Constructions of exotic 4-manifolds with non-trivial fundamental group

In this thesis, we investigate exotic 4-manifolds. The existence of infinite exotic families of 4-manifolds highlights the discrepancy between the topological and the smooth category in the 4-dimensional realm. By an exotic family we mean a collection of smooth manifolds whose elements are pairwise homeomorphic but not diffeomorphic. After Michael Freedman's celebrated result on the topological classification of closed simply connected 4-manifolds in 1982, several examples of infinite families of exotic simply connected 4-manifolds have been constructed. In recent times, there has been increasing interest in building new exotic 4-manifolds with non-trivial fundamental group. In my thesis, I focus on the construction of such examples. The thesis is arranged into five main chapters and an appendix. In Chapter 1 we collect all the background notions that are needed to construct the new exotic 4-manifolds in the thesis. In particular, we list the cut-and-paste construction tools and we survey criteria to identify the homeomorphism types, as well as invariants to distinguish the diffeomorphism types. Chapter 2 is based on a joint work with Rafael Torres and Daniele Zuddas, in which we pin down the homeomorphism type of some closed oriented 4-manifolds with fundamental group $\mathbb{Z}$ and $\mathbb{Z}^2$. Such examples come in infinite families and are produced via torus-surgery-based constructions starting from some closed simply connected 4-manifolds with small Euler characteristics appearing in the work of Akhmedov--Park, Baldridge--Kirk and Torres. These diffeomorphism types are distinguished by their Seiberg--Witten invariants and any two homeomorphic 4-manifolds in these collections become diffeomorphic after connected summing with a single copy of a product of two 2-spheres. The homeomorphism types of the examples in our main result are pinned down by the study of the equivariant intersection form, invoking classification results by Freedmann--Quinn, Stong--Wang and Hambleton--Kreck--Teichner. Our main contribution is a mechanism to compute the equivariant intersection form of 4-manifolds that are obtained via torus surgeries along embedded 2-tori with trivial normal bundle. The content of Chapter 3 is taken from a joint work with Rafael Torres . We introduce a simple cut-and-paste mechanism that, given a pair $(M_1,M_2)$ of homeomorphic but not diffeomorphic 4-manifolds, produces for every integer p a pair $(M_1(p), M_2(p))$ of homeomorphic and potentially not diffeomorphic 4-manifolds. Such a mechanism alters the fundamental group of the initial pair and we show that it unveils new exotic irreducible smooth structures on closed 4-manifolds with finite cyclic fundamental group, which include $\mathbb{Q}$-homology real projective 4-spaces. Chapter 4 is based on a joint work with Rafael Torres, where we build an exotic structure Y on the total space $\mathbb{RP}^2 \widetilde{\times} S^2$ of the non-trivial 2-sphere bundle over the real projective plane. Such smooth structure is produced by Gluck twist on a smoothly embedded 2-sphere S inside $\ \mathbb{RP}^2\times S^2$. We also show that there is a 5-dimensional cobordism with a single 2-handle between the 4-manifold Y and a mapping torus that was used by Cappell--Shaneson to construct an exotic $\mathbb{RP}^4$. We further show that twisting an embedded real projective plane inside Y produces a manifold that is homeomorphic but not diffeomorphic to the circle sum of two copies of $\mathbb{RP}^4$. The results in Chapter 5 are taken from a single author paper. It contains a discussion of $\text{Pin}^{\pm}$-structures on non-orientable Lefschetz fibrations. More precisely, we study necessary and sufficient conditions for a 4-dimensional Lefschetz fibration over the 2-disk to admit a $\text{Pin}^{+}$ or a $\text{Pin}^-$-structure, extending the work of Stipsicz in the oriented setting. As a corollary, we get existence results of $\text{Pin}^{\pm}$-structures on closed nonorientable 4-manifolds and on Lefschetz fibrations over the 2-sphere. In particular, we show via three explicit examples how to read-off $\text{Pin}^{\pm}$-structures from the Kirby diagram of a 4-manifold. We also provide a proof of the well-known fact that any closed 3-manifold admits a $\text{Pin}^-$-structure and we find a criterion to check whether or not it admits a $\text{Pin}^+$-structure in terms of a given handlebody decomposition. In particular, the main motivation for studying $\text{Pin}^+$-structures on 4-manifolds in this chapter is the fact that in some cases they might help to detect exotic phenomena in the nonorientable realm. For example, this is the case for the smooth structures produced in Chapter 3 and Chapter 4. We conclude with an appendix, based on a short survey article containing a proof of Dold--Whitney criterion for the parallelizability of a smooth closed connected orientable 4-manifold. This follows from a stronger result due to Dold and Whitney on the classification of oriented sphere bundles over a 4-complex. Our main contribution is to outline in detail an argument due to R. Kirby, using the classification of $SO(4)$-bundles over the 4-sphere by means of their Euler and first Pontryagin classes as a main tool.