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In this thesis, a bundle F →(M,ω) → B is said to be Lagrangian if (M,ω) is a 2n- dimensional symplectic manifold and the fibres are compact and connected Lagrangian submanifolds of (M,ω), i.e. ω |F = 0 for all F. This condition implies that the fibres and the base space are n-dimensional. Such bundles arise naturally in the study of a special class of dynamical systems in Hamiltonian mechanics, namely those called completely integrable Hamiltonian systems. A celebrated theorem due to Liouville [39], Mineur [46] and Arnol`d [2] provides a semi-global (i.e. in the neighbourhood of a fibre) symplectic classification of Lagrangian bundles, given by the existence of local action-angle coordinates. A proof of this theorem, due to Markus and Meyer [41] and Duistermaat [20], shows that the fibres and base space of a Lagrangian bundle are naturally integral affine manifolds, i.e. they admit atlases whose changes of coordinates can be extended to affine transformations of Rn which preserve the standard cocompact lattice Zn Rn. This thesis studies the problem of constructing Lagrangian bundles from the point of view of affinely at geometry. The first step to study this question is to construct topological universal Lagrangian bundles using the affine structure on the fibres. These bundles classify Lagrangian bundles topologically in the sense that every such bundle arises as the pullback of one universal bundle. However, not all bundles which are isomorphic to the pullback of a topological universal Lagrangian bundle are Lagrangian, as there exist further smooth and symplectic invariants. Even for bundles which admit local action-angle coordinates (these are classified up to isomorphism by topological universal Lagrangian bundles), there is a cohomological obstruction to the existence of an appropriate symplectic form on the total space, which has been studied by Dazord and Delzant in [18]. Such bundles are called almost Lagrangian. The second half of this thesis constructs the obstruction of Dazord and Delzant using the spectral sequence of a topological universal Lagrangian bundle. Moreover, this obstruction is shown to be related to a cohomological invariant associated to the integral affine geometry of the base space, called the radiance obstruction. In particular, it is shown that the integral a ne geometry of the base space of an almost Lagrangian bundle determines whether the bundle is, in fact, Lagrangian. New examples of (almost) Lagrangian bundles are provided to illustrate the theory developed.

Let Knots be the abelian monoid of isotopy classes of knots S1 ⊂ S3 under connected sum, and let C be the topological knot concordance group of knots modulo slice knots. Cochran-Orr-Teichner [COT03] defined a filtration of C: C ⊃ F(0) ⊃ F(0.5) ⊃ F(1) ⊃ F(1.5) ⊃ F(2) ⊃ . . .The quotient C/F(0.5) is isomorphic to Levine’s algebraic concordance group AC1 [Lev69]; F(0.5) is the algebraically slice knots. The quotient C/F(1.5) contains all metabelian concordance obstructions. The Cochran-Orr-Teichner (1.5)-level two stage obstructions map the concordance class of a knot to a pointed set (COT (C/1.5),U). We define an abelian monoid of chain complexes P, with a monoid homomorphism Knots → P. We then define an algebraic concordance equivalence relation on P and therefore a group AC2 := P/ ~, our second order algebraic knot concordance group. The results of this thesis can be summarised in the following diagram: . That is, we define a group homomorphism C → AC2 which factors through C/F(1.5). We can extract the two stage Cochran-Orr-Teichner obstruction theory from AC2: the dotted arrows are morphisms of pointed sets. Our second order algebraic knot concordance group AC2 is a single stage obstruction group.