The research in this thesis was initially motivated by an outstanding problem posed by Argyros and Haydon. They used a generalised version of the Bourgain-Delbaen construction to construct a Banach space $XK$ for which the only bounded linear operators on $XK$ are compact perturbations of (scalar multiples of) the identity; we say that a space with this property has very few operators. The space $XK$ possesses a number of additional interesting properties, most notably, it has $ell_1$ dual. Since $ell_1$ possesses the Schur property, weakly compact and norm compact operators on $XK$ coincide. Combined with the other properties of the Argyros-Haydon space, it is tempting to conjecture that such a space must necessarily have very few operators. Curiously however, the proof that $XK$ has very few operators made no use of the Schur property of $ell_1$. We therefore arrive at the following question (originally posed in cite{AH}): must a HI, $mathcal{L}_{infty}$, $ell_1$ predual with few operators (every operator is a strictly singular perturbation of $lambda I$) necessarily have very few operators? We begin by giving a detailed exposition of the original Bourgain-Delbaen construction and the generalised construction due to Argyros and Haydon. We show how these two constructions are related, and as a corollary, are able to prove that there exists some $delta > 0$ and an uncountable set of isometries on the original Bourgain-Delbaen spaces which are pairwise distance $delta$ apart. We subsequently extend these ideas to obtain our main results. We construct new Banach spaces of Bourgain-Delbaen type, all of which have $ell_1$ dual. The first class of spaces are HI and possess few, but not very few operators. We thus have a negative solution to the Argyros-Haydon question. We remark that all these spaces have finite dimensional Calkin algebra, and we investigate the corollaries of this result. We also construct a space with $ell_1$ Calkin algebra and show that whilst this space is still of Bourgain-Delbaen type with $ell_1$ dual, it behaves somewhat differently to the first class of spaces. Finally, we briefly consider shift-invariant $ell_1$ preduals, and hint at how one might use the Bourgain-Delbaen construction to produce new, exotic examples.

Kommungränser har en minskad betydelse för kommuner och för människors rörelsemönster. Ett ökat behov av samarbete mellan kommuner skapas då planeringsfrågor i större utsträckning överskrider de administrativa gränserna och utmaningar för en kommun stannar i regel inte vid dess formella gräns. Det förekommer behov av att agera på andra skalnivåer än den som definieras av kommunens gränser för att kunna hantera vissa frågor. Uppsatsen tar utgångspunkt i en fallstudie, det mellankommunala samarbetet Skåne Nordost. Forskningsöversikten för uppsatsen omfattar utvecklingen av government och nätverksstyrning i den svenska planeringen över tid och knyter an till relevant forskning där förutsättningar, motiv och begränsningar för kommunal samverkan redogörs. Det empiriska materialet i uppsatsen baseras främst på den kvalitativa innehållsanalysen av transkriberade intervjuer med två politiker och fyra tjänstemän från två utvalda kommuner i Skåne Nordost, men kompletteras även med valda planeringsdokument. Presentationen av resultatet och uppsatsens syfte omfattar motiv och drivkrafter bakom mellankommunala samarbeten, kommunalekonomiska behov som hanteras, vad som uppnås samt hur mellankommunala samarbeten förhåller sig till rådande bestämda kompetenser. Utifrån frågeställningarna, forskningsöversikten och uppsatsens analytiska ramverk med kategorierna aktörer, relationer, institutionella ramverk och beslutsprocessen samt de teoretiska utgångspunkterna, governance och soft planning, har sedan resultatet analyserats. Studien pekar på förändringar i omvärlden som en drivkraft bakom mellankommunala samarbeten. Genom länssammanslagningen som ägde rum i Skåne var kommunerna i Skåne Nordost tvungna att samarbeta för att bli större och kunna hävda sig mot Region Skåne och andra delar av regionen. Motiv som framkommer bakom mellankommunala samarbeten är att erhålla en gemensam röst för att bli starkare i flera frågor, resurser kan utnyttjas mer effektivt, få tillgång till rätt kompetens och specialistkunskaper och gemensam marknadsföring. Vidare lyfts behov som hanteras i mellankommunala samarbeten som kan uppfylla mervärde och storskalig nytta. Finansiellt stora behov med ett stort antal involverade aktörer så som infrastruktur, bredband, vattenförsörjning, kompetensutveckling, arbetsmarknad, näringsliv, integration samt turism, men även VA, renhållning och räddningstjänst är inte geografiskt avgränsade i respektive kommun utan är gränsöverskridande. Varje kommun måste hantera dessa, men klarar vanligtvis inte det enskilt. Dessa uppfattas som okontroversiella kommunalekonomiska behov som generellt hanteras i dessa samarbeten. Vidare tydliggörs det att samarbetet har uppnått att skapa ett forum där kommunerna träffas, utbyter idéer och utför handlingar som annars inte hade realiserats, vilket kan vara generellt för mellankommunala samarbeten. Slutligen framgår det att Skåne Nordost skapat en egen formell organisation, men utgörs samtidigt av ett informellt nätverk präglat av konsensus och ömsesidigt företroende utan sanktioner. Mellankommunala samarbeten som har en organisationsstruktur likt Skåne Nordost är i stor utsträckning förenliga med governance och soft planning.

published_or_final_version / Urban Planning / Master / Master of Science in Urban Planning

published_or_final_version / Urban Planning / Master / Master of Science in Urban Planning

We say that a covering space of the surface S over X has the Birman--Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a covering space has this property. We give new explicit examples of irregular branched covering spaces that do not satisfy the necessary condition as well as explicit covering spaces that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.

If X is a simply connected CW complex, then it has a unique (up to isomorphism) minimal Sullivan model. There is an important rational homotopy invariant, called the rational Lusternik–Schnirelmann of X, denoted cat0(X), which has an algebraic formulation in terms of the minimal Sullivan model of X. We study another such numerical invariant called the rational retraction index of X, denoted r0(X), which is defined in terms of the minimal Sullivan model of X and satisfies 0 ≤ r0(X) ≤ cat0(X). It was introduced by Cuvilliez et al. as a tool to estimate the rational Lusternik–Schnirelmann category of the total space of a fibration. In this thesis we compute the rational retraction index on a range of rationally elliptic spaces, including for example spheres, complex projective space, the biquotient Sp(1) \ Sp(3) / Sp(1) × Sp(1), the homogeneous space Sp(3)/U(3) and products of these. In particular, we focus on formal spaces and formulate a conjecture to answer a question posed in the original article of Cuvilliez et al., “If X is formal, what invariant of the algebra H∗(X;Q) is r0(X)?”

Student Number : 0413210T - MSc dissertation - School of Mathematics - Faculty of Science / The aim of this work is to do a literature study on spaces of martingales on Riesz spaces and Banach lattices, using [16, 19, 20, 17, 18, 2, 30] as a point of departure. Convergence of martingales in the classical theory of stochastic processes has many applications in mathematics and related areas. Operator theoretic approaches to the classical theory of stochastic processes and martingale theory in particular, can be found in, for example, [4, 5, 6, 7, 13, 15, 26, 27]. The classical theory of stochastic processes for scalar-valued measurable functions on a probability space ( ,#6;, μ) utilizes the measure space ( ,#6;, μ), the norm structure of the associated Lp(μ)-spaces as well as the order structure of these spaces. Motivated by the existing operator theoretic approaches to classical stochastic processes, a theory of discrete-time stochastic processes has been developed in [16, 19, 20, 17, 18] on Dedekind complete Riesz spaces with weak order units. This approach is measure-free and utilizes only the order structure of the given Riesz space. Martingale convergence in the Riesz space setting is considered in [18]. It was shown there that the spaces of order bounded martingales and order convergent martingales, on a Dedekind complete Riesz space with a weak order unit, coincide. A measure-free approach to martingale theory on Banach lattices with quasi-interior points has been given in [2]. Here, the groundwork was done to generalize the notion of a filtration on a vector-valued Lp-space to the M-tensor product of a Banach space and a Banach lattice (see [1]). In [30], a measure-free approaches to martingale theory on Banach lattices is given. The main results in [30] show that the space of regular norm bounded martingales and the space of norm bounded martingales on a Banach lattice E are Banach lattices in a natural way provided that, for the former, E is an order continuous Banach lattice, and for the latter, E is a KB-space. The definition of a ”martingale” defined on a particular space depends on the type of space under consideration and on the ”filtration,” which is a sequence of operators defined on the space. Throughout this dissertation, we shall consider Riesz spaces, Riesz spaces with order units, Banach spaces, Banach lattices and Banach lattices with quasi-interior points. Our definition of a ”filtration” will, therefore, be determined by the type of space under consideration and will be adapted to suit the case at hand. In Chapter 2, we consider convergent martingale theory on Riesz spaces. This chapter is based on the theory of martingales and their properties on Dedekind complete Riesz spaces with weak order units, as can be found in [19, 20, 17, 18]. The notion of a ”filtration” in this setting is generalized to Riesz spaces. The space of martingales with respect to a given filtration on a Riesz space is introduced and an ordering defined on this space. The spaces of regular, order bounded, order convergent and generated martingales are introduced and properties of these spaces are considered. In particular, we show that the space of regular martingales defined on a Dedekind complete Riesz space is again a Riesz space. This result, in this context, we believe is new. The contents of Chapter 3 is convergent martingale theory on Banach lattices. We consider the spaces of norm bounded, norm convergent and regular norm bounded martingales on Banach lattices. In [30], filtrations (Tn) on the Banach lattice E which satisfy the condition 1[n=1 R(Tn) = E, where R(Tn) denotes the range of the filtration, are considered. We do not make this assumption in our definition of a filtration (Tn) on a Banach lattice. Our definition yields equality (in fact, a Riesz and isometric isomorphism) between the space of norm convergent martingales and 1Sn=1R(Tn). The aforementioned main results in [30] are also considered in this chapter. All the results pertaining to martingales on Banach spaces in subsections 3.1.1, 3.1.2 and 3.1.3 we believe are new. Chapter 4 is based on the theory of martingales on vector-valued Lp-spaces (cf. [4]), on its extension to the M-tensor product of a Banach space and a Banach lattice as introduced by Chaney in [1] (see also [29]) and on [2]. We consider filtrations on tensor products of Banach lattices and Banach spaces as can be found in [2]. We show that if (Sn) is a filtration on a Banach lattice F and (Tn) is a filtration on a Banach space X, then 1[n=1 R(Tn Sn) = 1[n=1 R(Tn) e M 1[n=1 R(Sn). This yields a distributive property for the space of convergent martingales on the M-tensor product of X and F. We consider the continuous dual of the space of martingales and apply our results to characterize dual Banach spaces with the Radon- Nikod´ym property. We use standard notation and terminology as can be found in standard works on Riesz spaces, Banach spaces and vector-valued Lp-spaces (see [4, 23, 29, 31]). However, for the convenience of the reader, notation and terminology used are included in the Appendix at the end of this work. We hope that this will enhance the pace of readability for those familiar with these standard notions.

In this thesis we study the homological behaviour of configuration spaces as the number of objects in the configuration goes to infinity. For unordered configurations of distinct points (possibly equipped with some internal parameters) in a connected, open manifold it is a well-known result, going back to G. Segal and D. McDuff in the 1970s, that these spaces enjoy the property of homological stability. In Chapter 2 we prove that this property also holds for so-called oriented configuration spaces, in which the points of a configuration are equipped with an ordering up to even permutations. There are two important differences from the unordered setting: the rate (or slope) of stabilisation is strictly slower, and the stabilisation maps are not in general split-injective on homology. This can be seen by some explicit calculations of Guest-Kozlowski-Yamaguchi in the case of surfaces. In Chapter 3 we refine their calculations to show that, for an odd prime p, the difference between the mod-p homology of the oriented and the unordered configuration spaces on a surface is zero in a stable range whose slope converges to 1 as p goes to infinity. In Chapter 4 we prove that unordered configuration spaces satisfy homological stability with respect to finite-degree twisted coefficient systems, generalising the corresponding result of S. Betley for the symmetric groups. We deduce this from a general “twisted stability from untwisted stability” principle, which also applies to the configuration spaces studied in the next chapter. In Chapter 5 we study configuration spaces of submanifolds of a background manifold M. Roughly, these are spaces of pairwise unlinked, mutually isotopic copies of a fixed closed, connected manifold P in M. We prove that if the dimension of P is at most (dim(M)−3)/2 then these configuration spaces satisfy homological stability w.r.t. the number of copies of P in the configuration. If P is a sphere this upper bound on its dimension can be increased to dim(M)−3.

We give a definition of the scanning map for configuration spaces that is equivariant under the action of the diffeomorphism group of the underlying manifold. We use this to extend the Bödigheimer-Madsen result for the stable splittings of the Borel constructions of certain mapping spaces from compact Lie group actions to all smooth actions. Moreover, we construct a stable splitting of configuration spaces which is equivariant under smooth group actions, completing a zig-zag of equivariant stable homotopy equivalences between mapping spaces and certain wedge sums of spaces. Finally we generalise these results to configuration spaces with twisted labels (labels in a fibre bundle subject to certain conditions) and extend the Bödigheimer-Madsen result to more mapping spaces.

Tensor products of Banach Spaces are studied. An introduction to tensor products is given. Some results concerning the reciprocal Dunford-Pettis Property due to Emmanuele are presented. Pelczyriski's property (V) and (V)-sets are studied. It will be shown that if X and Y are Banach spaces with property (V) and every integral operator from X into Y* is compact, then the (V)-subsets of (X⊗F)* are weak* sequentially compact. This in turn will be used to prove some stronger convergence results for (V)-subsets of C(Ω,X)*.