Communications in Mathematical Physics - Volume 301 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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We shall take k to be the field of rational numbers, and write C∗ (−) and C ∗ (−) for the singular chain complex and the singular cochain complex, respectively. We grade C ∗ (−) negatively. The chain complex C∗ (M) has a partially defined product given by intersection of transversal chains, and a coproduct given by the Alexander-Whitney approximation of the diagonal embedding M → M × M. In fact, C∗ (M) is a partially defined (noncommutative) DG open Frobenius algebra over Z; however, over Q, one may define the Frobenius algebra structure fully, due to a result by P.

Arch. Rat. Mech. Anal. 3, 271–288 (1959) 31. : On the existence of global solutions to two-dimensional Navier-Stokes equations of a compressible viscous fluid. Siberian Math. J. 36(6), 1108–1141 (1995) 32. : Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. 51, 229–240 (1998) Communicated by P. Constantin Commun. Math. Phys. A. A. edu Received: 13 March 2009 / Accepted: 18 July 2010 Published online: 7 October 2010 – © The Author(s) 2010.

The proof is divided into several steps. 2. Proof of the DG Lie algebra. The product on V is graded commutative, hence if we shift the degree of C down by 1, the induced pairing ε(a ·b) : C[1]⊗C[1] → k is graded skew-symmetric. Therefore the bracket { , } defined by (8) is graded skew-symmetric. We now show the Jacobi identity: for any α = N ([a1 | · · · |an ]), β = N ([b1 | · · · |bm ]), γ = N ([c1 | · · · |c p ]) ∈ L, {{α, β}, γ } = ±ε(ai b j )ε(ak cl )N ([a1 | · · · |b j+1 | · · · |b j−1 | · · · |cl+1 | · · · |cl−1 | · · · |an ]) (10) i, j,k,l ±ε(ai b j )ε(bk cl )N ([a1 | · · · |b j+1 | · · · |cl+1 | · · · |cl−1 | · · · |b j−1 | · · · |an ]).

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