I'm interested whether there is a simple proof for Novikov's conjecture for higher signatures in case of fundamental group $\pi(M) = \mathbb{Z}^k$. I guess that something can be found in Kasparov's work "On the homotopy invariance of the rational Pontryagin numbers", but it seems to be almost impossible to find it anywhere.
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$\begingroup$ What is "simple"? Did you read the original proofs by Novikov and Farrell-Hsiang? See references on p.202 of indiana.edu/~jfdavis/papers/d_manc.pdf in "Manifold aspects of the Novikov Conjecture" by J.Davis. $\endgroup$– Igor BelegradekCommented Jun 9, 2017 at 11:22
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$\begingroup$ I recently stumbled across a paper by G. Lusztig called "Novikov's higher signature and families of elliptic operators". It contains a proof of said case of Novikov's conjecture based on the Atiyah-Singer index theorem for families. Is that simple enough for you? Here's the link: projecteuclid.org/euclid.jdg/1214430829 $\endgroup$– Stefan BehrensCommented Jun 9, 2017 at 18:00
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