41

One usually considers the analogue of Morse theory in algebraic geometry to be the theory of vanishing cycles and Lefschetz pencils. Because of the nature of algebraic functions, Morse theory must be a little more complicated. A Morse function on a compact manifold lets us build the manifold up step by step, starting with a local minimum from which the ...


23

I find myself more than a little confused by this question. First, the "Lectures on K(X)" are not about Morse Theory. It is true that I gave a lecture on Morse Theory at Bott's Seminar in 1963, but I did not take and write up notes of lectures by Bott (at least not as far as I can recall---but that was half a century ago). The lecture I gave was on extending ...


22

This paper by Cohen and Norbury discussed Steenrod operations, including Adem relations and Cartan formulae. Here's the abstract: In this paper we define and study the moduli space of metric-graph-flows in a manifold M. This is a space of smooth maps from a finite graph to M, which, when restricted to each edge, is a gradient flow line of a smooth (...


21

Massey products are discussed in Section 1.3 of Fukaya, Kenji. Morse homotopy, $A_{\infty}$-category, and Floer homologies. Proceedings of GARC Workshop on Geometry and Topology '93 (Seoul, 1993), 1--102, available here (pdf). The Massey products are obtained by counting gradient flow graphs with four external edges and one (finite-, possibly ...


19

A Morse function is a map of a manifold to the real line locally equivalent to: $$f(x_1,\ldots, x_n)=-x_1^2\ldots -x_k^2+ x_{k+1}^2+\ldots+x_n^2$$ for some $k$. In other words, for which the singularities are as simple as possible. While a Lefschetz pencil is a map of a smooth projective variety to the projective line local analytically given by $f=x_1^2+\...


17

The answer to your question is yes, of course. The theory has been around at least since the late 60s! See Wasserman's paper A Wasserman. Equivariant differential topology, Topology 1969; 8(2):127-150. I think the first "big application" is due to Atiyah and Bott, who used it to study (what else?) Yang-Mills on surfaces. See the first section of this ...


17

It's 6: you need one critical point of index 0 and index 3, and two of index 1 and index 2. It's at least 6, since the only 3-manifolds with lower Morse number are the 3-sphere (2) and lens spaces (4). It's at most 6, since there it has Heegaard genus at most 2. This is true for all 3-manifolds obtained as surgery along a 2-bridge knot: one place I know ...


16

By a result of Milnor, the space of maps from a finite CW complex to any CW complex is homotopy equivalent to a CW complex. This gives a general reason why spaces like $\Omega^k M$ have a CW structure. There is a well-known construction from which you can, at least in principle, get an explicit cell structure for spaces of the form $\Omega^k \Sigma^k X$, ...


15

If you understand intuition behind the fact that the Euler characteristic is the alternating sum of the betti numbers, then I think you can grasp the Morse inequalities. A Morse function gives rise to a CW structure on the manifold, by considering the unstable manifolds of index $k$ critical points as giving $k$-cells attached to the $k-1$-skeleton. The ...


14

In high dimensions ($\geq 5$) the most basic examples arise on manifolds with nonempty boundary, where one requires that diffeomorphisms restrict to the identity on the boundary. The simplest case is a diffeomorphism of $S^1\times D^{n-1}$ that is pseudoisotopic to the identity but not isotopic to the identity (always fixing the boundary). I described this ...


14

If you are looking for the classical approach to Morse theory, I feel nothing beats Milnor's book on the subject: Milnor, J. Morse theory. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963 For the Morse homological approach, i.e. counting flowlines, I really like Weber's paper on the subject: Weber, Joa The ...


14

(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps. I will retract this for now. I do recall being told this, but I am not aware at this point in time where the gaps in his paper are, if any. (I do stand by my belief that a number papers in this area are incomplete.) (2). The result you seek can be deduced in the following ...


13

The second of the theorems you quoted is considerably harder to prove. The gist of the proof is as follows. Consider the closure $\overline{D(p)}$ of $D(p)$ in $M$. Then Lizhen Qin proves that it admits a resolution in the sense of semi-algebraic geometry. More precisely he constructs a compact space $\widehat{D(p)}$ and a continuous surjective map ...


13

It is still an open and very interesting question in dimension 4. Akbulut (The Dolgachev surface. Disproving the Harer-Kas-Kirby conjecture. Comment. Math. Helv. 87 (2012), no. 1, 187–241) showed that the Dolgachev surface (and subsequently other elliptic surfaces in the same homotopy type) has a handle decomposition with no 1 or 3 handles. In the ...


12

The Morse function on $(S^1)^3$ that gives the genus 3 Heegaard splitting (the standard one) is basically just a smoothed "distance from the 1-skeleton function". So if you think of $S^1$ as the unit circle in $\mathbb C$, then $$f : (S^1)^3 \to \mathbb R$$ is given by $$f(z_1,z_2,z_3) = |z_1-1|^2 + |z_2-1|^2 + |z_3-1|^2$$ where we're taking the norm/...


11

Forman's papers and the books by Kozlov and Orlik-Welker that you see cited at Daniel's Wikipedia link are good starting points for the more combinatorial tradition of PL Morse theory. For more geometric approaches see Bestvina's PL Morse theory and the ancient Piecewise linear critical levels and collapsing by Kearton and Lickorish. Unfortunately, the ...


11

For TOP Morse function a reference is the classical book of Kirby and Siebenmann "Foundational essays on topological manifolds, smoothings, and triangulations" (1977). The key point is to consider the local standard coordinate charts given by the Morse lemma in the smooth category, and use this to define the TOP Morse functions. These are strictly related to ...


11

Let's consider CS functional for concreteness. The problem is that CS is neither Morse nor Morse-Bott (because its critical points are flat connections and character varieties of 3-manifold groups could be rather bad). The trick is to perturb CS to a Morse function. This was done first by Taubes ("Casson's invariant and gauge theory") and then developed into ...


11

The problem with the CS functional is that the Morse indices of its critical points are infinite. In particular, this functional cannot be perfect. The Floer complex does not compute the homology of any particular space (though it might compute the homology of a certain spectrum). On 4-manifolds the YM functional has some analytic deficiencies: it ...


11

Hormander's approach to solving the $\bar \partial$ problem is basically this, and his paper is from 1965, predating Witten's work by a couple of decades! By varying the "weight" function $h$, you can get families of estimates on the solution of $\bar \partial$ problem. Check out Hormander's 1965 ACTA paper for more details. Really a fabulous paper. He ...


11

In looking for counter-examples it helps to notice that $M$ and $N$ have CW structures with the same numbers of $i$ cells, and hence they have the same Euler characteristic. It turns out that there is a Morse function on the Klein bottle with critical points of index $0, 1, 1, 2$, so together with the standard height function on the 2-torus we get a counter-...


10

There's a generalization of Morse-Bott called Morse-Bott-Kirwan that you can read about in Kirwan's book. Basically this condition guarantees that the unstable sets are manifolds, but not the stable sets, so the negative of a function that's Morse-Bott-Kirwan may not be. If one defines a "Yang-Mills functional" very generally to be the norm-square of a ...


10

I want to mention an approach described in Kronheimer and Mrowka's book Monopoles and Three-Manifolds. In section 2, they give an outline of Morse theory, including Morse homology for manifolds with boundary and functoriality in Morse theory. The nice thing is we can recover the induced map $f_* : H_* (M) \rightarrow H_* (N)$ from a chain map between Morse ...


10

To any continuous dynamical system $\Phi_t$ on a reasonable space $X$ (say a compact metric space) there is an associated Morse like theory, namely the Conley index theory, C. Conley: Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Series, vol. 38, Amer. Math. Soc., 1978. Conley and Zehnder, in Sect. 3 in their paper Morse type ...


10

Why not go to Witten's actual paper? He explains it well, and here is a sketch: The operator $\Delta_s$ depends on a Morse function $f:M\to\mathbb{R}$, and it takes on the form $$\Delta_s = \Delta+ s^2|df|^2 + s\cdot F$$ where $F$ is something depending on derivatives of $f$ (basically the Hessian). The point is that for $s>>0$ this is dominated by $|...


10

The expression $(x^2+y^2)^2 + x^5 + y^5$ cannot be written in the form $(z^2+w^2)^2$ for any smooth functions $z$ and $w$ of $x$ and $y$. (Just look at the Taylor series expansion.) Similarly, $n>1$, the function $(x^2+y^2)^n + x^{2n+1} + y^{2n+1}$ cannot be written in the form $(z^2+w^2)^n$ for any smooth functions $z$ and $w$ of $x$ and $y$.


10

It is not hard to construct a smooth function $f$ on $\mathbb R$ such that $f \ge 0$ with $f(x) = 0$ if and only if $x$ is in the Cantor set $E$. If $F$ is an antiderivative of $f$, the critical values of $F$ will be an uncountably infinite perfect set.


9

I no longer think 3-dimensional lens spaces are a productive strategy. What you need is to have a manifold $M$ as a level-set of the Morse function and you need a non-trivial diffeomorphism of $M$ to be pseudo-isotopic to the identity. The idea is that roughly, between any two consecutive critical levels of your Morse function (modulo the degeneracies ...


9

Steenrod operations in Floer homology are constructed in the thesis of Matthias Schwarz: http://www.math.uni-leipzig.de/~schwarz/diss.pdf Perhaps a similar construction is feasible for Morse homology?


9

You only need $C^2$. See Nirenberg's book Topics in Nonlinear Functional Analysis, Theorem 3.1.1. He attributes this version of the Morse lemma to the late great Lars Hormander, Fourier Integral Operators I.


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