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Novikov’s Conjecture

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Abstract

We describe Novikov’s “higher signature conjecture,” which dates back to the late 1960s, as well as many alternative formulations and related problems. The Novikov Conjecture is perhaps the most important unsolved problem in high-dimensional manifold topology, but more importantly, variants and analogues permeate many other areas of mathematics, from geometry to operator algebras to representation theory.

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Notes

  1. 1.

    Spheres have stably trivial tangent bundle and no interesting cohomology, so one’s first guess might be that the theory of vector bundles and the signature theorem might be irrelevant to studying homotopy spheres. Milnor, however, showed that one can construct lots of manifolds with the homotopy type of a 7-sphere as unit sphere bundles in rank-4 vector bundles over S 4. He also showed that the signature of an 8-manifold bounded by such a manifold yields lots of information about the homotopy sphere.

  2. 2.

    The same does not hold for the torsion part of the Pontrjagin classes, as one can see from calculations with lens spaces [87, Sect. 3].

  3. 3.

    A precise statement to this effect may be found in [31, Theorem 6.5]. It says for example that if M is a closed simply connected manifold and dimM is not divisible by 4, then for any set of elements \(x_{j} \in H^{4j}(M, \mathbb{Q})\), \(1 \leq j \leq \left \lfloor \frac{\dim M} {4} \right \rfloor \), there is a positive integer R such that for any integer m, there is a homotopy equivalence of manifolds φ m :  M′ m  → M such that \(p_{j}(M'_{m}) =\varphi _{ m}^{{\ast}}{\bigl (p_{j}(M) + m\,R\,x_{j}\bigr )}\).

  4. 4.

    Once the dimension is bigger than 4!

  5. 5.

    It turns out that (2) coincides with the analogue of (1) in the topological, rather than smooth, category, but the difference between these is rather small since all homotopy groups of Top∕O are torsion.

  6. 6.

    The group operation is the connected sum; inversion comes from reversing the orientation.

  7. 7.

    Just for the experts: one needs to use the −∞ decoration on the L-spectra here.

  8. 8.

    It is known that the natural map C ∗(π) ↠ C r ∗(π) is an isomorphism if and only if π is amenable.

  9. 9.

    In the extreme case where π is a torsion group, \(\mathcal{E}\pi = \text{pt}\), while if π is nontrivial, E π is necessarily infinite dimensional.

  10. 10.

    Recall that two varieties are said to be birationally equivalent if there are rational maps between them which are inverses of each. Since rational maps do not have to be everywhere defined (this is why we denote rational maps below by dotted lines), two varieties are birationally equivalent if and only if they have Zariski-open subsets which are isomorphic as varieties.

  11. 11.

    It takes a bit of work to make sense of φ ∗ here, since φ may not be everywhere defined, but this can be done. The point is that by the factorization theorem for birational maps [1], we can factor φ into a sequence of blow-ups and blow-downs, and φ ∗ is clearly defined for a blow-down (since it is a continuous map) and is an isomorphism in this case by the Baum-Fulton-MacPherson variant of Grothendieck-Riemann-Roch [12]. In the case of a blow-up, let φ ∗ be given by the inverse of the map induced by the reverse blow-down.

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Acknowledgements

Work on this paper was partially supported by the United States National Science Foundation, grant number 1206159. I would like to thank Greg Friedman, Daniel Kasprowski, Andrew Ranicki, and Shmuel Weinberger for useful feedback on an earlier draft of this paper.

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  1. Department of Mathematics, University of Maryland, College Park, MD, 20742-4015, USA

    Jonathan Rosenberg

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    John Forbes Nash, Jr.

  2. Department of Mathematics, Princeton University, Princeton, New Jersey, USA

    Michael Th. Rassias

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Rosenberg, J. (2016). Novikov’s Conjecture. In: Nash, Jr., J., Rassias, M. (eds) Open Problems in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-32162-2_11

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