Ph.D. Program in Mathematics.Karakurt, Çağrı.Şavk, Oğuz.2025-04-142025-04-142023Ph.D. Program in Mathematics. EQE 2023 D37 PhD (Thes CSE 2023 G75https://digitalarchive.library.bogazici.edu.tr/handle/123456789/21627In this thesis, we discuss several results on the bounding and cobordism relations between homology 3-spheres and contractible 4-manifolds. The majority of our work is based on constructive methodology by providing new examples of homology 3-spheres bounding Mazur or Poénaru type contractible 4-manifolds, rational homology 4-balls, and homology planes. In particular, they simultaneously provide new zero order elements in the homology cobordism group 3Z and the rational homology cobordism group 3Q . We also properly address obstructive methodology, searching for infinite order elements in 3Z with complicated behaviour and focusing on combinatorial techniques for the computation of powerful invariants originating from classical, involutive, and connected Heegaard Floer homology theories. As an application, we present two new families of Seifert fibered spaces generating infinite rank summands in 3Z . They both bound almost simple linear graphs, as does the first family of Dai, Hom, Stoffregen and Truong.Homology theory.Cobordism theory.xv, 116 leaves