Novikov conjecture

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The Novikov conjecture attempts to answer the following question: Which expressions of the rational Pontryagin characteristic classes are homotopy invariants and how should we classify them?

Since the beginning of topology in the late 19th century, topological invariants were defined using additional structures. Poincare defined homology (Betti numbers) through combinatorics and the invention of simplicial complexes. Using smooth skew symmetric tensor fields (i.e. differential forms), Poincare also defined cohomology (now, "de Rham cohomology"). Even definition of the dimension of a manifold requires the use of either coordinate structure or combinatorial form, after which it must be proved that this quantity is invariant under continuous homeomorphisms (i.e. is topologically invariant).

By definition, characteristic classes of smooth manifolds are invariant under diffeomorphisms. They were discovered in the late 1930s by Stiefel and Whitney (classes \(w_k\) in the cohomology groups \(H^k\) of manifold modulo 2), and by Pontryagin (classes \(p_k\) in the cohomology groups \(H^{4k}\) with integral coefficients). There are also Chern classes \(c_k\) for complex manifolds and bundles depending on complex structure.

Pontryagin classes can be expressed through the Riemann curvature tensor of any Riemannian metric as differential forms, whose integrals along the cycles yield a "rational" homological Pontryagin class: these integrals are integers which do not change under the small variations of metric (i.e. are diffeomorphism invariant). They define only the image of class \(p_k\) in the groups \(H^{4k}(M,Q)\) with rational coefficients (i.e. modulo torsion). However, it turned out that the dimension of closed manifolds, homology groups (rings), Stiefel-Whitney classes and Pontryagin classes modulo 12 are each homotopy invariant. At the same time the very fact that the homotopy groups of spheres \(\pi_{n+3}(S^n)\) are finite for \(n>4\) (in fact they are equal to \(Z/24Z\)) implies that rational Pontryagin classes are not homotopy invariant, even for the direct product of spheres.

Only one expression from the rational Pontryagin classes is homotopy invariant for simply connected manifolds of dimension \(4k\). Consider the celebrated Hirzebruch polynomial \(L_k(p_1,...,p_k)\) (the \(L\)-genus). The integral along the whole \(4k\)-dimensional closed manifold \[<L_k,[M^{4k}]>=\tau(M^{4k})\] is equal to the signature \(\tau\) of the quadratic form naturally defined on the middle-dimensional homology group by the intersection index or by the ring structure of cohomology--it is an obvious homotopy invariant. For example, \[L_0=1, L_1=(1/3)p_1\] and \(L_2=(1/45)(p_1^2-7p_2)\); let us define all of them. Take the formal Hirzebruch series \(u/\tanh (u)\). Consider the product \[\prod_{i=1}^Nu_i/\tanh(tu_i)=\sum_{j=0}^{\infty}t^{2k}L_k +O(t^{2N})\] where \(N\rightarrow \infty\). By definition, the Pontryagin class \(p_k\) enters this expression as an elementary symmetric polynomial in the variables \(u_1^2,...,u_N^2\).

As was clarified by the theory of formal groups in complex cobordism theory, the function \(\tanh (u)\) appears here because its inverse function \[g(v)=\tanh^{-1}=1/2[\log(1+v)-\log(1-v)]\] is a logarithm of the formal group associated with its signature--i.e. its derivative is a generating function for the signatures of complex projective spaces \[dg(v)/dv=1+\sum_{i>0}\tau(CP^{2i})v^{2i} =\sum_{j\geq 0}v^{2j}\] Such a formula is true for the Hirzebruch series of all multiplicative characteristics from the Pontryagin-Chern classes: other functions appear instead of \(\tanh\). There are several very important characteristics between them (Todd genus, A-genus, Elliptic genus, et cetera).

Such expressions are extremely important for topology and Riemannian geometry (cobordisms, homotopy groups of spheres, discovery of nontrivial differential structures on spheres and topological non-invariance of the torsion part of Pontryagin classes), algebraic geometry (Riemann-Roch theorems), and for partial differential equations (the index of elliptic operators) as was revealed in the 1950s and early 1960s. These polynomials play a fundamental role in the proof of the topological invariance of rational Pontryagin classes. Much later, characteristic classes also became important for quantum field theory.

The topological invariance of rational Pontryagin classes was proved by Novikov in 1965/66. Let us formulate two corollaries of this result for the closed simply connected manifolds:

1. Homotopy equivalent manifolds can be non-homeomorphic \((n>5)\) (which disproves the Hurewicz conjecture going back to the 1930s);
2. There exists only a finite number of differential structures on the manifold \((n>4)\).

Development of these ideas brought solution to several fundamental problems of multidimensional topology. Unfortunately, some of these works remain unfinished.

The proof is based on an idea quite similar to that of Grothendieck etale topology. The new approach to the homeomorphism problems requires work with special torical nonsimply connected open and closed submanifolds and their coverings, applying technique developed by algebraic and differential topology, in particular, in the process of solving the classification problems. The first version of the Novikov conjecture appeared in 1965 as a byproduct of this work. The conjecture was proved step-by-step between 1965 and 1970, starting with the work of Novikov himself and finished by several famous topologists. This version of the Novikov conjecture posited that all integrals of the classes \(L_k\) along the cycles which are the intersections of cycles of codimension one are homotopy invariant.

The general Novikov conjecture was finally formulated in 1970. Personal discussion of S.Novikov with A.Borel in Princeton (1967) was very important for understanding what form the general conjecture might have. The general Novikov conjecture claims:

1. Consider every closed orientable manifold with fundamental group \(G\) and every cohomology class \(y\in H^*(G,Q)\) defining the corresponding class in cohomology of every manifold with fundamental group \(G\). Then, the expressions
\[<y\bigcup L_k(p_1,...,p_k),[M^n]>\]
integrated along the whole manifold are homotopy invariant. (They are called "the higher signatures.")
2. There are no other homotopy invariant expressions.

Many mathematicians attempted to prove this conjecture, and a large branch of functional analysis (the Fredholm representations) and of K-theory were developed for this purpose. In many cases the Novikov conjecture is proved, such as for the discrete subgroups of Lie groups, hyperbolic groups in the sense of Gromov, as well as other classes of groups). However, the Novikov conjecture has not been proved for all groups. Such a proof has been a very active area of research during the past 40 years, since 1970.

A very interesting analog of this conjecture was found for the spin manifolds with positive scalar curvature, namely, the Gromov-Lawson-Rosenberg conjecture--where the \(L\)-genus should be replaced by the \(A\)-genus and similar expressions \(<y\bigcup A(p_1,...,p_k),[M]>\) play the role of obstructions to the existence of a metric with positive scalar curvature. However, such a family of obstructions may be incomplete in the nonsimply connected case.

Recommended reading

Ferry S., Ranicki A., & Rosenberg J. (1995). A History and Survey of the Novikov Conjecture, London Mathematical Society Notes, v 226,227, Proceedings of the Conference, Cambridge Univ Press, Cambridge, 1995

Kreck M. & Luck W. (2005). The Novikov Conjecture: Geometry and Algebra (Kindle Edition), Birkhauser, Basel. (A Review of this book can be found in the Bulletin of the London Mathematical Society 42 (2010), pp. 181-190)

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