Pontrjagin Classes Research Articles

Instanton number of noncommutative U(n) gauge theory

We show that the integral of the first Pontrjagin class is given by an integer and it is identified with instanton number of the U(n) gauge theory on noncommutative ${\bf R^4}$. Here the dimension of the vector space $V$ that appear in the ADHM construction is called Instanton number. The calculation is done in operator formalism and the first Pontrjagin class is defined by converge series. The origin of the instanton number is investigated closely, too.

Calculation of the Pontrjagin class for U(1) instantons on non-commutative Bbb R4

In noncommutative spaces, it is unknown whether the Pontrjagin class gives integer, as well as, the relation between the instanton number and Pontrjagin class is not clear. Here we define ``Instanton number'' by the size of $B_{\alpha}$ in the ADHM construction. We show the analytical derivation of the noncommuatative U(1) instanton number as an integral of Pontrjagin class (instanton charge) with the Fock space representation. Our approach is for the arbitrary converge noncommutative U(1) instanton solution, and is based on the anti-self-dual (ASD) equation itself. We give the Stokes' theorem for the number operator representation. The Stokes' theorem on the noncommutative space shows that instanton charge is given by some boundary sum. Using the ASD conditions, we conclude that the instanton charge is equivalent to the instanton number.

Homotopy complex projective spaces with Pin(2)-action

Let M be a manifold homotopy equivalent to the complex projective space CP m . Petrie conjectured that M has standard total Pontrjagin class if M admits a non-trivial action by S 1. We prove the conjecture for m<12 under the assumption that the action extends to a nice Pin(2)-action with fixed point. The proof involves equivariant index theory for Spin c -manifolds and Jacobi functions as well as classical results from the theory of transformation groups.

On the Topology of Scalar-Flat Manifolds

Let M be a simply connected closed manifold of dimension greater than 4 which does not admit a metric with positive scalar curvature. We give necessary conditions for M to admit a scalar-flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of M. As a consequence, any simply connected scalar-flat manifold of dimension greater than 4 with vanishing first Pontrjagin class admits a metric with positive scalar curvature. We also describe some relations between scalar-flat metrics, almost complex structures and the free loop space.

Weak $Spin(9)$-structures on 16-dimensional Riemannian manifolds

The aim of the present paper is the investigation of $Spin(9)$-structures on 16-dimensional manifolds from the point of view of topology as well as holonomy theory. First we construct several examples. Then we study the necessary topological conditions resulting from the existence of a $Spin(9)$-reduction of the frame bundle of a 16-dimensional compact manifold (Stiefel-Whitney and Pontrjagin classes). We compute the homotopy groups $\pi_i (X^{84})$ of the space $X^{84}= SO(16) / Spin(9)$ for $i \le 14$. Next we introduce different geometric types of $Spin(9)$-structures and derive the corresponding differential equation for the unique self-dual 8-form $\Omega^8$ assigned to any type of $Spin(9)$-structure. Finally we construct the twistor space of a 16-dimensional manifold with $Spin(9)$-structure and study the integrability conditions for its universal almost complex structure as well as the structure of the holomorphic normal bundle.

A note on Pontrjagin forms

Let P P be a principal O ( 2 m ) O(2m) bundle over a manifold M M of dimension 4 m 4m , and let p p be its m m -dimensional Pontrjagin class. In this paper, we aim at answering the following question: Which representatives of the class p p can be realised as the Pontrjagin form of some connection on P P ?

Absolute torsion and eta-invariant

In a recent joint work with V. Turaev [6], we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion, it has the advantage of having a well-determined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector bundles, and not only for unimodular ones, as is classical Reidemeister torsion. In this paper I show that the sign behavior of the absolute torsion, under a continuous deformation of the flat bundle, is determined by the eta-invariant and the Pontrjagin classes. This result has a twofold significance. Firstly, it justifies the definition of the absolute torsion by establishing a relation to the well-known geometric invariants of manifolds. Viewed differently, the result of this paper allows to express (partially) the eta-invariant, which is defined using analytic tools, in terms of the absolute torsion, having a purely topological definition. The result may find applications in studying the spectral flow by methods of combinatorial topology.

Pinching, Pontrjagin classes, and negatively curved vector bundles

We prove several finiteness results for the class $M_{a,b,G,n}$ of $n$-manifolds that have fundamental groups isomorphic to $G$ and that can be given complete Riemannian metrics of sectional curvatures within $[a,b]$ where $a\le b<0$. In particular, if $M$ is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class $M_{a,b,\pi_1(M), n}$ are total spaces of vector bundles over $M$. Furthermore, given a word-hyperbolic group $G$ and an integer $n$ there exists a positive $\epsilon=\epsilon(n,G)$ such that the tangent bundle of any manifold in the class $M_{-1-\epsilon, -1, G, n}$ has zero rational Pontrjagin classes.

$Spin^{\rm c} $ -manifolds with Pin (2)-action

We study $Spin^{\rm c} $ -manifolds with Pin(2)-action. The main tool is a vanishing theorem for certain indices of twisted $Spin^{\rm c}$ -Dirac operators. This theorem is used to show that the Witten genus vanishes on such manifolds provided the first Chern class and the first Pontrjagin class are torsion. We apply the vanishing theorem to cohomology complex projective spaces and give partial evidence for a conjecture of Petrie. For example we prove that the total Pontrjagin class of a cohomology $\mathbb{C}P^{2n}$ with $S^3$ -action has standard form if the first Pontrjagin class has standard form. We also determine the intersection form of certain 4-manifolds with Pin(2)-action.

Structures Riemanniennes L p et K-Homologie

We construct analytically the signature operator for a new family of topological manifolds. This family contains the quasi-conformal manifolds and the topological manifolds modeled on germs of homeomorphisms of R^n possessing a derivative which is in L^p, with p > n(n+1)/2. We obtain an unbounded Fredholm module which defines a class in the K-homology of the manifold, the Chern character of which is the Hirzebruch polynomial in the Pontrjagin classes of the manifold. This generalizes previous works of N. Teleman for Lipschitz manifolds and of A. Connes, N. Teleman and D. Sullivan for quasi-conformal manifolds of even dimension.

Stably extendible vector bundles over the quaternionic projective spaces

We show that, if a quaternionic k-dimensional vector bundle l' over the quaternionic projective space Hpn is stably extendible and its non-zero top Pontrjagin class is not zero mod 2, then l' is stably equivalent to the Whitney sum of k quaternionic line bundles provided k S; n.

Anomalous D-brane and orientifold couplings from the boundary state

We compute scattering amplitudes involving both R-R and NS-NS fields in the presence of a D-brane or orientifold plane. These provide direct evidence for the anomalous couplings in the D-brane and orientifold actions. The D9-brane and O9-plane are found to couple to the first Pontrjagin class with the expected relative strength.

Continuously controlled algebraic $K$ -theory of spaces and the Novikov conjecture

Here the L-polynomial is a certain rational polynomial in the Pontrjagin classes. This theorem is surprising from many points of view: the left hand side is obviously a homotopy invariant and an integer, whereas the right hand side a priory is only a smooth invariant and a rational number. This led Novikov to the following conjecture: If M has fundamental group π and x is an element in H∗(Bπ), f the classifying map, could it be that the higher signatures, x ∪ L(M)[M ] are homotopy invariants of the manifold M? From this point of view Hirzebruch’s theorem is a verification of the Novikov conjecture for simply connected manifolds. Later Wall [17, section 17H] realized that the Novikov conjecture can be expressed using the assembly map h∗(Bπ;L(Z)) → L∗(Zπ). The Novikov conjecture is equivalent to the assembly map being a rational monomorphism. Over the years it has turned out that there are lots of assembly maps. In algebraic Ktheory, in C∗-theory and in A-theory among others. It has become common practice to call the statement that the assembly map is a rational monomorphism, the Novikov conjecture in that theory. In the case of C∗-theory, monicity of the assembly map implies, but is not equivalent to the classical Novikov conjecture. In this paper we treat the A-theory case. We wish to extend the results in [6] and [7] to A-theory, using a variation of the continuously controlled A-theory in [13, 15] to replace the continuously controlled K-theory in [1]. One of the main problems here is, that as a computational device, slightly discontinuous maps were allowed in [1] and [6], and Atheory does not respond nicely to that. The answer is to work with spaces that are so locally contractible, that the slightly discontinuous maps (eventually continuous maps) can be replaced by continuous maps. Otherwise the strategy is to follow [7, 6] and [13, 15], and we shall assume the reader has some familiarity with these papers. We prove the following theorem

C*-Algebras and Controlled Topology

This paper is an attempt to explain some aspects of the relationship between the K-theory of C-algebras, on the one hand, and the categories of modules that have been developed to systematize the algebraic aspects of controlled topology, on the other. It has recently become apparent that there is a substantial conceptual overlap between the two theories, and this allows both the recognition of common techniques, and the possibility of new methods in one theory suggested by those of the other. In this first part we will concentrate on defining the C-algebras associated to various kinds of controlled structure and giving methods whereby their K-theory groups may be calculated in a number of cases. From a ‘revisionist’ perspective, this study originates from an attempt to relate two approaches to the Novikov conjecture. The Novikov conjecture states that a certain assembly map is injective. The process of assembly can be thought of as the formation of a ‘generalized signature’ [38], and therefore to understand the connection between different approaches to the conjecture is to understand the connection between different definitions of the ‘generalized signature’. Now, broadly speaking, there are two approaches to the Novikov conjecture in the literature. One approach considers the original assembly map of Wall in L-theory, and attempts to prove it to be injective by investigating homological properties of the L-theory groups (which properties themselves may be derived algebraically, or geometrically, by relating them to surgery problems). The other approach proceeds via analysis, considering the assembly map to be the formation of a generalized index of the Atiyah-Singer signature operator [3]. This approach ultimately leads to the consideration of assembly on the K-theory of C-algebras. Nevertheless it can be shown that the injectivity of assembly on the C-algebra level implies (modulo 2-torsion) the injectivity of assembly on the L-theory level. All this is explained in the paper of Rosenberg [37], to which we also refer for an extensive bibliography on the Novikov conjecture. To proceed with the background to this paper. In the late seventies and eighties it occurred both to the topologists and independently to the analysts that a more flexible generalized signature theory might be developed if one neglected the group structure of the fundamental group π in question and considered only its “large scale” or “coarse” structure induced by some translation invariant metric; up to “large scale equivalence” the choice of such a metric is irrelevant. Some references are [26, 29, 11, 12, 14, 16, 9, 33, 34, 35, 36, 41]. Since the two theories were based on the same idea, it was inevitable that they would eventually come into interaction, but this did not happen for some while. It was the insight of Shmuel Weinberger, and especially his note [40] relating the index theory of [36] to Novikov’s theorem on the topological invariance of the rational Pontrjagin classes, which provoked the discussions among the present authors which eventually led to the writing of this paper. This paper is intended to be foundational, setting down some of the language in which one can talk about the relationship between analysis and controlled topology. Many of its ideas have already been worked out in the special case of bounded control in the work of the first and third authors and G. Yu [18, 20, 19]. But it seems that there is much to be gained by considering more general kinds of control, more general coefficients, “spacification” of the theory and so on, and it

Self-Dual Gauge Fields in Eight Dimensions

Self-dual gauge fields in even dimensional spaces are characterised by a set of constraints on the eigenvalues of F. We derive a new topological bound \int_{M} ( F,F )^2 \geq \int_{M} p_1^2 on a compact 8-manifold M where p_1 is the first Pontrjagin class of the SO(n) Yang-Mills bundle. Self-dual fields realise the lower bound.

Rigidity of higher elliptic genera

We prove some general rigidity theorems for both elliptic and higher elliptic genera under a natural condition on the first equivariant Pontrjagin classes. We also obtain the vanishing of some higher elliptic genera.

A note on the homotopy invariance of Pontrjagin classes

We prove that the classical result of Wu that, for every n ⩾ 1, the integral Pontrjagin class p n modulo 3 is homotopy invariant is the only and best possible result, ‘only’ in the sense that no other Pontrjagin classes of stable vector bundles—rational, integral, multiple of integral or integral mod p, p ≠ 3, are homotopy invariant and ‘best’ in the sense that p n mod 3 r is not homotopy invariant if r > 1.

Self-dual Yang-Mills fields in eight dimensions

Strongly self-dual Yang-Mills fields in even dimensional spaces are characterised by a set of constraints on the eigenvalues of the Yang-Mills fields $F_{\mu \nu}$. We derive a topological bound on ${\bf R}^8$, $\int_{M} ( F,F )^2 \geq k \int_{M} p_1^2$ where $p_1$ is the first Pontrjagin class of the SO(n) Yang-Mills bundle and $k$ is a constant. Strongly self-dual Yang-Mills fields realise the lower bound.

Finite domination and Novikov rings

IN order to distinguish between the combinatorial properties of finite simplicial complexes and the topology of compact polyhedra and compact manifolds it is necessary to consider infinite simplicial complexes, non-compact polyhedra, open manifolds, and algebraic Kand L-theory. The classic cases are the Milnor Hauptvermutung counterexamples of non-combinatorial homeomorphisms of compact polyhedra, the proof by Novikov of the topological invariance of the rational Pontrjagin classes, and the structure theory of Kirby and Siebenmann for high-dimensional compact topological manifolds. The open manifolds arise geometrically as tame ends: in the applications it is necessary to close them. The obstruction theory for closing tame ends of open manifolds is also the obstruction theory for deciding if a finitely dominated space is homotopy equivalent to a finite C W complex.

Topological invariance of integral Pontrjagin classes mod p

It is proved that integral Pontrjagin classes P k mod p are topological invariant if p is odd, k < n( p) and they are not topological invariant if k ⩾ n( p), where n(p) is the smallest value of k for which p divides e k and e k is the smallest positive integer such that e k p k is topological invariant. For example, p k mod p is topological invariant for p = 3, 5, 11 etc. for every k ⩾ 1 but not topological invariant for p = 7 and k ⩾ 2.