Fiberwise Poincaré–Hopf Theory and Exotic Smooth Structures on Manifold Bundles

Jain, Yajit Kumar

doi:https://doi.org/10.21985/n2-pa1r-6c66

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Fiberwise Poincaré–Hopf Theory and Exotic Smooth Structures on Manifold Bundles

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We prove the Rigidity Conjecture of Goette and Igusa, which states that, after rationalizing, there are no stable exotic smoothings of manifold bundles with closed even dimensional fibers. The key ingredients of the proof are fiberwise Poincaré–Hopf theorems generalizing earlier such results about the Becker–Gottlieb transfer. These theorems show how to compute the smooth structure class, an invariant of smooth structures on fiber bundles, using the data of a fiberwise generalized Morse function. We use these results to prove a duality theorem for the smooth structure class, from which the conjecture directly follows. This duality theorem generalizes Milnor’s duality theorems for Reidemeister and Whitehead torsion, as well as similar results for higher Franz–Reidemeister torsion due to Igusa.

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