45

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 ...


19

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 $\Omega^k \Sigma^k X$, ...


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 ...


17

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 ...


15

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

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 ...


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

(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 ...


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

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

The formal answer is yes. Moreover, the function does not have to be Morse, just any smooth function. Indeed let $f\colon M\to \mathbb{R}$ be any smooth function. Let us fix an imbedding $i\colon M\to \mathbb{R}^{n-1}$; for large $n$ it always exists. Consider the imbedding $(i\times f)\colon M\to \mathbb{R}^{n-1}\times \mathbb{R}=\mathbb{R}^n$. This is ...


9

This is trivially true: take an embedding $g:M\to \mathbb R^{n-1}$ and consider $(g,f):M\to \mathbb R^{n-1}\times \mathbb R\to \mathbb R$.


9

Warning: this is not an immersion (it has twelve Whitney-umbrella-like pinch points) Here is a relatively simple explicit realization: the $z$ coordinate for the parametric surface$$(x,y,z)=(\sin(2u),\sin(2v),\sin(u)\sin(v)\sin(u-v));$$after an affine shift leaving $z$ unchanged the graph looks like this: It is thus similar to the Steiner's Roman surface ...


9

Assuming that by `cohomology' you mean integer coefficients, there is a general result saying what you want for simply connected manifolds of dimension $5$ or more, without the Kähler condition. According to Smale (Generalized Poincare’s conjecture in dimensions greater than four, Ann. Math. 74, No, 2, 391-406 (1961)) for $n > 5$ and Barden (Simply ...


9

A generic perturbation of the metric makes the flow Morse Smale, (stable and unstable manifolds meet transversally). In this situation, the unstable manifold do form a CW-complex. The unstable manifold of a critical point $x$ of index $k$ is an embedded disk of dimension equal to the Morse index; its closure is made adding a union of unstable manifolds of ...


9

You can do the next best think. To a Forman-Morse function $f$ one can associate a flow on the manifold whose stationary points are precisely the barycenters of the faces of your simplicial decomposition. The Conley index of the barycenter of a critical face has the homotopy type of a sphere of the dimension of that face. The Conley index of the ...


9

A recent breakthrough result which uses Morse theory in a substantial manner is Watanabe's disproof of Smale conjecture in dimension 4. In it, he provides a method to compute Kontsevich's configuration space integrals by counting certain broken flowlines for gradients of Morse functions. These Morse-theoretic invariants are used to prove that certain 4-...


8

If the Morse function $f$ is perfect, then, for any choice of metric $g$, the attaching maps cannot be homeomorphism. Indeed if the Morse function was perfect, then the boundary operator of the associated Morse-Smale complex is trivial. If the attaching map was a homeomorphism, then the boundary operator cannot be zero. This can be seen from Thm. 4....


8

Yes; see Smale, On the structure of manifolds (Amer. J. Math. 84 1962 387–399) where it's shown that in high dimensions, you can eliminate handles under various connectivity assumptions. The h-cobordism theorem is a special case. For index greater than 1, one usually doesn't do handle-trading, because you can in fact just do handle cancellation, which is ...


8

There is a standard way to get a Heegaard splitting that works more generally for 3-manifolds fibering over $S^1$. Take two copies of the fiber surface, and "tube" them together on either side. The two copies of $F_g$ split up the manifold into two copies of $F_g\times I$. Adding a 1-handle to one $F_g\times I$ removes a 2-handle from the other side, ...


7

In dimension $3$ we know from Perelman's work that a simply connected $3$-manifold is a sphere. In dimension $4$ I quote Kirby in 1989 "It is not known if a simply connected $4$-manifold needs $1$-handles and/or $3$-handles but the Dolgachev surface is a good candidate for needing them." page 8, R. Kirby: The Topology of $4$-Manifolds, Lect. Notes ...


7

In section 11 of this paper I show that a discrete Morse function on a simplicial complex leads to a dynamical description of Forman's theory. More precisely there is a canonical flow associated to the function such that the (open) faces of the barycentric subdivision are invariant sets. The stationary points of this flow are the barycenters of the ...


7

These lecture notes were actually mainly devoted to the Morse Complex in the infinite dimensional setting; but they were thought to be suitable for finite dimensional manifolds as well (btw, you don't need to pay for them).


7

Any Morse function on $S^2$ may be realized by an embedding $S^2\hookrightarrow \mathbb{R}^3 \to \mathbb{R}$. For a Morse function $F:S^2\to \mathbb{R}$, take the equivalence relation with equivalence classes given by the components of the level sets of the Morse function $F$. The quotient of the sphere by this equivalence relation is a cubic tree $\mathcal{...


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