Formal Immersions and the Smale Hirsch Theorem

If $x\in S$ has $f(x)<c$, continuity of $f$ and openness of $(-\mathrm{\infty },c)$ give an open neighbourhood $U$ of $x$ with $f(U)\subseteq (-\mathrm{\infty },c)$, so $U\subseteq S$. Hence every point of the open sublevel is interior; the local normal form below excludes points of the level from the interior.

Let $x\in f^{-1}(c)$. By [F1] the differential $d{f}_x$ is nonzero, so $f$ is a submersion at $x$; by [L1] choose adapted source coordinates with last coordinate $f-c$. To see that these are coordinates, start with the submersion normal form and replace its final coordinate by ${\psi }^{-1}(u_m)-c$, whose derivative is nonzero. Then $S\cap U$ corresponds to $\{u_m\le 0\}$ and the level to $\{u_m=0\}$. Every neighbourhood of a level point also meets $\{f>c\}$, so no level point is interior to $S$.

The charts of step 1.2 around points of the level together with the ordinary charts of $M$ around the interior points of step 1.1 cover $S$. Their transition maps are restrictions of smooth transition maps of the ambient manifold $M$ (each boundary chart extends to an ambient smooth chart), hence smooth in the local-extension sense; therefore they define a smooth $m$-manifold structure with boundary on $S$ whose boundary is the level $f^{-1}(c)$ and whose interior is the open sublevel, and [L2] makes that boundary closed and embedded.

By hypothesis $S$ is compact. If the level is empty, step 1.1 applies at every point and $S=S^{\circ }$, so $S$ is a compact manifold without boundary; conversely, if the boundary is empty then the level is empty. The assertion for a Morse function with compact sublevels is the special case in which the regular values are exactly the non-critical values.

Fix $x\in M$. Since $f$ is an immersion at $x$ and $\dim M=\dim N=n$, [L1] gives charts near $x$ and $f(x)$ in which $f$ reads as the identity on an open subset of ${\mathbb{R}}^n$; hence some open neighbourhood $U_x$ of $x$ is mapped diffeomorphically onto an open subset $f(U_x)$ of $N$. Therefore $f(M)={\bigcup }_{x\in M}f(U_x)$ is open in $N$, and $f$ is a local diffeomorphism.

Since $M$ is nonempty and compact, [L2] makes $f(M)$ nonempty and compact, and since $N$ is Hausdorff, [L2] makes $f(M)$ closed in $N$.

An open and closed subset of a locally connected space is a union of components; in particular, if $N$ is connected then the nonempty clopen set $f(M)$ equals $N$. Applying this with $N={\mathbb{R}}^n$ and [L3] gives $f(M)={\mathbb{R}}^n$.

But ${\mathbb{R}}^n$ is not compact: the open cover by the balls of radius $k$, $k\ge 1$, has no finite subcover, because a finite union of bounded sets is bounded while ${\mathbb{R}}^n$ is unbounded. This contradicts step 2.1 with $f(M)={\mathbb{R}}^n$. Hence no immersion $M\to {\mathbb{R}}^n$ exists; since a smooth map $f$ is an immersion exactly when $(f,df)$ is a formal immersion, no formal immersion of this nonempty $M$ into ${\mathbb{R}}^n$ is holonomic, and equidimensional immersions into a general target $N$ have image a union of components of $N$ by step 3.1.

Fix a neighbourhood specified in the first atlases and a map $h$ belonging to it. On each of its nonempty compact pieces $K_i$, the maximum derivative error of $h$ is strictly smaller than the prescribed tolerance, so there is a positive residual margin. Around each point of $K_i$, choose a coordinate ball with compact closure inside the old source chart, a source chart of the second atlas, and the inverse image under $h$ of a target chart of the second atlas intersected with the old target chart. Finitely many smaller balls cover $K_i$ by [L1]. The resulting closed pieces $C_{ij}$ are compact, cover $K_i$, and lie entirely in these chart overlaps; unlike intersections of $K_i$ with open chart domains, they really are compact.

Choose compact target neighbourhoods of $h(C_{ij})$ inside the target overlaps, and impose sufficiently small zeroth-order conditions to keep nearby maps there. Source transition derivatives are bounded on the $C_{ij}$; target transition derivatives are bounded and uniformly continuous on these fixed target neighbourhoods. Repeated chain and product rules write each derivative in an old chart as a finite sum of products of new-coordinate derivatives and transition derivatives evaluated at the nearby map. Uniform continuity of the latter, together with boundedness of the former near $h$, implies that sufficiently small new-coordinate errors through order $r_i$ make each old-coordinate error smaller than the residual margin from step 1.1. This comparison holds uniformly on each $C_{ij}$.

Intersect the finitely many new-chart conditions furnished by step 2.1. They give a weak neighbourhood of $h$ contained in the original neighbourhood, since the $C_{ij}$ cover each $K_i$. Thus every first-atlas neighbourhood is open in the second-atlas topology. Interchanging the atlases proves equality. This also applies to any supplied countable locally finite atlas; no existence claim about such an atlas is needed here.

Give the set of restricted jet data on a compact $K$ the topology generated by the finite-order chartwise uniform conditions on compact subsets of $K$. Every such condition pulls back to a weak open condition on $C^{\mathrm{\infty }}(M,Q)$, and every weak basic condition is a pullback of one of them, using its compact piece $K_i$. These two inclusions prove the asserted initial-topology description. Retaining jets makes this description meaningful even for a singleton or a compact set with empty interior.

Suppose $\mathrm{\Phi }$ is continuous. Choose a compact neighbourhood $K^{\mathrm{\prime }}$ of $K$ inside $U\cap \mathrm{\Phi }(p_0{)}^{-1}(V)$, using finitely many small closed coordinate balls from [L1]. The zeroth-order weak neighbourhood requiring the image of $K^{\mathrm{\prime }}$ to remain in $V$ pulls back to a neighbourhood $W$ of $p_0$. This single $W$ works for every derivative order.

Conversely assume the adjoint and its local jet functions are jointly continuous. Fix $p_0$ and finite weak-neighbourhood data. Joint continuity of the adjoint and compactness of each $K_i$ first give a parameter neighbourhood on which its image stays in the target chart: take finitely many product neighbourhoods covering $\{p_0\}\times K_i$ and intersect their parameter factors. On that neighbourhood the maximum $G_i(p,x)$ of the finitely many derivative errors through order $r_i$ is continuous and vanishes when $p=p_0$. For each $x\in K_i$, choose a product neighbourhood on which $G_i<{\epsilon }_i$; finitely many source factors cover $K_i$, and the intersection of their parameter factors makes this inequality hold on all of $K_i$. Intersect also over the finitely many $i$. The resulting neighbourhood maps into the specified weak neighbourhood. This finite-cover argument works for every topological $P$; no first-countability or sequential argument is used.

Fix any multi-index $\gamma$ and any $(p_1,x_1)\in W\times \alpha (K)$. Continuity of $\mathrm{\Phi }$ at $p_1$ supplies, for every $\epsilon >0$, a parameter neighbourhood on which the $\gamma$-derivative differs uniformly on $\alpha (K^{\mathrm{\prime }})$ from that of $\mathrm{\Phi }(p_1)$ by less than $\epsilon \mathrm{/}2$. The latter derivative is continuous in $x$, since $\mathrm{\Phi }(p_1)$ is smooth, and differs from its value at $x_1$ by less than $\epsilon \mathrm{/}2$ near $x_1$. The triangle inequality proves joint continuity at $(p_1,x_1)$. Hence (i) holds on $W$, for all orders.

If $P$ is smooth and the adjoint is smooth, then its local $x$-derivatives are jointly continuous by [L2], so step 1.2 applies. Empty compact pieces impose no conditions. This proves (ii) and completes both implications.

Empty $M$ and $m=0$ have automatic injectivity; if $M$ is nonempty and $m>n$, both loci are empty and open. For (i) in the remaining case, fix an arbitrary immersion $f:M\to N$. Around each source point take a source chart mapped by $f$ into a target chart, and a smaller compact coordinate ball whose interior contains that point. Compactness selects finitely many such pieces $K_i\subset U_i$ covering $M$, with $f(U_i)\subset V_i$. Write $J_i(x)=D({\psi }_i\circ f\circ {\phi }_i^{-1})({\phi }_i(x))$; its entries are continuous and bounded on $K_i$.

For each $i$ and $x\in K_i$ some $m\times m$ minor of $J_i(x)$ is nonzero by [L1], so the maximum ${\delta }_i(x)$ of the absolute determinants of the finitely many minors is a continuous strictly positive function on $K_i$; by [L3] it has a positive minimum $c_i$. Let $B_i\ge 1$ bound the absolute values of all entries of $J_i$ on $K_i$. The determinant of an $m\times m$ matrix is a polynomial in the entries, so there is ${\eta }_i>0$, depending only on $m$, $B_i$, $c_i$, such that any matrix $J^{\mathrm{\prime }}$ with $\mathrm{\mid }{J}^{\mathrm{\prime }}-J_i(x)\mathrm{\mid }<{\eta }_i$ entrywise satisfies $\mathrm{\mid }\det J^{\mathrm{\prime }}-\det J_i(x)\mathrm{\mid }<c_i\mathrm{/}2$ for the maximizing minor, hence has a nonzero $m\times m$ minor. Choosing the finitely many ${\eta }_i$ uses no choice, and may be taken in the form $2^{-k}$ with the least suitable $k$.

Let $\mathcal{U}$ be the basic weak open set of maps $g:M\to N$ determined by the data $(U_i,{\phi }_i)$, $(V_i,{\psi }_i)$, $K_i$, $r_i=1$, ${\epsilon }_i={\eta }_i$. Its definition constrains the partial derivatives of first order of ${\psi }_i\circ g\circ {\phi }_i^{-1}$ on ${\phi }_i(K_i)$ to differ from those of $f$ by less than ${\eta }_i$; in particular every entry of the Jacobian of $g$ differs from the corresponding entry of $J_i$ by less than ${\eta }_i$ at every point of $K_i$. By step 2.1 every such $g$ has rank $m$ at every point of $M$, so by [L1] every $g\in \mathcal{U}$ is an immersion. Hence $\mathrm{Imm}(M,N)$ contains the basic neighbourhood $\mathcal{U}$ of $f$ and is open.

For (ii), independently fix an arbitrary fibrewise injective pair $(f,F)$, whose base map need not be an immersion. Choose compact pieces $K_i$ and induced bundle charts over source and target chart domains. In such charts a bundle map has the form $(x,v)\mapsto (g(x),A_i(x)v)$. The compact set of vectors $(x,e_j)$ with $x\in K_i$, $1\le j\le m$, is a valid compact test set in $TM$; zeroth-order control of the images of these vectors controls every column of $A_i$. Zeroth-order control of $g$ keeps these images in the same target bundle chart. Each matrix $A_i$ has rank $m$ by the injectivity of $F$, so its own maximum of absolute $m$-minors has a positive minimum on $K_i$. Apply the polynomial determinant estimate of step 2.1 to these matrices, independently of $df$. This gives a neighbourhood of the pair $(f,F)$, among all bundle-map pairs, on which all fibre maps remain injective. Restricting that neighbourhood to the fixed-base fibre proves its openness as well.

The smooth function $H:E\to \mathbb{R}$, $H(v)=\mathrm{\parallel }v{\mathrm{\parallel }}_h^2$, has vertical derivative $w\mapsto 2h(v,w)$. At $H(v)=1$, evaluating on $w=v$ gives $2$, so $1$ is regular. The local normal-form argument of [L1] supplies the subspace smooth half-space charts on $H\le 1$, without needing compactness. Its boundary is $H=1$ and its interior is $H<1$. For $q=0$, $H=0$ and the disk bundle is simply $M$.

These charts are restrictions of smooth ambient charts, so projection and zero section remain smooth. The boundary is closed and embedded by [L1], with the asserted dimension when the total dimension is positive.

If $M$ is compact, cover it by finitely many compact coordinate pieces $K_i$ lying inside bundle-trivialization domains. For $q>0$, on the compact set $K_i\times S^{q-1}$ the function $(u,z)\mapsto z^{\mathsf{T}}A(u)z$ has a positive minimum $c_i$ by [L2]. Thus $v^{\mathsf{T}}A(u)v\le 1$ implies $\mathrm{\mid }v\mathrm{\mid }\le c_i^{-1\mathrm{/}2}$. The disk bundle over $K_i$ is a closed bounded subset of a Euclidean coordinate product, hence compact by [L2]. Their finite union is $D_h(E)$, which is therefore compact; its closed boundary is compact too. Empty pieces are omitted. For $q=0$ compactness is just compactness of $M$.

Choose $h$ as in [F1]. Choose any $x_0\in M$. The nonempty compact sublevel $\{h\le h(x_0)\}$ has compact image in $\mathbb{R}$, by pulling any image cover back to a cover of the sublevel; the identity function on that image has a minimum by [F5], and all other points have larger values; thus $h$ attains a global minimum $m_0$; choose a regular value $b_0\ge m_0$ with $m_0<b_0<m_0+1$, and for every $j\ge 1$ choose a regular value $b_j\in (b_0+j,b_0+j+1)$. Each interval is nonempty and contains a regular value by [F2], and the countably many selections are licensed by [F1]$$'s ${\mathrm{A}\mathrm{C}}_{\omega }$. The sequence $(b_j)$ is strictly increasing with $b_j\to \mathrm{\infty }$.

For each $j$ put $K_j:=h^{-1}((-\mathrm{\infty },b_j])$. By [F3] each $K_j$ is a nonempty compact smooth $m$-manifold with boundary $h^{-1}(b_j)$ and interior $h^{-1}((-\mathrm{\infty },b_j))$; since $b_j<b_{j+1}$ are regular, $K_j\subseteq \mathrm{int}{K}_{j+1}$, and the $K_j$ exhaust $M$ because $h$ is exhaustive.

Fix $j$ and let $Z$ be a cap of $K_j$, that is a compact connected component of $M\setminus \mathrm{int}{K}_j$. Its boundary in $M$ is $\mathrm{\partial }Z=Z\cap h^{-1}(b_j)$: a point of $Z$ with $h>b_j$ has a ball around it contained in the open set $\{h>b_j\}\subseteq M\setminus \mathrm{int}{K}_j$ and, being connected, that ball lies in the component $Z$, while a ball around a point of $Z\cap h^{-1}(b_j)$ meets $\{h<b_j\}$ because $b_j$ is a regular value. If $\mathrm{\partial }Z=\mathrm{\varnothing }$, then every point of $Z$ is interior to $Z$ in $M$, so $Z$ is open in $M$, while $Z$ is closed in $M$ because it is a component of the closed set $M\setminus \mathrm{int}{K}_j$; connectivity of $M$ then forces $Z=M$; but then $M$ would be compact, contradicting that $M$ has no compact component. Hence $\mathrm{\partial }Z\mathrm{\ne }\mathrm{\varnothing }$.

The cap $Z$ is a compact smooth $m$-manifold with boundary $\mathrm{\partial }Z$: at a point with $h>b_j$ it is open in $M$ by the ball argument of step 3.1, and at a point of the level the local normal form of $h$ at the regular value $b_j$ (Local normal form for submersions) exhibits a neighbourhood of $Z$ as a half-space. Hence $\mathrm{\partial }Z$ is a nonempty closed $(m-1)$-submanifold of the compact $(m-1)$-manifold $h^{-1}(b_j)$ (The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold, Embedded smooth submanifolds with boundary), hence a union of components of $h^{-1}(b_j)$. Two distinct caps have disjoint boundaries: a point of $\mathrm{\partial }{Z}_1\cap \mathrm{\partial }{Z}_2$ has a neighbourhood in $M\setminus \mathrm{int}{K}_j$ that is connected (a half-ball at the level set) and meets both caps, contradicting that they are distinct components. Since $h^{-1}(b_j)$ is compact and locally connected it has finitely many components by [F4], so sending a cap to the nonempty set of level components in its boundary injects the caps into the power set of a finite set: there are finitely many caps $Z_1,\dots ,Z_r$ of $K_j$.

Put $K_j^{+}:=K_j\cup Z_1\cup \cdots \cup Z_r$. Give $K_j^{+}$ the smooth structure with boundary carried by the ambient charts of $M$: a point of $\mathrm{int}{K}_j$ or of $\mathrm{int}{Z}_i$ has an open neighbourhood in $M$ contained in $K_j^{+}$ and serves as an interior chart; at a seam point in $\mathrm{\partial }{Z}_i\subseteq \mathrm{\partial }{K}_j$ the local regular-level chart has its lower half in $K_j$ and its upper half in $Z_i$, so their union contains a full ambient neighbourhood and the seam point is interior too; a point of $\mathrm{\partial }{K}_j\setminus (\mathrm{\partial }{Z}_1\cup \cdots \cup \mathrm{\partial }{Z}_r)$ has a half-space chart inherited from a boundary chart of $K_j$, and a sufficiently small such chart avoids the caps because the caps meet $\mathrm{\partial }{K}_j$ exactly in the closed sets $\mathrm{\partial }{Z}_i$; transitions are restrictions of transition maps of $M$. Hence $K_j^{+}$ is a compact smooth $m$-manifold with boundary, with $\mathrm{\partial }{K}_j^{+}=\mathrm{\partial }{K}_j\setminus (\mathrm{\partial }{Z}_1\cup \cdots \cup \mathrm{\partial }{Z}_r)$ and $\mathrm{int}{K}_j^{+}=\mathrm{int}{K}_j\cup Z_1\cup \cdots \cup Z_r$.

The complement $M\setminus \mathrm{int}{K}_j^{+}$ is obtained from $M\setminus \mathrm{int}{K}_j$ by deleting the components $Z_1,\dots ,Z_r$, so every connected component of it is a connected component of $M\setminus \mathrm{int}{K}_j$ other than the $Z_i$, hence is noncompact by the definition of a cap. Therefore $K_j^{+}$ has no cap.

Define $M_0:=K_0^{+}$, which is nonempty, compact, and cap-free by step 6.1. Given a cap-free compact $M_j$, let $k_j$ be the least integer with $M_j\subseteq \mathrm{int}{K}_{k_j}$, which exists because the compact $M_j$ is contained in $M={\bigcup }_k\mathrm{int}{K}_k$; set $M_{j+1}:=K_{k_j}^{+}$. Then $M_j\subseteq \mathrm{int}{K}_{k_j}\subseteq \mathrm{int}{M}_{j+1}$ and $M_{j+1}$ is compact and cap-free by step 6.1. The indices $k_j$ strictly increase, since $K_{k_j}\subseteq M_{j+1}$ forces $k_{j+1}>k_j$; hence $M={\bigcup }_j{M}_j$ and each $M_j$ is a nonempty compact $m$-manifold with boundary.

Let $C$ be a connected component of the band $M_{j+1}\setminus \mathrm{int}{M}_j$ and suppose $C\cap \mathrm{\partial }{M}_{j+1}=\mathrm{\varnothing }$; then $C\subseteq \mathrm{int}{M}_{j+1}\setminus \mathrm{int}{M}_j$. At a point $x\in \mathrm{\partial }C$ we have $\mathrm{\partial }C\subseteq \mathrm{\partial }{M}_j$, and since $\mathrm{\partial }{M}_j\subseteq M_j\subseteq \mathrm{int}{M}_{j+1}$ there is a ball $B$ around $x$ contained in $\mathrm{int}{M}_{j+1}$; the set $B\cap (M\setminus \mathrm{int}{M}_j)$ is a half-ball, hence connected, and meets $C$, so it lies in $C$. Thus $C$ is open and closed in $M\setminus \mathrm{int}{M}_j$, and it is compact, so it is a compact component of $M\setminus \mathrm{int}{M}_j$, that is a cap of $M_j$, contradicting cap-freeness. Hence $C\cap \mathrm{\partial }{M}_{j+1}\mathrm{\ne }\mathrm{\varnothing }$. The band is compact and locally connected and therefore has finitely many components by [F4].