Consider an analytic germ f:(Cm, 0)→ (C, 0) (m[ges ]3) whose critical locus is a 2-dimensional complete intersection with an isolated singularity (icis). We prove that the homotopy type of the Milnor fiber of f is a bouquet of spheres, provided that the extended codimension of the germ f is finite. This result generalizes the cases when the dimension of the critical locus is zero [8], respectively one [12]. Notice that if the critical locus is not an icis, then the Milnor fiber, in general, is not homotopically equivalent to a wedge of spheres. For example, the Milnor fiber of the germ f:(C4, 0)→(C, 0), defined by f(x1, x2, x3, x4) = x1x2x3x4 has the homotopy type of S1×S1×S1. On the other hand, the finiteness of the extended codimension seems to be the right generalization of the isolated singularity condition; see for example [9–12, 17, 18].

In the last few years different types of ‘bouquet theorems’ have appeared. Some of them deal with germs f:(X, x)→(C, 0) where f defines an isolated singularity. In some cases, similarly to the Milnor case [8], F has the homotopy type of a bouquet of (dim X−1)-spheres, for example when X is an icis [2], or X is a complete intersection [5]. Moreover, in [13] Siersma proved that F has a bouquet decomposition F∼F0∨Sn∨…∨Sn (where F0 is the complex link of (X, x)), provided that both (X, x) and f have an isolated singularity. Actually, Siersma conjectured and Tibăr proved [16] a more general bouquet theorem for the case when (X, x) is a stratified space and f defines an isolated singularity (in the sense of the stratified spaces). In this case F∼∨iFi, where the Fi are repeated suspensions of complex links of strata of X. (If (X, x) has the ‘Milnor property’, then the result has been proved by Lê; for details see [6].)

In our situation, the space-germ (X, x) is smooth, but f has big singular locus. Surprisingly, for dim Sing f−1(0)[les ]2, the Milnor fiber is again a bouquet (actually, a bouquet of spheres, maybe of different dimensions). This result is in the spirit of Siersma's paper [12], where dim Sing f−1(0) = 1. In that case, there is only a rather small topological obstruction for the Milnor fiber to be homotopically equivalent to a bouquet of spheres (as explained in Corollary 2.4). In the present paper, we attack the dim Sing f−1(0) = 2 case. In our investigation some results of Zaharia are crucial [17, 18].

Let $G=G(q)$ be a finite almost simple exceptional group of Lie type over the field of $q$ elements, where $q=p^a$ and $p$ is prime. The main result of the paper determines all maximal subgroups $M$ of $G(q)$ such that $M$ is an almost simple group which is also of Lie type in characteristic $p$, under the condition that ${\rm rank}(M) > {1\over 2}{\rm rank}(G)$. The conclusion is that either $M$ is a subgroup of maximal rank, or it is of the same type as $G$ over a subfield of $\F_q$, or $(G,M)$ is one of $(E_6^\e(q),F_4(q))$, $(E_6^\e(q),C_4(q))$, $(E_7(q),\,^3\!D_4(q))$. This completes work of the first author with Saxl and Testerman, in which the same conclusion was obtained under some extra assumptions.

Asymptotic formulae for the number of rational points of bounded height on flag varieties have earlier been established. In the paper these asymptotic formulae are recovered by a new method for varieties in biprojective space defined over ℚ that are isomorphic to the flag variety of lines in hyperplanes.

The result is obtained by an application of Heath-Brown's new form of the circle method. It serves as a pointer to the investigation of rational points of bounded height on varieties in multiprojective space.

Let f and g be two analytic function germs without common branches. We define the Jacobian quotients of (g, f), which are ‘first order invariants’ of the discriminant curve of (g, f), and we prove that they only depend on the topological type of (g, f). We compute them with the help of the topology of (g, f). If g is a linear form transverse to f, the Jacobian quotients are exactly the polar quotients of f and we affirm the results of D. T. Lê, F. Michel and C. Weber.

The main result is that a finitely generated group that is isomorphic to all of its finite index subgroups has free Abelian first homology, and that its commutator subgroup is a perfect group. A number of corollaries on the structure of such groups are obtained, including a method of constructing all such groups for which the commutator subgroup has a trivial centralizer. As an application, conditions are presented for the covering spaces of compact manifolds that determine when the fundamental groups of the base spaces are free Abelian.

A profinite group $G$ of finite cohomological dimension with (topologically) finitely generated closed normal subgroup $N$ is studied. If $G$ is pro-$p$ and $N$ is either free as a pro-$p$ group or a Poincaré group of dimension 2 or analytic pro-$p$, it is shown that $G/N$ has virtually finite cohomological dimension ${\rm cd}(G)\,{-}\,{\rm cd}(N)$. Some other cases when $G/N$ has virtually finite cohomological dimension are also considered.

If $G$ is profinite, the case of $N$ projective or the profinite completion of the fundamental group of a compact surface is considered.

We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality.

In [2] we discussed almost complex curves in the nearly Kähler S6. These are surfaces with constant Kähler angle 0 or π and, as a consequence of this, are also minimal and have circular ellipse of curvature. We also considered minimal immersions with constant Kähler angle not equal to 0 or π, but with ellipse of curvature a circle. We showed that these are linearly full in a totally geodesic S5 in S6 and that (in the simply connected case) each belongs to the S1-family of horizontal lifts of a totally real (non-totally geodesic) minimal surface in [Copf ]P2. Indeed, every element of such an S1-family has constant Kähler angle and in each family all constant Kähler angles occur. In particular, every minimal immersion with constant Kähler angle and ellipse of curvature a circle is obtained by rotating an almost complex curve which is linearly full in a totally geodesic S5.

Let $G$ be a group and $K$ a field. For any finite-dimensional $KG$-module $V$ and any positive integer $n$, let $L^n(V)$ denote the $n$th homogeneous component of the free Lie $K$-algebra generated by (a basis of) $V$. Then $L^n(V)$ can be considered as a $KG$-module, called the $n$th Lie power of $V$. The paper is concerned with identifying this module up to isomorphism. A simple formula is obtained which expresses $L^n(V)$ in terms of certain linear functions on the Green ring. When $n$ is not divisible by the characteristic of $K$ these linear functions are Adams operations. Some results are also obtained which clarify the relationship between Adams operations defined by means of exterior powers and symmetric powers and operations introduced by Benson. Some of these results are put into a more general setting in an appendix by Stephen Donkin.

A procedure is given to reduce the interpolation spaces on an ordered pair generated by the function parameter $t^{\theta}(1+|\hspace{-2pt}\log t|)^{-b}$ to the classical real interpolation spaces. Applications are given for Lorentz–Zygmund function spaces, Besov spaces of generalized smoothness and Lorentz–Zygmund operator spaces.

This paper brings together two apparently unrelated results about locally compact groups G by giving them a common proof. The first concerns the number of topologically left invariant means on L∞(G), while the second states that the topological centre of the largest semigroup compactification of G is simply G itself. On the way, we introduce as vital tools some new compactifications of the half line ([0, ∞), +), we produce a right invariant pseudometric on a compactly generated G for which the bounded sets are precisely the relatively compact sets, and we receive striking confirmation that some algebraic properties of semigroups can be transferred by maps which are quite far from being homomorphisms.

The paper shows that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (that is, is tangent to a smooth variation through harmonic maps). This provides one of the few known answers to the problem of integrability, which was raised in different contexts of geometry and analysis. It implies that the Jacobi fields form the tangent bundle to each component of the manifold of harmonic maps from $S^2$ to ${\bb C} P^2$ thus giving the nullity of any such harmonic map; it also has a bearing on the behaviour of weakly harmonic $E$ -minimizing maps from a 3-manifold to ${\bb C} P^2$ near a singularity and the structure of the singular set of such maps from any manifold to ${\bb C} P^2$ .

Let X be a smooth projective surface of non-negative Kodaira dimension. Bogomolov [1, Theorem 5] proved that c21 [les ] 4c2. This was improved to c21 [les ] 3c2 by Miyaoka [12, Theorem 4] and Yau [19, Theorem 4]. Equality c21 [les ] 3c2 is attained, for example, if the universal cover of X is a ball (if κ(X) = 2 then this is the only possibility). Further generalizations of inequalities for Chern classes for some singular surfaces with (fractional) boundary were obtained by Sakai [16, Theorem 7.6], Miyaoka [13, Theorem 1.1], Kobayashi [6, Theorem 2; 7, Theorem 12], Wahl [18, Main Theorem] and Megyesi [10, Theorem 10.14; 11, Theorem 0.1].

In [8] we introduced Chern classes of reflexive sheaves, using Wahl's local Chern classes of vector bundles on resolutions of surface singularities. Here we apply them to obtain the following generalization of the Bogomolov–Miyaoka–Yau inequality.

Let $G$ be a finite group and suppose that $P$ is a soluble $\{2, 3\}^\prime$ -subgroup of $G$ . The reader will lose only a little by assuming that $P$ is a subgroup of prime order $p > 3$ . Define \[ \Sigma_G(P) = \{A \le G \mid A \hbox{ is soluble and } A = \langle P, P^a\rangle\hbox{ for some }a \in A\}. \] This set is partially ordered by inclusion and we let \[ \Sigma^*_G(P) \] denote the set of maximal members of $\Sigma_G(P)$ .

The paper proves an almost-orthogonality principle for maximal operators over arbitrary sets of directions in ${\bb R}^2$. Namely, the Lp-bounds for an operator of this type are obtained from the corresponding Lp-bounds of the maximal functions associated to a certain partition of the set of directions, and from the particular structure of this partition. Applications to several types of maximal operators are provided.

It is shown that for any meromorphic function $f$ the Julia set $J(f)$ has constant local upper and lower box dimensions, $\overline{d}(J(f))$ and $\underline{d}(J(f))$ respectively, near all points of $J(f)$ with at most two exceptions. Further, the packing dimension of the Julia set is equal to $\overline{d}(J(f))$. Using this result it is shown that, for any transcendental entire function $f$ in the class $B$ (that is, the class of functions such that the singularities of the inverse function are bounded), both the local upper box dimension and packing dimension of $J(f)$ are equal to 2. The approach is to show that the subset of the Julia set containing those points that escape to infinity as quickly as possible has local upper box dimension equal to 2.

Let $M$ be a module over the ring $R$. Extensive use is made of Krull codimension to study further the Artinian-finitary automorphism group $\[ F_1\hbox{\rm Aut}_RM \,{=}\, \{g\,{\in}\hbox{ Aut}_RM \,{:}\, M(g - 1) \hbox{ is } R\hbox{-Artinian}\} \]$ of $M$ over $R$. Substantial progress is made where either $M$ is residually Noetherian or $R$ is commutative. There are some group-theoretic consequences of the two main structure theorems.

We study here some foundational aspects of the classification problem for torsion-free abelian groups of finite rank. These are, up to isomorphism, the subgroups of the additive groups (ℚn, +), for some n = 1, 2, 3,…. The torsion-free abelian groups of rank [les ] n are the subgroups of (ℚn, +).

A computer construction of the Monster is used to prove that PSL(2, 59) is a subgroup of the Monster.

The paper considers a mixing Smale space, the relations of stable and unstable equivalence on such a space and the C*-algebras which are constructed from them. In general, these associations are not functorial. However, it is shown that, if one restricts to the class of s-resolving, finite-to-one factor maps, then the construction of the stable C*-algebra is contravariant, while that of the unstable C*-algebra is covariant. The paper also discusses the constructions of these C*-algebras for Smale spaces which are not mixing.