Abstract For each integer we construct an oriented closed simply connected 4‐manifold admitting a smoothly embedded closed connected surface of self‐intersection number such that the complement of the surface has non‐trivial fundamental group. This answers a question of Kronheimer in Kirby's 1997 problem list. The proof combines a topological construction with homological properties of simple groups such as Thompson's group and certain sporadic finite simple groups.

We give examples of spin $4$-manifolds with boundary that do not admit metrics of positive scalar curvature and nonnegative mean curvature. These manifolds in fact have the stronger property that the conformal Laplacian with appropriate boundary conditions is never positive. The obstruction to the positivity of the conformal Laplacian is given by a real-valued $\xi$-invariant associated to the APS theorem for the twisted Dirac operator. We use analytic techniques related to the prescribed scalar curvature problem in conformal geometry to move beyond earlier work where the metric is a product near the boundary.

Let X be a smooth simply connected closed 4manifold with definite intersection form. We show that any automorphism of the intersection form of X is realized by a diffeomorphism of X#(S(2)xS(2)). This extends and completes Wall's foundational result from 1964.

We construct infinite rank summands isomorphic to ℤ^∞ in the higher homotopy and homology groups of the diffeomorphism groups of certain 4-manifolds. These spherical families become trivial in the homotopy and homology groups of the homeomorphism group; an infinite rank subgroup becomes trivial after a single stabilization by connected sum with S^2 × S^2. The stabilization result gives rise to an inductive construction, starting from non-isotopic but pseudoisotopic diffeomorphisms constructed by the second author in 1998. The spherical families give ℤ^∞ summands in the homology of the classifying spaces of specific subgroups of those diffeomorphism groups. The non-triviality is shown by computations with family Seiberg-Witten invariants, including a gluing theorem adapted to our inductive construction. As applications, we we obtain infinite generation for higher homotopy and homology groups of spaces of embeddings of surfaces and 3-manifolds in various 4-manifolds, and for the space of positive scalar curvature metrics on standard PSC 4-manifolds.

We construct closed, aspherical, smooth 4-manifolds that are homeomorphic but not diffeomorphic. These provide counterexamples to a smooth analog of the Borel conjecture in dimension four. Our technique is to apply the `reflection group trick' of the first author to pairs of exotic 4-manifolds with boundary constructed by the second author and Piccirillo.

We construct a number of topologically trivial but smoothly non-trivial families of embeddings of 3-manifolds in 4-manifolds. These include embeddings of homology spheres in S^4 that are not isotopic but have diffeomorphic complements, and families (parameterized by high-dimensional spheres) of embeddings of any 3-manifold that embeds in a blown-up K3 surface. In each case, the families are constructed so as to be topologically trivial in an appropriate sense. We also illustrate a general technique for converting a non-trivial family of embeddings into a non-trivial family of submanifolds.

For each integer $n$ we construct a simply connected $4$-manifold $X$ admitting a smoothly embedded surface $\Sigma$ of self intersection number $n$ such that the complement of the surface has non-trivial fundamental group. This answers a question of Kronheimer in Kirby's 1997 problem list. The proof combines a topological construction with homological properties of simple groups such as Thompson's group $V$ and certain sporadic finite simple groups.

Given an involution on a rational homology 3-sphere Y with quotient the 3-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg–Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants associated to spin structures on Y. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology.

The main result of this paper is that any 3-dimensional manifold with a finite group action is equivariantly invertibly homology cobordant to a hyperbolic manifold; this result holds with suitable twisted coefficients as well. The following two consequences motivated this work. First, there are hyperbolic equivariant corks (as defined in previous work of the authors) for a wide class of finite groups. Second, any finite group that acts on a homology 3-sphere also acts on a hyperbolic homology 3-sphere. The theorem has other corollaries, including the existence of infinitely many hyperbolic homology spheres that support free Zp-actions that do not extend over any contractible manifolds, and (from the non-equivariant version of the theorem) infinitely many that bound homology balls but do not bound contractible manifolds. In passing, it is shown that the invertible homology cobordism relation on 3-manifolds is antisymmetric, and thus a partial order.

The Inoue surfaces are certain non-Kaehler complex surfaces that have the structure of a T^3 bundle over the circle. We study the Inoue surfaces S_M with the Tricerri metric and the canonical spin^c structure, and the corresponding chiral Dirac operators twisted by a flat ℂ^*–connection. The twisting connection is determined by z ∈ℂ^*, and the points for which the twisted Dirac operators 𝒟^±_z are not invertible are called spectral points. We show that there are no spectral points inside the annulus α^-1/4 < |z| < α^1/4, where α >1 is the only real eigenvalue of the matrix M that determines S_M, and find the spectral points on its boundary. Via Taubes' theory of end-periodic operators, this implies that the corresponding Dirac operators are Fredholm on any end-periodic manifold whose end is modeled on S_M.

We show that the periodic $\eta$-invariants introduced by Mrowka--Ruberman--Saveliev~\cite{MRS3} provide obstructions to the existence of cobordisms with positive scalar curvature metrics between manifolds of dimensions $4$ and $6$. The proof combines a relative version of the Schoen--Yau minimal surface technique with an end-periodic index theorem for the Dirac operator. As a result, we show that the bordism groups $\Omega^{\spin,+}_{n+1}(S^1 \times BG)$ are infinite for any non-trivial group $G$ which is the fundamental group of a spin spherical space form of dimension $n=3$ or $5$.

In this article, we describe a number of applications of instanton Floer homology, focusing on two particular areas that have seen much recent activity.

We define a new topological invariant of an embedded torus in a homology S-1 x S-3, analogous to the Levine-Tristram invariant of a knot. We compare it to an invariant of smooth tori, defined recently by Echeverria using gauge theory for singular connections.

What is the simplest smooth simply connected $4$ -manifold embedded in $\mathbb {C}\mathbb {P}^3$ homologous to a degree $d$ hypersurface $V_d$ ? A version of this question associated with Thom asks if $V_d$ has the smallest $b_2$ among all such manifolds. While this is true for degree at most $4$ , we show that for all $d \geq 5$ , there is a manifold $M_d$ in this homology class with $b_2(M_d) < b_2(V_d)$ . This contrasts with the Kronheimer–Mrowka solution of the Thom conjecture about surfaces in $\mathbb {C}\mathbb {P}^2$ and is similar to results of Freedman for $2n$ -manifolds in $\mathbb {C}\mathbb {P}^{n+1}$ with $n$ odd and greater than $1$ .

Let $K$ be a knot in an integral homology 3-sphere $Y$, and $\Sigma$ the corresponding $n$-fold cyclic branched cover. Assuming that $\Sigma$ is a rational homology sphere (which is always the case when $n$ is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of $\Sigma$. The proof relies on a careful analysis of the Seiberg--Witten equations on 3-orbifolds and of various $\eta$-invariants. We give several applications of our formula: (1) we calculate the Seiberg--Witten and Furuta--Ohta invariants for the mapping tori of all semi-free actions of $Z/n$ on integral homology 3-spheres; (2) we give a novel obstruction (in terms of the Jones polynomial) for the branched cover of a knot in $S^3$ being an $L$-space; (3) we give a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers.

In this paper, we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the G-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold.

We show that infinitely many of the simply connected 4-manifolds constructed by Levine and Lidman that do not admit PL spines actually admit topological spines.

Any two homologous surfaces of the same genus embedded in a smooth 4-manifold X with simply-connected complements are shown to be smoothly isotopic in the connected sum of X and the product of a 2-sphere with itself, if the surfaces are ordinary, and in the connected sum of X with the non-trivial sphere bundle over the sphere if they are characteristic.

Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4 4 -manifold X X with the homology of S 1 × S 3 S^1 \times S^3 . Specifically, we show that for any smoothly embedded 3 3 -manifold Y Y representing a generator of H 3 ( X ) H_3(X) , a suitable version of the Heegaard Floer d d invariant of Y Y , defined using twisted coefficients, is a diffeomorphism invariant of X X . We show how this invariant can be used to obstruct embeddings of certain types of 3 3 -manifolds, including those obtained as a connected sum of a rational homology 3 3 -sphere and any number of copies of S 1 × S 2 S^1 \times S^2 . We also give similar obstructions to embeddings in certain open 4 4 -manifolds, including exotic R 4 \mathbb {R}^4 s.