Homotopy Type in Terms of Critical Values

Theorem

Let $f$ be a smooth real valued function on a manifold $M$. Let $a<b$ and suppose that the set $f^{-1}[a,b]$, consisting of all $p\in M$ with $a\le f(p)\le b$ is compact, and contains no critical points of $f$. Define $M^a=f^{-1}(-\infty ,a]=\{p\in M:f(p)\le a\}$. Then $M^a$ is diffeomorphic to $M^b$. Furthermore, $M^a$ is a deformation retract of $M^b$, so that the inclusion map $M^a\to M^b$ is a homotopy equivalence.

Proof The idea of the proof is to push $M^b$ down to $M^a$ along the orthogonal trajectories of the hypersurfaces $f=\text{constant}$. Choose a Riemannian metric on $M$; and let $⟨X,Y⟩$ denote the inner product of two tangent vectors, as determined by this metric. The gradient of $f$ is the vector field $\text{grad }f$ on $M$ which is characterized by the identity $⟨X,\text{grad }f⟩=X(f)$ ($=$ directional derivative of $f$ along $X$) for any vector field $X$. This vector field $\text{grad }f$ vanishes precisely at the critical points of $f$. In classical notation, in terms of local coordinates $u^1,\dots ,u^n$, the gradient has components ${\sum }_j{g}^{ij}\frac{\partial f}{\partial u^j}$. If $c:R\to M$ is a curve with velocity vector $\frac{dc}{dt}$, note the identity $⟨\frac{dc}{dt},\text{grad }f⟩=\frac{d(f\circ c)}{dt}.$ Let $\rho :M\to R$ be a smooth function which is equal to $\frac{1}{⟨\text{grad }f,\text{grad }f⟩}$ throughout the compact set $f^{-1}[a,b]$; and which vanishes outside of a compact neighborhood of this set. Then the vector field $X$, defined by $X_q=\rho (q)(\text{grad }f{)}_q$ satisfies the conditions of Lemma. Hence $X$ generates a $1$-parameter group of diffeomorphisms

${\varphi }_t:M\to M.$

For fixed $q\in M$ consider the function $t\to f({\varphi }_t(q))$. If ${\varphi }_t(q)$ lies in the set $f^{-1}[a,b]$, then $\frac{df({\varphi }_t(q))}{dt}=⟨\frac{d{\varphi }_t(q)}{dt},\text{grad }f⟩=⟨X,\text{grad }f⟩=+1.$ Thus the correspondence $t\to f({\varphi }_t(q))$ is linear with derivative $+1$ as long as $f({\varphi }_t(q))$ lies between $a$ and $b$. Now consider the diffeomorphism ${\varphi }_{b-a}:M\to M$. Clearly this carries $M^a$ diffeomorphically onto $M^b$. This proves the first half of the theorem. Define a $1$-parameter family of maps $r_t:M^b\to M^b$ by $r_t(q)=\{\begin{array}{ll}q& \text{if }f(q)\le a\\ {\varphi }_{t(a-f(q))}(q)& \text{if }a\le f(q)\le b.\end{array}$ Then $r_0$ is the identity, and $r_1$ is a retraction from $M^b$ to $M^a$. Hence $M^a$ is a deformation retract of $M^b$. This completes the proof. REMARK: The condition that $f^{-1}[a,b]$ is compact cannot be omitted.

Theorem

Let $f:M\to \mathbb{R}$ be a smooth function, and let $p$ be a non-degnerate critical point with index $\lambda$. Setting $f(p)=c$, suppose that $f^{-1}[c-ϵ,c+ϵ]$ is compact, and contains no critical point of $f$ other than $p$, for some $ϵ>0$. Then, for all sufficiently small $ϵ$, the set $M^{c+ϵ}$ has the homotopy type of $M^{c-ϵ}$ with a $\lambda$-cell attached.

Proof Sketch The idea of the proof of this theorem is introducing a new function $F:M\to \mathbb{R}$ which coincides with the height function $f$ except that $F<f$ in a small neighborhood of $p$. Thus the region $F^{-1}(-\infty ,c-ϵ]$ will consist of $M^{c-ϵ}$ together with a region $H$ near $p$. We can construct such $F$ so that there’s no critical point in the region $F^{-1}[c-ϵ,c+ϵ]$. Choosing a suitable cell $e^{\lambda }\subset H$, a direct argument (i.e. pushing in along the horizontal lines) will show that $M^{c-ϵ}\cup e^{\lambda }$ is a deformation retraction of $M^{c-ϵ}\cup H$. Then apply the previous theorem, we get $M^{c-ϵ}\cup H$ is a deformation retract of $M^{c+ϵ}$.

Construction of $F$

From Morse Lemma, choose a coordinate $u^1,\dots ,u^n$ in a neighborhood $U$ of $p$ so that the identity $f=c-(u^1{)}^2-\cdots -(u^{\lambda }{)}^2+(u^{\lambda +1}{)}^2+\cdots +(u^n{)}^2$ holds throughout $U$. Thus the critical point $p$ will have coordinates $u^1(p)=\cdots =u^n(p)=0.$ Choose $ϵ>0$ sufficiently small so that

• The region $f^{-1}[c-ϵ,c+ϵ]$ is compact and contains no critical points other than $p$.
• The image of $U$ under the diffeomorphic imbedding $(u^1,\dots ,u^p):U\to {\mathbb{R}}^n$ contains a closed ball. $\{(u^1,\dots ,u^p):\sum (u^i{)}^2\le 2ϵ\}$ Now define $e^{\lambda }$ , a $\lambda$-cell to be the set of points in $U$ with $(u^1{)}^2+\dots (u^{\lambda }{)}^2\le ϵ$ and $u^{\lambda +1}(p)=\cdots =u^n(p)=0.$ Note that $e^{\lambda }\cap M^{c-ϵ}$ is precisely the boundary of $e^{\lambda }$, so $e^{\lambda }$ is attached to $M^{c-ϵ}$ as required. Construct a new smooth function $F:M\to \mathbb{R}$ as follows. Let $\mu :\mathbb{R}\to \mathbb{R}$ be a $C^{\infty }$ function satisfying the conditions:
• $\mu (0)>ϵ$
• $\mu (r)=0$ for $r\ge 2ϵ$.
• $-1<{\mu }^{\prime }(r)\le 0$ for all $r$, where ${\mu }^{\prime }(r)=\frac{\text{d}\mu }{\text{d}r}$. Let $F$ coincide with f outside of the coordinate neighborhood $U$, and let $F=f-\mu ((u^1{)}^2+\dots (u^{\lambda }{)}^2+2u^{\lambda +1}(p)+\cdots +2u^n(p)),$ within $U$.

The region $F^{-1}(-\infty ,c+ϵ]$ coinsides with the region $M^{c+ϵ}=f^{-1}(-\infty ,c+ϵ]$

The critical points of $F$ are the same as those of $f$.

The region $F^{-1}(-\infty ,c-ϵ]$ is a deformation retract of $M^{c+ϵ}$.

$M^{c-ϵ}\cup e^{\lambda }$ is a deformation retraction of $M^{c-ϵ}\cup H$

Remark

• More generally suppose that there are $k$ non-degenerate critical points $p_1,\dots ,p_k$ with indices ${\lambda }_1,\dots ,{\lambda }_k$ in $f^{-1}(c)$. Then a similar proof shows that $M^{c+ϵ}$ has the homotopy type of $M^{c-ϵ}\cup e^{{\lambda }_1}\cup \cdots \cup e^{{\lambda }_k}$.
• A modification of the proof of above theorem can shows that the set $M^c$ is also a deformation retract of $M^{c+ϵ}$. In fact $M^c$ is a deformation retract of $F^{-1}(-\infty ,c]$, which is a deformation retract of $M^{c+ϵ}$. Combining these facts, $M^{c-ϵ}\cup e^{\lambda }$ is a deformation retract of $M^c$.

Theorem

If $f$ is a differentiable function on a manifold $M$ with no degenerate critical points and if each $M^a$ is compact, then $M$ has the homotopy type of $CW$-complex, with one cell of dimension $\lambda$ for each critical point of index $\lambda$.

Whitehead

Let ${\phi }_0$ and ${\phi }_1$ be homotopic maps from the sphere $\dot{e^{\lambda }}$ to $X$. Then the identity map of $X$ extends to a homotopy equivalence: $k:X{\cup }_{{\phi }_0}{e}^{\lambda }\to X{\cup }_{{\phi }_1}{e}^{\lambda }$

Lemma

Let $\phi :{\dot{e}}^{\lambda }\to X$ be an attaching map. Any homotopy equivalence $f:X\to Y$ extends to a homotopy equivalence $F:X{\cup }_{\phi }{e}^{\lambda }\to Y{\cup }_{f(\phi )}{e}^{\lambda }$

Proof of theorem Let $c_1<c_2<c_3<\dots$ be the critical values of $f:M\to \mathbb{R}$. The sequence $\{c_i\}$ has no cluster point since each $M^a$ is compact. The set $M^a$ is is vacuous for $a<c_1$. Suppose $a\ne c_1,c_2,c_3,\dots$ and that $M^a$ is of the homotopy type of a $CW$-complex. Let $c$ be the smallest $c_i>a$. From above theorems, $M^{c+ϵ}$ has the homotopy type of $M^{c-ϵ}{\cup }_{{\phi }_1}{e}^{{\lambda }_1}\cup \cdots {\cup }_{{\phi }_{j(c)}}{e}^{{\lambda }_{j(c)}}$ for certain maps ${\phi }_1,\dots ,{\phi }_{j(c)}$ where $ϵ$ is small enough, and there is a homotopy equivalence $h:M^{c-ϵ}\to M^a$. We have assumed that there is a homotopy equivalence $h^{\prime }:M^a\to K$, where $K$ is a $CW$-complex. Then each $h^{\prime }\circ h\circ {\phi }_j$ is homotopic by cellular approximation to the map ${\psi }_j:{\dot{e}}^{{\lambda }_j}\to ({\lambda }_j-1)-\text{skeleton of}K$. Then $K{\cup }_{{\psi }_1}{e}^{{\lambda }_1}\cdots {\cup }_{{\psi }_{j(c)}}{e}^{{\lambda }_{j(c)}}$ is a $CW$-complex, and has the same homotopy type as $M^{c+ϵ}$ by the above lemma. By induction it follows that each $M^{a^{\prime }}$ has the homotopy type of a $CW$-complex. If $M$ is compact, the proof completes, if $M$ is not compact but all critical points lie in one of the compact sets $M^a$, then a similar proof previous theorem gives $M^a$ is a deformation retract of $M$. If there are infinitely many critical points then the above construction gives us an infinite sequence of homotopy equivalences $\begin{array}{ccccccc}{M}^{a_1}& \subset & M^{a_2}& \subset & M^{a_3}& \subset & \dots \\ \downarrow & & \downarrow & & \downarrow & & \\ K_1& \subset & K_2& \subset & K_3& \subset & \dots ,\end{array}$ each extending the previous one. Let $K$ denote the union of the $K_i$ in the direct limit topology, i.e., the finest possible compatible topology, and let $g:M\to K$ be the limit map. Then $g$ induces isomorphisms of homotopy groups in all dimensions. We need only apply Theorem 1 of Combinatorial homotopy I to conclude that $g$ is a homotopy equivalence. Whitehead’s theorem states that if $M$ and $K$ are both dominated by CW-complexes, then any map $M\to K$ which induces isomorphisms of homotopy groups is a homotopy equivalence. Certainly $K$ is dominated by itself. To prove that $M$ is dominated by a CW-complex it is only necessary to consider $M$ as a retract of tubular neighborhood in some Euclidean space. This completes the proof of Theorem.