Isotopies of 3-manifolds

The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and \operatorname{CAT}(0) groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete \mathcal{VH} complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually finite fundamental group. For the class of complete directed \mathcal{VH} complexes, we prove that there are exactly two complexes with six squares having a non-residually finite fundamental group. In particular, this positively answers to a question of Wise. Our approach relies on the connection between square complexes and automata discovered by Glasner and Mozes, where complete \mathcal{VH} complexes with one vertex correspond to bireversible automata. We prove that the square complex associated to a bireversible automaton with two states or over the binary alphabet generating an infinite automaton group has a non-residually finite fundamental group. We describe automaton groups associated to CSCs with four squares and get two simple automaton representations of the free group F_2 .

Chapter 26 - Wall's Finiteness Obstruction

The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is nonnegative for all closed simply connected manifolds of dimension ≥ 5. We extend this to show that Yamabe invariant is nonnegative for all closed manifolds of dimension ≥ 5 with fundamental group of odd order having all Sylow subgroups abelian. The main new geometric input is a way of studying the Yamabe invariant on Toda brackets. A similar method of proof shows that all closed oriented manifolds of dimension ≥ 5 with non-spin universal cover, with finite fundamental group having all Sylow subgroups elementary abelian, admit metrics of positive scalar curvature, once one restricts to the complement of manifolds whose homology classes are toral. The exceptional toral homology classes only exist in dimensions not exceeding the rank of the fundamental group, so this proves important cases of the Gromov-Lawson-Rosenberg Conjecture once the dimension is sufficiently large.

This paper is concerned with the homotopy theory and simple homotopy theory of connected finite 2 dimensional CW complexes with finite cyclic fundamental group. The main theorem presents a complete classification of such complexes up to homotopy type, and this theorem has the corollary that homotopy type and simple homotopy type coincide for these complexes. This work is motivated by the general problem of describing the sets HT(n) and SHT(n) of homotopy types and simple homotopy types of n-complexes, that is, connected finite 2 dimensional CW-complexes with a given fundamental group n. A visually satisfying description of HT (n) or SHT (n) is that of either set as a graph whose edges connect the type of each n-complex X to the type of its sum X v S 2 with the 2-sphere S 2. These graphs are actually trees; they clearly contain no circuits, and they are connected because any two n-complexes have the same type once each is summed with an appropriate number of copies of the 2-sphere S 2. To re establish this latter observation ofJ. H. C. Whitehead ([20, Theorem 12]), note that each n-complex has the simple homotopy type of one modeled in an obvious fashion on some finite presentation of the fundamental group ~z (see Proposition 1). But two finite presentations of the same group n differ by a finite sequence of Tietze operations, two of which leave the simple homotopy type of the associated topological model unchanged, while two alter the simple homotopy type by an S 2 summand. Of special interest in each of these trees are the roots and the junctions. The roots are the (simple) homotopy types that do not admit a factorization involving an S 2 summand; they generate the rest of the types in the tree under the operation of forming sum with S 2. The junctions are the (simple) homotopy types that admit two or more inequivalent factorizations involving an S 2 summand; they determine the shape of the tree. Each junction is a 2-dimensional instance of non-cancellation of the 2-sphere S 2 with respect to the sum operation. When the group n is a free group F of finite rank or is the finite cyclic group Z~ of prime order q, complete descriptions of the trees HT (n) and SHT (n) can be derived from the literature, as follows. A result of C. T. C. Wall ([17, Proposition 3.3]) can be specialized to read that for a free group F of finite rank r every F-complex has the homotopy type of a sum of r

An irreducible open 3-manifold W is R 2 -irreducible if it contains no non-trivial planes, i.e. given any proper embedded planein W some component of W −� must have closure an embedded halfspace R 2 ×(0, ∞). In this paper it is shown that if M is a connected, P 2 -irreducible, open 3-manifold such that �1(M) is finitely generated and the universal covering space f M of M is R 2 -irreducible, then either f M is homeomorphic to R 3 or �1(M) is a free product of infinite cyclic groups and fundamental groups of closed, connected surfaces other than S 2 or P 2 . Given any finitely generated group G of this form, uncountably many P 2 -irreducible, open 3-manifolds M are constructed with �1(M) ∼ G such that the universal covering space f M is R 2 -irreducible and not homeomorphic to R 3 ; the f M are pairwise non- homeomorphic. Relations are established between these results and the conjecture that the universal covering space of any irreducible, orientable, closed 3-manifold with infinite fundamental group must be homeomorphic to R 3 .

A question dating back to Waldhausen [10] and discussed in various contexts by Thurston (see [9]) is the problem of the extent to which irreducible 3-manifolds with infinite fundamental group must contain surface groups. To state our results precisely, it is convenient to make the definition that a map i: S 9<M of a closed, orientable connected surface S is essential if it is injective at the level of fundamental groups and the group i*rr1 (S) cannot be conjugated into a subgroup 7rr(coM) of -rr(M), where &oM is a component of OM. This latter condition is equivalent to the statement that the image of the surface S cannot be freely homotoped into OM. One of the main results of this paper is the following:

The dominant topologies in the Euclidean path integral for quantum gravity differ sharply according to the sign of the cosmological constant. For , saddle points can occur only for topologies with vanishing first Betti number and finite fundamental group. For , on the other hand, the path integral is dominated by topologies with extremely complicated fundamental groups; while the contribution of each individual manifold is strongly suppressed, the `density of topologies' grows fast enough to overwhelm this suppression. The value is thus a sort of boundary between phases in the sum over topologies. I discuss some implications for the cosmological constant problem and the Hartle-Hawking wavefunction.

We study the existence of pluricanonical forms on an algebraic variety of general type with infinite fundamental group. We have a nonvanishing of m th plurigenus of the variety by that of a general fibre of the so-called Shafarevich map of the variety.

We apply G. Prasad's volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of SO(1, n).As a result we prove that for any even dimension n there exists a unique compact arithmetic hyperbolic n-orbifold of the smallest volume.We give a formula for the Euler-Poincaré characteristic of the orbifolds and present an explicit description of their fundamental groups as the stabilizers of certain lattices in quadratic spaces.We also study hyperbolic 4-manifolds defined arithmetically and obtain a number theoretical characterization of the smallest compact arithmetic 4-manifold.

We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogue remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth action on an acyclic 5-manifold are the alternating groups A_5 and A_6, and deduce from this a short list of finite groups, closely related to the finite subgroups of SO(5), which are the candidates for orientation-preserving actions on acyclic 5-manifolds.

Let $X$ be a finite CW-complex with finite fundamental group. We show that the group ${\text {aut}}(X)$ of homotopy classes of self-homotopy equivalences of $X$ is commensurable to an arithmetic group. If in addition $X$ is an oriented manifold then the subgroup ${\text {au}}{{\text {t}}_t}(X)$ of homotopy classes of tangential homotopy equivalences is commensurable to an arithmetic group. Moreover if $X$ is a smooth manifold of dimension $\geqslant 5$ then the subgroup ${\text {diff}}(X)$ of ${\text {aut}}(X)$ the elements of which are represented by diffeomorphisms is also commensurable to an arithmetic group.

We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection.

Groups associated to a knot in S3 which classify the knot have been constructed by Conway and Gordon [4], Simon [12] and Whitten[16]; see also [1] and [19] for certain special classes of knots. In this note we show that for a knot K in S3 of the groups π1(S3/K)/〈malb〉 classify the knot; here m resp. l denote a meridian resp. longitude of the knot and 〈〉 denotes normal closure. This is based on Thurston's orbifold geometrization theorem [13–15] and the following result (see also [8, 19]):Theorem 1. Let O1 and O2 be good closed irreducible 3-orbifolds which possess a decomposition into geometric pieces (along Euclidean 2-suborbifolds, see [2, 15]) and have infinite (orbifold-) fundamental group. Then O1 and O2 are diffeomorphic (as orbifolds) if and only if π1 O1 and π1 02 are isomorphic.

Let (S, º) be a finite simple group, s i , i = 1, ... , n , be fixed (not necessarily distinct) elements of S , and let E α 1 , E α 2 , ... , E α k +1 be a random realisation of a chain of states of a simple homogeneous irreducible Markov chain with the set of states { E 1 , E 2 , ... , E n }. We study convergence conditions and limit distributions for the sequences of random products of the form η ( k ) = s α 1 º s α 2 º ... º s α k +1 . The convergence conditions are formulated in terms of some homomorphism from the fundamental group of the transition graph of the Markov chain to the structural group of the semigroup S .

We present an algorithm for the following problem: given a context-free grammar for the word problem of a virtually free group G, compute a finite graph of groups G with finite vertex groups and fundamental group G. Our algorithm is non-deterministic and runs in doubly exponential time. It follows that the isomorphism problem of context-free groups can be solved in doubly exponential space. Moreover, if, instead of a grammar, a finite extension of a free group is given as input, the construction of the graph of groups is in NP and, consequently, the isomorphism problem in PSPACE.