Universal quadratic forms and Whitney tower intersection invariants

The first part of this paper exposits a simple geometric description of the Kirby-Siebenmann invariant of a 4--manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higher-order intersection invariants of Whitney towers studied by the authors, particularly for the 4--ball. In the second part of this paper, a general theory of quadratic forms is developed and then specialized from the non-commutative to the commutative to finally, the symmetric settings. The intersection invariant for twisted Whitney towers is shown to be the universal symmetric refinement of the framed intersection invariant. As a corollary we obtain a short exact sequence that has been essential in the understanding of Whitney towers in the 4--ball.Comment: This paper subsumes the second half (Section 7) of the previously posted paper "Universal Quadratic Forms and Untwisting Whitney Towers" (http://arxiv.org/abs/1101.3480

Fine Structure of Class Groups \cl^{(p)}\Q(\z_n) and the Kervaire--Murthy Conjectures II

There is an Mayer-Vietoris exact sequence involving the Picard group of the integer group ring $\Z C_{p^n}$ where $C_{p^n}$ is the cyclic group of order $p^n$ and $\zeta_{n-1}$ is a primitive $p^n$-th root of unity. The "unknown" part of the sequence is a group. $V_n$. $V_n$ splits as $V_n\cong V_n^+\oplus V_n^-$ and $V_n^-$ is explicitly known. $V_n^+$ is a quotient of an in some sense simpler group $\mathcal{V}_n$. In 1977 Kervaire and Murthy conjectured that for semi-regular primes $p$, V_n^+ \cong \mathcal{V}_n^+ \cong \cl^{(p)}(\Q (\zeta_{n-1}))\cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}, where $r(p)$ is the index of regularity of $p$. Under an extra condition on the prime $p$, Ullom calculated $V_n^+$ in 1978 in terms of the Iwasawa invariant $\lambda$ as $V_n^+ \cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}\oplus (\mathbb{Z}/p^{n-1} \mathbb{Z})^{\lambda-r(p)}$. In the previous paper we proved that for all semi-regular primes, \mathcal{V}_n^+\cong \cl^{(p)}(\Q (\zeta_{n-1})) and that these groups are isomorphic to (\mathbb{Z}/p^n \mathbb{Z})^{r_0}\oplus (\mathbb{Z}/p^{n-1} \mathbb{Z})^{r_1-r_0} \oplus \hdots \oplus (\mathbb{Z}/p \mathbb{Z})^{r_{n-1}-r_{n-2}} for a certain sequence $\{r_k\}$ (where $r_0=r(p)$). Under Ulloms extra condition it was proved that V_n^+ \cong \mathcal{V}_n^+ \cong \cl^{(p)}(\Q(\z_{n-1})) \cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}\oplus (\mathbb{Z}/p^{n-1}\mathbb{Z})^{\lambda-r(p)}. In the present paper we prove that Ullom's extra condition is valid for all semi-regular primes and it is hence shown that the above result holds for all semi-regular primes.Comment: 7 pages, Continuation of NT/020728

Concordance of Zp×ZpZ_p\times\Z_p actions on S4S^4

We consider locally linear Z_p x Z_p actions on the four-sphere. We present simple constructions of interesting examples, and then prove that a given action is concordant to its linear model if and only if a single surgery obstruction taking to form of an Arf invariant vanishes. We discuss the behavior of this invariant under various connected-sum operations, and conclude with a brief discussion of the existence of actions which are not concordant to their linear models

Stabilisation, bordism and embedded spheres in 4--manifolds

It is one of the most important facts in 4-dimensional topology that not every spherical homology class of a 4-manifold can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions to constructing such a sphere vanish if one modifies the ambient 4-manifold by adding products of 2-spheres, a process which is usually called stabilisation. In this paper, we extend this result to non-simply connected 4-manifolds and show how it is related to the Spin^c-bordism groups of Eilenberg-MacLane spaces.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-10.abs.htm

A survey on modular Hadamard matrices

AbstractWe provide constructions of 32-modular Hadamard matrices for every size n divisible by 4. They are based on the description of several families of modular Golay pairs and quadruples. Higher moduli are also considered, such as 48,64,128 and 192. Finally, we exhibit infinite families of circulant modular Hadamard matrices of various types for suitable moduli and sizes

The singular linear preservers of non-singular matrices

Given an arbitrary field K, we reduce the determination of the singular endomorphisms $f$ of M_n(K) that stabilize GL_n(K) to the classification of n-dimensional division algebras over K. Our method, which is based upon Dieudonn\'e's theorem on singular subspaces of M_n(K), also yields a proof for the classical non-singular case.Comment: 12 pages, some minor corrections from the first versio

Higher order intersection numbers of 2-spheres in 4-manifolds

This is the beginning of an obstruction theory for deciding whether a map f:S^2 --> X^4 is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall's self-intersection number mu(f) which tells the whole story in higher dimensions. Our second order obstruction tau(f) is defined if mu(f) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of pi_1(X) modulo S_3-symmetry (rather then just one copy modulo S_3-symmetry). It generalizes to the non-simply connected setting the Kervaire-Milnor invariant which corresponds to the Arf-invariant of knots in 3-space. We also give necessary and sufficient conditions for moving three maps f_1,f_2,f_3:S^2 --> X^4 to a position in which they have disjoint images. Again the obstruction lambda(f_1,f_2,f_3) generalizes Wall's intersection number lambda(f_1,f_2) which answers the same question for two spheres but is not sufficient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in dimension 3, our new invariant corresponds to the Milnor invariant mu(1,2,3), generalizing the Matsumoto triple to the non simply-connected setting.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-1.abs.htm

Optimally small sumsets in finite abelian groups

AbstractLet G be a finite abelian group of order g. We determine, for all 1⩽r,s⩽g, the minimal size μG(r,s)=min|A+B| of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μG(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μG are recalled in the Introduction

On the product of vector spaces in a commutative field extension

Let $K \subset L$ be a commutative field extension. Given $K$-subspaces $A,B$ of $L$, we consider the subspace spanned by the product set $AB=\{ab \mid a \in A, b \in B\}$. If $\dim_K A = r$ and $\dim_K B = s$, how small can the dimension of be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on $\dim_K$ turns out, in this case, to be provided by the numerical function $$\kappa_{K,L}(r,s) = \min_{h} (\lceil r/h\rceil + \lceil s/h\rceil -1)h,$$ where $h$ runs over the set of $K$-dimensions of all finite-dimensional intermediate fields $K \subset H \subset L$. This bound is closely related to one appearing in additive number theory.Comment: Submitted in November 200