homology . These are related to the invariants for closed 3-manifolds by gluing formulas for the Floer homology of a 3-manifold described as the union along the boundary of two 3-manifolds with boundary. The three-manifold Floer homologies also come equipped with a distinguished element of the homology if the three-manifold is equipped with a contact structure. Kronheimer and Mrowka first introduced the contact element in the Seiberg–Witten case. Ozsvath and Szabo constructed it for Heegaard Floer homology using Giroux's relation between contact manifolds and open book decompositions, and it comes for free, as the homology class of the empty set, in embedded contact homology. (Which, unlike the other three, requires a contact homology for its definition. For embedded contact homology see . These theories all come equipped with a priori relative gradings; these have been lifted to absolute gradings (by homotopy classes of oriented 2-plane fields) by Kronheimer and Mrowka (for SWF), Gripp and Huang (for HF), and Hutchings (for ECH). Cristofaro-Gardiner has shown that Taubes' isomorphism between ECH and Seiberg–Witten Floer cohomology preserves these absolute gradings. Instanton Floer homology This is a three-manifold invariant connected to Donaldson theory introduced by Floer himself. It is obtained using the Chern–Simons functional on the space of connections on a principal SU(2)-bundle over the three-manifold (more precisely, homology 3-spheres). Its critical points are flat connections and its flow lines are instantons, i.e. anti-self-dual connections on the three-manifold crossed with the real line. Instanton Floer homology may be viewed as a generalization of the Casson invariant because the Euler characteristic of the Floer homology agrees with the Casson invariant. Soon after Floer's introduction of Floer homology, Donaldson realized that cobordisms induce maps. This was the first instance of the structure that came to be known as a topological quantum field theory. Seiberg–Witten Floer homology Seiberg–Witten Floer homology or monopole Floer homology is a homology theory of smooth 3-manifolds (equipped with a spinc structure). It may be viewed as the Morse homology of the Chern–Simons–Dirac functional on U(1) connections on the three-manifold. The associated gradient flow equation corresponds to the Seiberg–Witten equations on the 3-manifold crossed with the real line. Equivalently, the generators of the chain complex are translation-invariant solutions to Seiberg–Witten equations (known as monopoles) on the product of a 3-manifold and the real line, and the differential counts solutions to the Seiberg–Witten equations on the product of a three-manifold and the real line,