Rational Pontryagin classes, local representations, and \(K^ G\)-theory. (English) Zbl 0883.57037

From the introduction: Suppose that \(X\) and \(Y\) are compact smooth manifolds and \(f:X\to Y\) is a smooth (homotopy) equivalence. In general, the map \(f\) does not preserve rational Pontryagin classes, which depend a priori upon the smooth structures on \(X\) and \(Y\), unless \(f\) happens to be a diffeomorphism. S. P. Novikov proved in 1965 that if \(f\) is a homeomorphism then rational Pontryagin classes are indeed preserved, and this remains the best general positive result on the subject. In 1981 Sullivan and Teleman jointly provided a proof of Novikov’s result using differential geometric and analytic techniques, and recently Shmuel Weinberger gave a “short and conceptually simple analytic proof” of Novikov’s theorem drawing upon new ideas in index theory for noncompact complete Riemannian manifolds. P. Baum and A. Connes [Adv. Stud. Pure Math. 5, 1-14 (1985; Zbl 0641.57008)] have studied the foliated version of the problem and obtained positive results in the presence of negative curved leaves. In the early 1970’s Ted Petrie developed a connection between the problem of preservation of Pontryagin classes and another classical problem. If \(G\) is a compact Lie group, \(X\) and \(Y\) are smooth \(G\)-manifolds, and \(f:X\to Y\) is a smooth \(G\)-map which restricts to a diffeomorphism on fixed point sets then what are the relations among the local representations at various fixed points \(x\), \(f(x)\) for \(x\in X^G\)? T. Petrie [Bull. Am. Math. Soc. 78, 105-153 (1972; Zbl 0247.57010), Topology 13, 363-374 (1974; Zbl 0328.57016)] considered a very special situation and took \(G=S^1\).
In the present paper we shall focus attention upon a more general case. Here is the main theorem:
Theorem. Suppose that \(X\) and \(Y\) are connected, simply connected \(\text{Spin}^c\)-manifolds of the same dimension. Let \(G\) be a compact connected Lie group with torsion free fundamental group which acts upon \(X\) and \(Y\) such that \(X^G\) and \(Y^G\) are nonempty and consist entirely of isolated fixed points. Suppose that \(f:X\to Y\) is a smooth \(G\)-map such that the induced map \(f^*: K^*_G(Y) \to K^*_G(X)\) is an isomorphism. Then:
(1) If \(X\) and \(Y\) are of the same even dimension then for each fixed point \(x\in X^G\), the local representation of \(G\) at \(x\) and at \(f(x)\) are equivalent.
(2) If \(f:X\to Y\) is an equivalence then \(f^*: H^*(Y; \mathbb{Q}) \to H^* (X;\mathbb{Q})\) preserves Pontryagin classes.
Although the statement of the main results of this paper would seem to be within the realm of classical algebraic topology, our proofs rely ultimately on the Universal Coefficient Spectral Sequence which is a spectral sequence which converges to the equivariant Kasparov group \(KK^G_* (A,B)\) for suitable \(G\)-\(C^*\)-algebras \(A\) and \(B\).


57S15 Compact Lie groups of differentiable transformations
57R20 Characteristic classes and numbers in differential topology
19L47 Equivariant \(K\)-theory
55T99 Spectral sequences in algebraic topology
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