By Dusa McDuff and Dietmar Salamon

J$-holomorphic curves revolutionized the research of symplectic geometry while Gromov first brought them in 1985. via quantum cohomology, those curves are actually associated with the various most fun new rules in mathematical physics. This publication offers the 1st coherent and entire account of the idea of $J$-holomorphic curves, the main points of that are shortly scattered in numerous learn papers. the 1st half the ebook is an expository account of the sphere, explaining the most technical elements. McDuff and Salamon supply whole proofs of Gromov's compactness theorem for spheres and of the life of the Gromov-Witten invariants. the second one 1/2 the e-book makes a speciality of the definition of quantum cohomology. The authors identify that this multiplication exists, and provides a brand new evidence of the Ruan-Tian outcome that's associative on applicable manifolds. They then describe the Givental-Kim calculation of the quantum cohomology of flag manifolds, resulting in quantum Chern periods and Witten's calculation for Grassmannians, which pertains to the Verlinde algebra. The Dubrovin connection, Gromov-Witten power on quantum cohomology, and curve counting formulation also are mentioned. The publication closes with an summary of connections to Floer idea

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If f ∈ HomC(A) (X • , Y • ) is null-homotopic with homotopy {s i }, then {F (s i )} is a null-homotopy for F (f ). Thus F induces a functor F : K(A) → K(B) between the homotopy categories. As an example, let X and Y be topological spaces and f : X → Y a continuous map, and take A = Sh(X), B = Sh(Y ). Then direct image f∗ : K(X) → K(Y ) and inverse image f ∗ : K(Y ) → K(X) are well-defined functors on the homotopy category of complexes of sheaves. 4 Derived Categories Let A be an abelian category.

To do this, we need to specify a translation automorphism T and a collection of distinguished triangles. Let T = [1], the shift on complexes. 3 The Triangulation of the Homotopy Category C n (u) = X n+1 ⊕ Y n , n 0 dX[1] n = dC(u) n+1 u dYn 33 . We have the natural inclusion i : Y • → C • (u) and projection p : C • (u) → The triangle X • [1]. p X • −→ Y • −→ C • (u) −→ X • [1] u i is called a standard triangle. Let Δ be the collection of all triangles in K(A) isomorphic to standard triangles. 1 The homotopy category K(A) together with the shift functor [1] : K(A) → K(A) and Δ as the collection of distinguished triangles forms a triangulated category.

We claim there is no quasiisomorphism t : Z • → X • with f t null-homotopic. Suppose we had such a t = {t n }n∈Z . Pick a cycle z ∈ Z 1 such that the class of t 1 (z) is the generator of H 1 (X • ) = Z/2. This implies that t 1 (z) is an odd integer. Now t 1 (2z) = 2t 1 (z) is even, so 0 in H 1 (X • ). As t is in Qis, it follows that the class of 2z is 0 in H 1 (Z • ) and 2z = dZ (z ) for some z ∈ Z 0 . 4 Derived Categories Z −1 dZ Z0 t0 s0 0 Z1 t1 s1 Z Z 1 2 1 Z 0 49 Z/3 We have 2t 1 (z) = t 1 (dZ (z )) = 2t 0 (z ) = 2s 1 (dZ (z )) and hence t 1 (z) = s 1 (dZ (z )) = s 1 (2z) = 2s 1 (z), contradicting t 1 (z) odd.