Manifolds = Bundles + Handles

The slogan in the title is one that I came up with for the book (or at least I think I did, though of course I may have stolen it from somewhere that I have now forgotten).  In my opinion it gives a nice picture of the basic ingredients of surgery theory: the tangent bundle to a manifold, representing its “locally linear” structure, and its handle decomposition, which leads to an algebraic calculus of how handles intersect one another, whose first manifestation is Poincare duality.  The Hirzebruch signature theorem is the basic example of a compatibility relation between these two kinds of structure, which any manifold must satisfy; and the fundamental result of surgery is that if this compatibility relation and its higher relatives actually are satisfied by a set of “bundle plus handle” data, then that data does in fact represent a manifold structure.

Anyhow, here is the next section of the book, which briefly introduces cobordism and the signature theorem, and calculates the first two L-polynomials by hand.  I think that for the amateur (i.e. me again) it is important to see first that what’s going on here is a simple “calculate enough examples to fill in the unknown constants”, before one introduces the L polynomial itself, which can seem just like an ad hoc mystery.  This is a bit like the heat equation proof of the index theorem: it is good, I think, to hack it out by hand for the \(\bar{\partial}\) operator on Riemann surfaces, before getting engaged with the comparing-coefficients calculations in Atiyah, Bott and Patodi.  Of course, in that case we now have the Getzler argument also, which yields the correct polynomial directly.  I don’t know whether there is something in the same spirit for the signature theorem?

References

Atiyah, M., R. Bott, and V. K. Patodi. “On the Heat Equation and the Index Theorem.” Inventiones Mathematicae 19, no. 4 (December 1, 1973): 279–330. doi:10.1007/BF01425417.
Getzler, E. “Pseudodifferential Operators on Supermanifolds and the Atiyah-Singer Index Theorem.” Communications on Mathematical Physics 92 (1983): 163–178.
This entry was posted in Book and tagged , , , , . Bookmark the permalink.

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>