Abstract
It is now time to reap the benefits of the theoretical work we sowed in the previous chapter. Most applications of Morse theory that we are aware of share one thing in common. More precisely, they rely substantially on the special geometric features of a concrete situation to produce an interesting Morse function, and then squeeze as much information as possible from geometrical data. Often this process requires deep and rather subtle incursions into the differential geometry of the situation at hand. The end result will display surprising local-to-global interactions.
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Notes
- 1.
With a bit of extra work one can prove that if X is affine algebraic , then f has only finitely many critical points, so X is homotopic to a compact CW complex. There exist, however, Stein manifolds for which f has infinitely many critical values.
- 2.
By Chow’s theorem, every complex submanifold of \({{\mathit \mathbb{C}\mathbb{P}}}^{\nu }\) can be described in this fashion [GH, I.3].
- 3.
This duality isomorphism does not require V ∞ to be smooth. Only V ∖ V ∞ needs to be smooth; V ∞ is automatically tautly embedded, since it is triangulable.
- 4.
The overuse of the letter ω in this example is justified only by the desire to stick with the physicists’ traditional notation.
- 5.
Warning: The existing literature does not seem to be consistent on the right choice of sign for {f,g}. We refer to [McS, Remark 3.3] for more discussions on this issue.
- 6.
\({{\mathit \mathbb{T}}}^{{\mathit n}}\) is a maximal torus for the subgroup SU(n + 1) ⊂ U(n + 1).
- 7.
In down-to-earth terms, we get rid of the useless factor i in the above formulæ.
- 8.
The sublattice L′⊂ L is called primitive if L∕L′ is a free Abelian group.
- 9.
The point of this emphasis is that only the singular cohomology H0 counts the number of path components. Other incarnations of cohomology count only components.
- 10.
The space of hyperplanes containing η and a vertex v of μ(M) is rather “thin.” The normals of such hyperplanes must be orthogonal to the segment [η,v], so that a generic hyperplane will not contain these vertices.
- 11.
We are using the following sequence of canonical isomorphisms of vector bundles over Cz,k:
$$\begin{array}{rcl}{ {\mathit T}}_{{{\mathit C}}_{{\mathit z},{\mathit k}}}{{\mathit M}}_{{\mathit k}} :={\mathit T}{{\mathit M}}_{{\mathit k}}/{\mathit T}{{\mathit C}}_{{\mathit z},{\mathit k}}&={\mathit T}{{\mathit M}}_{{\mathit k}}/({\mathit T}{{\mathit M}}_{{\mathit k}}\cap {\mathit T}{{\mathit C}}_{{\mathit z}})\cong({\mathit TM}+{\mathit T}{{\mathit C}}_{{\mathit z}})/{\mathit T}{{\mathit C}}_{{\mathit z}},& \\ {{\mathit T}}_{{{\mathit C}}_{{\mathit z}}}{\mathit M}& :={\mathit TM}/{\mathit T}{{\mathit C}}_{{\mathit z}}=({\mathit TM}+{\mathit T}{{\mathit C}}_{{\mathit z}})/{\mathit T}{{\mathit C}}_{{\mathit z}}.& \\ \end{array}$$ - 12.
Compare this result with the harmonic oscillator computations in Example 3.46.
- 13.
The minus sign in the above formula comes from the fact that the Euler class of the tautological line bundle over \({\mathbb{C}\mathbb{P}}^{1}\cong{S}^{2}\) is the opposite of the generator of \({H}^{2}({\mathbb{C}\mathbb{P}}^{1})\) determined by the orientation of \({\mathbb{C}\mathbb{P}}^{1}\) as a complex manifold.
- 14.
For example, any compact CW-complex is an ENR or the zero set of an analytic map \(F : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{m}\) is an ENR. For more details we refer to the appendix of [Ha].
- 15.
The eigenvalues λk belong to \({\mathit \mathbb{Z}}\) since \({\mathrm{{\mathit e}}}^{({\mathit t}+2\pi )\dot{{{\mathit A}}_{{\mathit p}}}}={ \mathrm{{\mathit e}}}^{{\mathit t}\dot{{{\mathit A}}}_{{\mathit p}}}\), \(\forall {\mathit t}\in {\mathit \mathbb{R}}\).
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Nicolaescu, L. (2012). Applications. In: An Invitation to Morse Theory. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1105-5_3
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DOI: https://doi.org/10.1007/978-1-4614-1105-5_3
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