Homology 3-spheres, homology cobordism and contractible 4-manifolds

In 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 el- ements in the homology cobordism group $\Theta^3_\mathbb{Z}$ and the rational homology cobordism group $\Theta^3_\mathbb{Q}$. We also properly address obstructive methodology, searching for infinite order elements in $\Theta^3_\mathbb{Z}$ with complicated behaviour and focusing on combinatorial tech- niques 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 $\Theta^3_\mathbb{Z}$. They both bound almost simple linear graphs, as does the first family of Dai, Hom, Stoffregen and Truong.