A pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

We show that there is a homotopy cofiber sequence of spectra relating Carlsson's deformation K-theory of a group G to its “deformation representation ring,” analogous to the Bott periodicity sequence relating connective K-theory to ordinary homology. We then apply this to study simultaneous similarity of unitary matrices.

We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus ≥ 2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.

In the paper, we give a lower bound on the genus of an amalgamation of two Heegaard splittings V1, ∪S1, W1 for M1 and V2, ∪S2, W2 for M2 along boundary components F1 and F2 under some conditions on the distances of the factor Heegaard splittings. A direct consequence is that under some circumstances, g(M) = g(M1) + g(M2) − g(F) holds, and another is that under some circumstances we have a better lower bound.

We will give an explicit formula of Ozsváth–Szabó's correction terms of lens spaces. Applying the formula to a restriction studied by P. Ozsváth and Z. Szabó in [12] and [13], we obtain several constraints of lens spaces which are constructed by a positive Dehn surgery in 3-sphere. Some of the constraints are results which are analogous to results which were known in [6] and [20] before. The constraints completely determine knots yielding L(p, 1) by positive Dehn surgery.

We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. We deduce that the ring is a perfect invariant, and prove a Chern class formula for weighted projective bundles.

We investigate the homology of finite index subgroups Gi of a given finitely presented group G. We fix a prime p, denote the field of order p by p and define dp(Gi) to be the dimension of H1(Gi; p). We will be interested in the situation where dp(Gi) grows fast as a function of the index [G : Gi]. Specifically, we say that a collection of finite index subgroups {Gi} has linear growth of mod p homology if infidp(Gi)/[G : Gi] is positive. This is a natural and interesting condition that arises in several different contexts. For example, the main theorem of [9] states that when G is a lattice in PSL(2, ℂ) with non-trivial torsion (equivalently, G is the fundamental group of a finite-volume hyperbolic 3-orbifold with non-empty singular locus), then G has such a sequence of subgroups. Another major class of groups G having such a collection of subgroups are those that are large. By definition, this means that G has a finite index subgroup that admits a surjective homomorphism onto a free non-abelian group. Large groups have many nice properties, for example super-exponential subgroup growth and infinite virtual first Betti number. One might wonder whether largeness is equivalent to the existence of some nested sequence of finite index subgroups {Gi} with linear growth of mod p homology for some prime p. If so, this would establish that lattices in PSL(2, ℂ) with non-trivial torsion are large, which would be a major breakthrough in low-dimensional topology.

We introduce a geometric invariant of knots in S3, called the first-order genus, that is derived from certain 2-complexes called gropes, and we show it is computable for many examples. While computing this invariant, we draw some interesting conclusions about the structure of a general Seifert surface for some knots.

Let β be a braid on n strands, with exponent sum w. Let Δ be the Garside half-twist braid. We prove that the coefficient of vw−n+1 in the Homfly polynomial of the closure of β agrees with (−1)n−1 times the coefficient of vw+n2−1 in the Homfly polynomial of the closure of βΔ2. This coincidence implies that the lower Morton–Franks–Williams estimate for the v–degree of the Homfly polynomial of is sharp if and only if the upper MFW estimate is sharp for the v–degree of the Homfly polynomial of .

We prove that for any group π with cohomological dimension at least n the nth power of the Berstein class of π is nontrivial. This allows us to prove the following Berstein–Svarc theorem for all n:

Theorem. For a connected complex X with dim X = cat X = n, we have ≠ 0 whereis the Berstein class of X.

Previously it was known for n ≥ 3.

We also prove that, for every map f: M → N of degree ±1 of closed orientable manifolds, the fundamental group of N is free provided that the fundamental group of M is.

We give a simple differential geometric description of the canonical Cartan (or tractor) bundle and connection in CR geometry, thus offering an alternative definition to the usual abstract Lie algebraic approach.

The purpose of this paper is to provide a simple relation between the Witten–Reshetikhin-Turaev SO(3) invariant and the Hennings invariant associated to quantum .

We prove that transversal non-simplicity is preserved under taking connect sum, generalizing Veŕtesi's result [11].

We discuss the minimal-to-maximal correspondence between surfaces and show that, under this correspondence, a congruence class of minimal surfaces in 3 determines an 2-family of congruence classes of maximal surfaces in 3. It is proved that further identifications among these classes may exist, depending upon the subgroup of automorphisms preserving the Hopf differential on the underlying Riemann surface. The space of maximal congruence classes inherits a quotient topology from 2. In the case of the Scherk minimal surface, the subgroup has order two and the quotient space is topologically a disc with boundary. Other classical examples are discussed: for the Enneper minimal surface, one obtains a non-Hausdorff space; for the minimal catenoid, a closed interval.

For closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold (M, 〈⋅, ⋅〉), where G-structures are considered as sections of the quotient bundle (M)/G. We deduce the corresponding first and second variation formulae and the characterising conditions for critical points by means of tools closely related to the study of G-structures. In this direction, we show the rôle in the energy functional played by the intrinsic torsion of the G-structure. Moreover, we analyse the particular case G=U(n) for 2n-dimensional manifolds. This leads to the study of harmonic almost Hermitian manifolds and harmonic maps from M into (M)/U(n).

The realization theorem asserts that for a finitely presented group G, the D(2) property and the realization property are equivalent as long as G satisfies a certain finiteness condition. We show that the two properties are in fact equivalent for all finitely presented groups.

A Hermitian form q on the dual space, *, of the Lie algebra, , of a simply connected complex Lie group, G, determines a sub-Laplacian, Δ, on G. Assuming Hörmander's condition for hypoellipticity, there is a smooth heat kernel measure, ρt, on G associated to etΔ/4. In a companion paper [6], we proved the existence of a unitary “Taylor” map from the space of holomorphic functions in L2(G, ρt) onto Jt0 (a subspace of) the dual of the universal enveloping algebra of . Here we give a very different proof of the surjectivity of the Taylor map under the assumption that G is nilpotent. This proof provides further insight into the structure of the Taylor map. In particular we show that the finite rank tensors are dense in Jt0 when the Lie algebra is graded and the Laplacian is adapted to the gradation. We also show how the Fourier–Wigner transform produces a natural family of holomorphic functions in L2(G, ρt), for appropriate t, when G is the complex Heisenberg group.

Let X be a smooth projective curve of genus g ≥ 2 over ℂ. A pair (E, φ) over X consists of an algebraic vector bundle E over X and a section φ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Here we prove that the third cohomology groups of the moduli spaces of τ-stable pairs with fixed determinant and rank n ≥ 2 are polarised pure Hodge structures, and they are isomorphic to H1(X) with its natural polarisation (except in very few exceptional cases). This implies a Torelli theorem for such moduli spaces. We recover that the third cohomology group of the moduli space of stable bundles of rank n ≥ 2 and fixed determinant is a polarised pure Hodge structure, which is isomorphic to H1(X). We also prove Torelli theorems for the corresponding moduli spaces of pairs and bundles with non-fixed determinant.

We provide a parametric construction in terms of minimal surfaces of the Euclidean submanifolds of codimension two and arbitrary dimension that attain equality in an inequality due to De Smet, Dillen, Verstraelen and Vrancken. The latter involves the scalar curvature, the norm of the normal curvature tensor and the length of the mean curvature vector.

For differential equations P(y(k), y)=0, where P is a polynomial, we prove that all meromorphic solutions having at least one pole are elliptic functions, possibly degenerate.