In this thesis, we study exotic smooth structures on 4-manifolds with finite fundamental groups. For an arbitrary finite group G, we construct an infinite family of smooth 4-manifolds with fundamental group G, which are all homeomorphic but mutually non-diffeomorphic, using the small symplectic manifold with arbitrary fundamental group constructed by S. Baldridge and P. Kirk, together with the methods of A. Akhmedov, R.&amp / #221 / . Baykur and D. Park for constructing infinite families of exotic simply connected 4-manifolds. In the final chapter, pairs of small exotic 4-manifolds with a cyclic fundamental group of any odd order are constructed.

In this thesis, we study the classification problem of Stein fillable tight contact structures on any Seifert fibered 3&minus / manifold M over S 2 with 4 singular fibers. In the case e0(M) &middot / &minus / 4 we have a complete classification. In the case e0(M) &cedil / 0 we have obtained upper and lower bounds for the number of Stein fillable contact structures on M.

In this thesis, we present real Lefschetz fibrations. We first study real Lefschetz fibrations around a real singular fiber. We obtain a classification of real Lefschetz fibrations around a real singular fiber by a study of monodromy properties of real Lefschetz fibrations. Using this classification, we obtain some invariants, called real Lefschetz chains, of real Lefschetz fibrations which admit only real critical values. We show that in case the fiber genus is greater then 1, the real Lefschetz chains are complete invariants of directed real Lefschetz fibrations with only real critical values. If the genus is 1, we obtain complete invariants by decorating real Lefschetz chains. For elliptic Lefschetz fibrations we define a combinatorial object which we call necklace diagrams. Using necklace diagrams we obtain a classification of directed elliptic real Lefschetz fibrations which admit a real section and which have only real critical values. We obtain 25 real Lefschetz fibrations which admit a real section and which have 12 critical values all of which are real. We show that among 25 real Lefschetz fibrations, 8 of them are not algebraic. Moreover, using necklace diagrams we show the existence of real elliptic Lefschetz fibrations which can not be written as the fiber sum of two real elliptic Lefschetz fibrations. We define refined necklace diagrams for real elliptic Lefschetz fibrations without a real section and show that refined necklace diagrams classify real elliptic Lefschetz fibrations which have only real critical values.

Let M be smooth symplectic closed manifold and L a closed Lagrangian submanifold of M. It was shown by Ozan that Ham(M,L): the relative Hamiltonian diffeomorphisms on M fixing the Lagrangian submanifold L setwise is a subgroup which is equal to the kernel of the restriction of the flux homomorphism to the universal cover of the identity component of the relative symplectomorphisms. In this thesis we show that Ham(M,L) is a non-simple perfect group, by adopting a technique due to Thurston, Herman, and Banyaga. This technique requires the diffeomorphism group be transitive where this property fails to exist in our case.

In this thesis, we define a new invariant of a Legendrian knot in a contact manifold using an open book decomposition supporting the contact structure. We define the support genus of a Legendrian knot L in a contact 3-manifold as the minimal genus of a page of an open book of M supporting the contact structure such that L sits on a page and the framings given by the contact structure and the page agree. For any topological link in 3-sphere we construct a planar open book decomposition whose monodromy is a product of positive Dehn twists such that the planar open book contains the link on its page. Using this, we show any topological link, in particular any knot in any 3-manifold M sits on a page of a planar open book decomposition of M and we show any null-homologous loose Legendrian knot in an overtwisted contact structure has support genus zero.

In this thesis, we write explicitly the open book decompositions of links of quotient surface singularities that support the corresponding unique Milnor fillable contact structures. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the corresponding unique Milnor fillable contact structures. That minimal page-genus is called Milnor genus. In this thesis we also investigate whether the Milnor genus is equal to the support genus for links of quotient surface singularities. We show that for many types of the quotient surface singularities the Milnor genus is equal to the support genus of the corresponding contact structure. For the remaining we are able to find an upper bound for the support genus which would be a step forward in understanding these contact structures.

Let G be a finite group and V be an orthogonal four-dimensional real representation space of G where the action of G is non-free. We give necessary and sufficient conditions for the existence of a G-equivariant vector field on the representation sphere of V in the cases G is the dihedral group, the generalized quaternion group and the semidihedral group in terms of decomposition of V into irreducible representations. In the case G is abelian, where the solution is already known, we give a more elementary solution.

Let N be a connected nonorientable surface of genus g with n punctures. Suppose that g is odd and g + n &gt / 6. We prove that the automorphism group of the complex of curves of N is isomorphic to the mapping class group M of N.