Local formulae for combinatorial Pontryagin classes

Abstract

Using direct analysis of the motional Stark effect (MSE) signals, an explicit measurement of the 'missing' bootstrap current density around the island location of a neoclassical tearing mode (NTM) is made for the first time. When the NTM is suppressed using co-electron cyclotron current drive, the measured changes in the current profile that restore the bootstrap current are also directly found from the MSE measurements. Additionally, direct analysis of helical perturbations in the MSE signals during slowly rotating 'quasi-stationary' modes shows the first explicit measurement of the deficit in the toroidal current density in the island O-point.

The construction of combinatorial manifolds with prescribed sets of links of vertices

“…Nevertheless, the obtained formulae are the first formulae for the higher Pontryagin classes that can be applied to an arbitrary combinatorial manifold with no additional structures and give an algorithm for computations. The approach is distinct from the approach used in [17]. Notice that for the first Pontryagin class the formula obtained in [17] is much easier than the formula obtained in the present paper.…”

“…Later their result was improved by R. MacPherson [24]. The easiest and most effective formula for the first Pontryagin class was obtained by the author [17] in 2004. For the higher Pontryagin classes there are two approaches to constructing combinatorial formulae.…”

“…The interest to such combinatorial data caused among others by the fact that the Pontryagin numbers of a manifold can be computed from the set of isomorphism classes of links of its vertices. The existence of functions computing the Pontryagin numbers from the set of isomorphism classes of links of vertices follows from a result of N. Levitt and C. Rourke [23] (see also [17], [18]). The problem of finding explicit formulae will be discussed below.…”

“…Cocycles of this complex give local formulae for A-valued additive invariants of cobordisms of oriented manifolds with singularities of class C. We prove that any Q-valued invariant admits a local formula unique up to a coboundary of the complex T * C (Q). In particular, we obtain a simpler and more direct proof of the author's theorem [17] claiming that the graded cohomology group of the complex T * (Q) = Hom(T * , Q) is additively isomorphic to the polynomial ring in Pontryagin classes with rational coefficients.…”

“…Thus we obtain a differential d : T n → T n−1 that turns the graded group T * into a chain complex. The complex T * was originally defined by the author in [17]. Let us now consider the transformation L T assigning to each oriented n-dimensional combinatorial manifold the sum of links of its vertices in the group T n .…”