{C}P^{k},\mathbb{C}P^{n})$ denote the space consisting of all holomorphic maps $f : \mathbb{C}P^{k}\to \mathbb{C}P^{n}$ of degree $d$. We shall compute the fundamental group of $\mathrm{Hol}_{d}(\mathbb{C}P^{k},\mathbb{C}P^{n})$ and study $\mathrm{PGL}_{n+1}(\mathbb{C})$-action on it.

Non-existence of black holes in certain Λ < 0 spacetimes

Assuming certain asymptotic conditions, we prove a general theorem on the non-existence of static regular black holes in spacetimes with a negative cosmological constant, given that the fundamental group of space is infinite. We use this to rule out the existence of regular negative mass AdS black holes with Ricci flat scri. For any mass, we also rule out a class of conformally compactifiable static black holes whose conformal infinity has positive scalar curvature and infinite fundamental group, subject to our asymptotic conditions. In a limited, but important, special case our result adds new support to the AdS/CFT inspired positive mass conjecture of Horowitz and Myers.

The spin holonomy group in general relativity

It has recently been shown by Goldberg et al that the holonomy group of the chiral spin-connection is preserved under time evolution in vacuum general relativity. Here, the underlying reason for the time-independence of the holonomy group is traced to the self-duality of the curvature 2-form for an Einstein space. This observation reveals that the holonomy group is time-independent not only in vacuum, but also in the presence of a cosmological constant. It also shows that once matter is coupled to gravity, the "conservation of holonomy" is lost. When the fundamental group of space is non-trivial, the holonomy group need not be connected. For each homotopy class of loops, the holonomies comprise a coset of the full holonomy group modulo its connected component. These cosets are also time-independent. All possible holonomy groups that can arise are classified, and examples are given of connections with these holonomy groups. The classification of local and global solutions with given holonomy groups is discussed.