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We study 4-dimensional homology cobordisms without 3-handles via Heegaard and instanton Floer homologies, character varieties, and Thurston geometries. We provide obstructions to such cobordisms arising from each of these theories, and illustrate some topological applications.

In this paper, we discuss Floer homology of Lagrangian submanifolds in an open symplectic manifold given as the complement of a smooth divisor. Firstly, a compactification of moduli spaces of holomorphic strips in a smooth divisor complement is introduced. Next, this compactification is used to define Lagrangian Floer homology of two Lagrangians in the divisor complement. The main new feature of this paper is that we do not make any assumption on positivity or negativity of the divisor.

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Kronheimer and Mrowka asked whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large. This question is answered affirmatively by studying a knot invariant derived from equivariant singular instanton theory, and which is closely related to the Chern-Simons functional. This also answers a conjecture of Livingston about slicing numbers. Also studied is the singular instanton Frøyshov invariant of a knot. If defined with integer coefficients, this gives a lower bound for the unoriented slice genus, and is computed for quasi-alternating and torus knots. In contrast, for certain other coefficient rings, the invariant is identified with a multiple of the knot signature. This result is used to address a conjecture by Poudel and Saveliev about traceless SU p2q representations of torus knots. Further, for a concordance between knots with non-zero signature, it is shown that there is a traceless representation of the concordance complement which restricts to non-trivial representations of the knot groups. Finally, some evidence towards an extension of the slice-ribbon conjecture to torus knots is provided.

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For each integer N ě 2, Mariño and Moore defined generalized Donaldson invariants by the methods of quantum field theory, and made predictions about the values of these invariants. Subsequently, Kronheimer gave a rigorous definition of generalized Donaldson invariants using the moduli spaces of anti-self-dual connections on hermitian vector bundles of rank N . In this paper, Mariño and Moore's predictions are confirmed for simply connected elliptic surfaces without multiple fibers and certain surfaces of general type in the case that N " 3. The primary motivation is to study 3-manifold instanton Floer homologies which are defined by higher rank bundles. In particular, the computation of the generalized Donaldson invariants are exploited to define a Floer homology theory for sutured 3-manifolds. π 1 pΣ N pKqq with Σ N pKq being the N -fold cyclic branched cover of Y , branched along K. This verifies the Covering Conjecture, which asserts that Σ N pKq, for a non-trivial knot K, is not a homotopy sphere [50, Problem 3.38]. A consequence of the Covering Conjecture is the Smith Conjecture, stating that a non-trivial knot is not the fixed point set of an orientation preserving homeomorphism f : S 3 Ñ S 3 of order N [50, Problem 3.38]. The Covering Conjecture and the Smith Conjecture are both theorems, proved by geometrization techniques [1].Kronheimer and Mrowka's sutured Up2q-instanton homology group, SHI 2 , can be employed to answer Question 1.1 affirmatively for N " 2 (and hence for any even N ) and any non-trivial knot K [59]. 3 Associated to any knot K, there is a sutured manifold pM pKq, αpKqq where M pKq is the knot complement and αpKq is the union of two oppositely oriented meridional curves. Kronheimer and Mrowka proved that if the dimension of SHI 2 pM pKq, αpKqq is greater than 1, then there is a nonabelian representation of the knot group of K that satisfies (1.2). Similar to foliations, SHI 2 also behaves well with respect to surface decomposition, and one can inductively construct non-trivial elements of SHI 2 pM pKq, αpKqq after simplifying pM pKq, αpKqq by a series of sutured decomposition. In particular, the dimension of SHI 2 pM pKq, αpKqq is at least two for a non-trivial knot K. It is also shown in [31,7] that if K is a knot with non-trivial Alexander polynomial, then the answer to Question 1.1 is positive for infinitely many values of N . In the light of the success of SHI 2 in addressing Question 1.1, it is natural to look for the generalization of SHI 2 for higher values of N .