Novikov conjecture

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Sergei Novikov (2010), Scholarpedia, 5(10):7912. doi:10.4249/scholarpedia.7912 revision #81220 [link to/cite this article]
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I have suggestions for improving the English spelling/grammar of the article, and also some suggestions for adding to the mathematical scope.

Spelling/grammar

====

I have indicated by * * the missing definite/indefinite articles in the article.

Even *the* definition

either *a* coordinate

or *a* combinatorial form

  • the* characteristic classes
  • The* Pontryagin classes

"under small variations" (remove "the" here!)

"They determine" instead "They define"

"Only one expression in" instead of "Only one expression from"

g(v) = tanh^{-1}(v)

"characteristics among them" instead of "characteristics between them"

"solved" instead of "brought solution"

  • the* Grothendieck etale topology

"toroidal" or "toric" instead of "torical"

Mathematics

=

The "first version of the Novikov conjecture" should actually be stated, as should the second version. It is even less clear from the article what "This version" actually is than the "first version".

The Borel conjecture http://www.maths.ed.ac.uk/~aar/surgery/borel.pdf (not stated as such by Borel) is the conjecture that every discrete group with Poincare duality is realized as the fundamental group of an aspherical manifold, and that any homotopy equivalence of such manifolds is homotopic to a homeomorphism (rigidity). The Novikov conjecture on the homotopy invariance of the higher signatures is viewd as the rational part of the Borel rigidity conjecture. The later Farrell-Jones conjecture on the relationship of equivariant homology and algebraic L-theory implies both the Borel and the Novikov conjectures, and has been proved in many cases of geometric interest. The survey paper of Bartels, Lueck and Reich http://wwwmath.uni-muenster.de/reine/inst/lueck/homepages/wolfgang_lueck/data/blr-appl.pdf is an excellent account of the state of play.

The information in the article of the direct personal connection between Borel and Novikov in Princeton in 1967 is of great historical interest. Could Prof. Novikov describe his visit to Princeton then in greater detail?

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