Morse theory

Morse theory relates the critical points of a Morse function to changes in the homotopy type of its sublevel sets. Crossing a nondegenerate critical value attaches a cell or handle of dimension equal to the Morse index. The theory also applies to geodesic energy through broken-geodesic approximations.

The free chain group in degree $i$ has one oriented generator for each index-$i$ critical point. A cellular chain complex obtained from the Morse handle-attachment theorem computes the integral homology. Under the Morse-Smale gradient flow condition its differential can instead be obtained by signed counts of connecting trajectories. Over a field, if $r_i$ is the rank of ${\partial }_i$, then $c_i=b_i+r_i+r_{i+1}$. This identity implies both weak and strong Morse inequalities.

Here $n(p,q)$ is the signed number of unparametrized trajectories of a Morse-Smale gradient flow from $p$ to $q$. Compactifying one-dimensional trajectory moduli spaces gives ${\partial }^2=0$. The generators coincide with the cells furnished by the unstable manifolds, and the trajectory count is their cellular incidence number, so the complex computes integral homology.

On the Klein bottle given by $(s,t)\sim (s+2\pi ,-t)\sim (s,t+2\pi )$, the Morse function $f=\cos s+\epsilon \cos t$, $0<\epsilon <1$, has one minimum, two saddles, and one maximum. In its descending Morse-Smale gradient flow, the two branches from either saddle to the minimum cancel. From the maximum to the fiber saddle, reflection of the fiber under the base identification reverses the transverse orientation, making the two contributions add. The contributions to the base saddle cancel. With a choice of generator orientations the displayed Morse chain complex follows. Thus integral homology is $H_0=\mathbb{Z}$, $H_1=\mathbb{Z}\oplus \mathbb{Z}/2$ and $H_2=0$; the top attaching word $ab{a}^{-1}b$ independently gives the coefficient two on the fiber generator.

For distinct ordered $a_0<\cdots <a_m$, the quotient $\sum a_j{x}_j^2/\sum x_j^2$ on Real projective space has one critical point $[e_i]$ of index $i$. Its descending flow is $[x_j(t)]=[e^{-2a_{j}t}{x}_j(0)]$. The unstable cell at $[e_i]$ is ${\mathbb{RP}}^i\setminus {\mathbb{RP}}^{i-1}$. Between adjacent critical points there are two trajectories, whose relative orientation is the degree $(-1{)}^i$ of the antipodal map on $S^{i-1}$. Thus, up to generator orientation, the differential is two in even degrees and zero in odd degrees. This gives the integral cellular homology of real projective space rather than only its mod-two ranks.

For a closed manifold and a Morse function, the displayed strong inequalities hold for every $j$, with equality at the top dimension. Indeed the Morse chain complex rank formula makes the alternating sum equal to $r_{j+1}\ge 0$. Adding the inequalities at $j$ and $j-1$ gives $c_j\ge b_j$. The weak inequalities, even together with equality of Euler characteristics, are less restrictive: $b=(1,0,0,1)$ and $c=(2,0,0,2)$ satisfy those numerical conditions but violate the strong inequality at $j=1$.

Crossing one nondegenerate critical point of Morse index $\lambda$ in a compact sublevel interval attaches a handle $D^{\lambda }\times D^{m-\lambda }$. Between critical levels a suitably normalized gradient flow identifies the sublevels. The Morse lemma supplies the local quadratic model for the attachment. Retracting each handle to its core disk gives a CW complex model with one $\lambda$-cell per critical point. Closed manifolds have finite such decompositions; a noncompact manifold requires a proper bounded-below exhaustion for the corresponding statement.

On a complete compact Riemannian manifold, nondegenerate critical points of fixed-endpoint geodesic energy yield a CW complex model of the path space, with one cell per geodesic and dimension equal to its Morse index. Finite-dimensional broken-geodesic models prove the assertion at bounded energy, and exhaustion gives the full path space. On $S^n$ with distinct nonantipodal endpoints the cell dimensions are $j(n-1)$, one in each such dimension.

A Morse-Bott critical manifold is a smooth submanifold $C$ of critical points such that the kernel of the Hessian matrix at every $x\in C$ is exactly $T_{x}C$. The Hessian is therefore nondegenerate in normal directions. Repeated based great-circle geodesics on a round sphere form such critical manifolds, parametrized by their unit initial directions.

The Morse index of a fixed-endpoint geodesic equals the number of conjugate points strictly between its endpoints, counted with multiplicity. Its nullity equals the multiplicity of the terminal conjugate point. A conjugate multiplicity is the dimension of the space of Jacobi fields vanishing at the initial point and at the point in question.

On the unit round sphere $S^n$, a geodesic of speed $L$ on $[0,1]$ has normal Jacobi equation $y^{\prime \prime }+L^{2}y=0$. Its conjugate parameters are $j\pi /L$, each with multiplicity $n-1$. If $L/\pi$ is not an integer, its index is $(n-1)\lfloor L/\pi \rfloor$. For $L=m\pi >0$, its index is $(m-1)(n-1)$ and its nullity is $n-1$.