Gluing manifolds along boundaries

This is a talk I gave for the Kiddie colloquium on November 8th, 2023.

Introduction#

Let $M, N$ be manifolds with boundary, and let $B \subseteq \partial M$ and $C \subseteq \partial N$ be some union of connected components. If we are given a diffeomorphism $\varphi \colon B \cong C$, how do we glue $M$ and $N$ along this diffeomorphism to obtain $$ M \amalg_{B,\varphi} N $$ as a smooth manifold?

What one usually does is first find a “collar neighborhood,” a neighborhood $U$ of $B$ inside $N$ that is diffeomorphic to $[0, \infty) \times B$, and similarly a collar neighborhood $V$ of $C$. Then we can take the diffeomorphism $\varphi$ to identify $$ U \amalg_{B, \varphi} V \cong B \times \mathbb{R}. $$ Once we do this, we can now form an atlas of $M \amalg_{B, \varphi} N$, and this becomes a smooth manifold. Then to make sure that this is a good construction, one proves the following.

Theorem 1. Collar neighborhoods exist, and for two choices of collar neighborhoods, the resulting smooth manifolds are diffeomorphic.

But what if we want to understand the space of all possible smooth structures? Somehow the fact that there is a unique gluing up to diffeomorphism is merely saying that the space is “connected” but doesn’t tell us much more. The goal is try and understand this space of gluings, prove that it is “contracible” in a particular sense, and deduce the above theorem.

Special thanks to Ciprian Bonciacat for the insight that the sheaf-theoretic approach would be useful.

Manifolds as ringed spaces#

Usually, in an introductory differential topology class, a smooth manifold is defined as a

• a topological manifold (second-countable and Hausdorff) and
• a collection of smooth charts covering the space, meaning transition maps are diffeomorphisms,
• satisfying the property that it is maximal among such collections.

This is all good, but this “maximal atlas” is really hard to understand and work with. So we will use a different definition, closer to the spirit of the definition of a scheme. For any topological space $X$, denote by $C^0(X, \mathbb{R})$ the $\mathbb{R}$-algebra of continuous real-valued functions on $X$.

Definition 2. A smooth $n$-manifold is the data of

• a second-countable Hausdorff space $M$,
• a $\mathbb{R}$-algebra subsheaf $\mathscr{O}_ M \subseteq C^0(-, \mathbb{R})$ on $M$,

such that $(M, \mathscr{O}_ M)$ is locally isomorphic to $(\mathbb{R}^n, C^\infty(-, \mathbb{R}))$.

For the less sheaf-minded people, here is a translation.

Definition 3. A smooth $n$-manifold is the data of

• a secound-countable Hausdorff space $M$,
• for each open subset $U \subseteq M$ a $\mathbb{R}$-subalgebra $\mathscr{O}_ M(U) \subseteq C^0(U, \mathbb{R})$,

such that

• for every inclusion $V \subseteq U$ of open subsets and $f \in \mathscr{O}_ M(U)$, the restriction $f \vert_V \in C^0(V, \mathbb{R})$ lies in $\mathscr{O}_ M(V)$,
• for any open cover $U = \bigcup U_i$ in $M$, a continuous function $f \colon U \to \mathbb{R}$ is in $\mathscr{O}_ M(U)$ if $f \vert_{U_i}$ is in $\mathscr{O}_ M(U_i)$ for all $i$,
• for every point $p \in M$, there exists an open neighborhood $U_p$ and a homeomorphism $\varphi_p \colon U_p \cong \mathbb{R}^n$ with the following property: for every open $V \subseteq U_p$, a continuous function $f \colon V \to \mathbb{R}$ is in $\mathscr{O}_ M(V)$ if and only if $$ f \circ \varphi_p^{-1} \colon \varphi_p(V) \to \mathbb{R} $$ is a smooth function on the open subset $\varphi_p(V) \subseteq \mathbb{R}^n$.

It’s not too difficult to see why this is the same definition as the usual one. Clearly, given a smooth manifold, we can define $\mathscr{O}_ M(U)$ by setting it to be the algebra of smooth $\mathbb{R}$-valued functions on $U$. The point is that conversely, if we know which functions are smooth, we know how to put charts.

We can also easily define a smooth map between smooth manifolds.

Definition 4. A smooth map $f$ from $(M, \mathscr{O}_ M)$ to $(N, \mathscr{O}_ N)$ is a continuous map $f \colon M \to N$ such that for every open $U \subseteq N$ and $g \in \mathscr{O}_ N(U)$, the composition $$ g \circ f \colon f^{-1}(U) \to \mathbb{R} $$ is in $\mathscr{O}_ M(f^{-1}(U))$.

Again, this is seen to be equivalent to the usual notion, by putting charts on $U$ and looking at coordinate functions.

Real ringed spaces with evaluation#

One benefit of working with sheaves of rings is that now we can define certain spaces that can’t be accessed through the more traditional notion of a manifold. Let’s first define the scope of generality we are going to work in.

Definition 5. A real ringed space with evaluation is a triple $(M, \mathscr{O}_ M, \mathrm{ev}_ M)$ of

• a topological space $M$,
• a sheaf $\mathscr{O}_ M$ of $\mathbb{R}$-algebras on $M$,
• and a morphism $\mathrm{ev}_ M \colon \mathscr{O}_ M \to C^0(-, \mathbb{R})$ of sheaves of $\mathbb{R}$-algebras on $M$.

Given two real ringed spaces with evaluation $(M, \mathscr{O}_ M, \mathrm{ev}_ M)$ and $(N, \mathscr{O}_ N, \mathrm{ev}_ N)$, we define a morphism $f \colon M \to N$ as a data of

• a continuous map $f \colon M \to N$,
• and a morphism of sheaves of $\mathbb{R}$-algebras $\mathscr{O}_ N \to f_\ast \mathscr{O}_ M$ on $N$ respecting the evaluation map.

For $f \in \mathscr{O}_ M(U)$ and $p \in U$, we will just write $f(p) \in \mathbb{R}$ for $\mathrm{ev}_ M(f)(p)$.

Example 6. A smooth manifold is naturally a real ringed space with evaluation, since we defined $\mathscr{O}_ M(U)$ as a subring of $C^0(U, \mathbb{R})$.

Here is a much more interesting example that allows us to talk about infinitesimals.

Example 7. The infinitesimal vector $D$ is the real ringed space with evaluation defined so that

• the underlying topological space is just $\lbrace \ast \rbrace$,
• the structure sheaf is given by $$ \mathscr{O}_ D(U) = \begin{cases} \mathbb{R}[\epsilon]/(\epsilon^2) & U = D \br 0 & U = \emptyset, \end{cases} $$
• and the evaluation of $f = a + b\epsilon$ on the unique point $\ast$ is just $f(\ast) = a$.

The tangent bundle#

How should we think about this space $D$? Well, it’s something that looks like a point but with a infinitesimal vector sticking out of it.

Proposition 8. Let $M$ be a smooth manifold. Then a map $D \to M$ of real ringed spaces with evaluation is the same thing as a point $p \in M$ together with a tangent vector $v \in T_p M$.

So a map from $D$ into a smooth manifold $M$ is the same thing as a point in the tangent bundle. But the tangent bundle isn’t just a set; it itself is a smooth manifold. Is there any way for us to see this smooth structure?

Definition 9. Let $X$ be a smooth manifold. We define the real ringed space with evaluation $D \dot{\times} X$ as the space where the underlying topological space is just $X$ and the sheaf is given by $$ \mathscr{O}_ {D \dot{\times} X}(U) = \mathscr{O}_ X(U)[\epsilon]/(\epsilon^2), $$ and where evaluation map is given by sending $\epsilon \to 0$.

The reason there is a dot above the multiplication symbol is because it’s not really a product in a particular category at this point. It’s just a construction that behaves like a product.

Proposition 10. Let $X, M$ be smooth manifolds. Then a map $D \dot{\times} X \to M$ canonically corresponds to a map $X \to TM$ of smooth manifolds.

We can generalize this picture to higher-order infinitesimals. Define for $n \ge 0$ a real ringed space with evaluation $D_n$ with

• underlying topological space $\lbrace \ast \rbrace$,
• sheaf $\mathscr{O}(\ast) = \mathbb{R}[\epsilon]/(\epsilon^{n+1})$,
• evaluation given by $\mathbb{R}[\epsilon]/(\epsilon^{n+1}) \twoheadrightarrow \mathbb{R}[\epsilon]/(\epsilon) = \mathbb{R}$.

Then a map $D_n \to M$ is the same as a point $p \in M$ together with an equivalence class of curves $\gamma \colon \mathbb{R} \to M$ with $\gamma(0) = p$, where the equivalence relation is given by $\gamma_1 \sim \gamma_2$ if $\gamma_1^{(k)}(0) = \gamma_2^{(k)}(0)$ for all $0 \le k \le n$, where we take the derivative after putting local coordinates around $p$. This space can be given a natural smooth manifold structure that can be probed by mapping a similar space $D_n \dot{\times} X$ to $M$.

Formal neighborhoods#

Let’s slowly come back to the problem of gluing manifolds together. Let $M$ be an $n$-manifold with boundary. Let $B \subseteq M$ be either

• a union of connected components of $\partial M$, or
• a closed $n-1$-submanifold of $M$ disjoint from $\partial M$.

In both settings, we want to define a “thickening” of $B$ inside $M$ that captures some infinitesimal information around $B$.

Proposition 11. Let $f \colon \mathbb{R}^n \to \mathbb{R}$ or $f \colon \mathbb{R}^{n-1} \times [0, \infty) \to \mathbb{R}$ be a smooth function, and let $k \ge 0$ be an integer. The following conditions are equivalent:

1. the smooth functions $f, \partial_n f, \dotsc, \partial_n^k f$ vanish on $\mathbb{R}^{n-1} \times \lbrace 0 \rbrace$,
2. there exists a smooth function $g$ on the domain of $f$ such that $g = x_n^{k+1} f$,
3. for every map $D_k \to \mathbb{R}^n$ of real ringed spaces with evaluation whose image (as maps of topological spaces) lies in $\mathbb{R}^{n-1} \times \lbrace 0 \rbrace$, the pullback of $f$ onto $D_k$ is zero,
4. for every $0 \le l \le k$ and $D_l \to \mathbb{R}^n$ of real ringed spaces with evaluation whose image (as maps of topological spaces) lies in $\mathbb{R}^{n-1} \times \lbrace 0 \rbrace$, the pullback of $f$ onto $D_l$ is zero.

We now define the formal neighborhood of $M$ at $B$. Roughly, this should be a space that knows things going on near $B$, precisely up to some fixed order $k$. That means that this space cannot see functions that vanishes on $B$ up to order higher than $k+1$. This motivates the following proposition and definition.

Proposition 12. Let $U \subseteq B$ be an open subset, and let $\tilde{U} \subseteq M$ be an open subset such that $\tilde{U} \cap B = U$. Consider the $\mathbb{R}$-algebra map $$ C^\infty(\tilde{U}, \mathbb{R}) \to \prod_{\substack{0 \le l \le k, \br f \colon D_l \to M, \br f(\ast) \in U}}^{} \mathscr{O}_ {D_l}(\ast) $$ sending $g \in \mathscr{O}_ M(\tilde{U})$ to the element in $\mathscr{O}_ {D_k}(\ast)$ obtained by pulling back along $f$. The image of this map only depends on $U$ and not $\tilde{U}$.

Definition 13. For every open subset $U \subseteq B$ and integer $k \ge 0$, define the $\mathbb{R}$-algebra $$ \mathscr{O}_ {M_B^{k\wedge}}(U) = \im\Bigl( C^\infty(\tilde{U}, \mathbb{R}) \to \prod_{f \colon D^l \to M}^{} \mathbb{R}[\epsilon]/(\epsilon^{l+1}) \Bigr) $$ as above.

The reason we are including $0 \le l \le k$ (rather than using $l = k$) is so that there is natural is sequence of $\mathbb{R}$-algebra maps $$ \dotsb \twoheadrightarrow \mathscr{O}_ {M_B^{2\wedge}}(U) \twoheadrightarrow \mathscr{O}_ {M_B^{1\wedge}}(U) \twoheadrightarrow \mathscr{O}_ {M_B^{0\wedge}}(U) = C^\infty(U, \mathbb{R}). $$ Moreover, if $U^\prime \subseteq U$ is an open subset, we get a natural restriction map $$ \mathscr{O}_ {M_B^{k\wedge}}(U) \to \mathscr{O}_ {M_B^{k\wedge}}(U^\prime). $$

Proposition 14. The association $U \mapsto \mathscr{O}_ {M_B^{k\wedge}}(U)$ is a sheaf, i.e., if $U = \bigcup_i U_i$ for $U_i \subseteq B$ opens, the equalizer of $$ \prod_{i}^{} \mathscr{O}_ {M_B^{k\wedge}}(U_i) \rightrightarrows \prod_{i,j}^{} \mathscr{O}_ {M_B^{k\wedge}}(U_i \cap U_j) $$ precisely $\mathscr{O}_ {M_B^{k\wedge}}(U)$.

Finally, we can define the formal neighborhood of $B$ in $M$.

Definition 15. Let $M$ be a manifold with boundary and $B \subseteq \partial M$ be a union of connected components. We define the formal neighborhood of order $k$ as the real ringed space with evaluation $M_B^{k\wedge}$ with

• underlying topological space $B$,
• sheaf of rings $\mathscr{O}_ {M_B^{k\wedge}}(-)$ as defined above,
• the evaluation map coming from the natural map $\mathscr{O}_ {M_B^{k\wedge}}(-) \to \mathscr{O}_ {M_B^{0\wedge}}(-) = C^\infty(-, \mathbb{R})$.

There is a natural sequence of ringed real spaces $$ B = M_B^{0\wedge} \to M_B^{1\wedge} \to M_B^{2\wedge} \to \dotsb $$ where the underlying topological space doesn’t change, but the structure sheaf $\mathscr{O}$ changes.

Remark 16. The definition $M_B^{k\wedge}$ as a real ringed space with evaluation also makes sense when $B \subseteq M$ is an arbitrary closed subset. However, we will not make use of this generality.

Example 17. When we consider $M = B \times [0, \infty)$ with the boundary $M$, the formal neighborhood $M_B^{k\wedge}$ can be identified with $D_k \dot{\times} B$.

Gluing manifolds along boundaries#

Let us come back to the original setup:

• $M$ and $N$ are two $n$-manifolds with boundary,
• $B \subseteq \partial M$ and $C \subseteq \partial N$ are unions of connected components,
• $\varphi \colon B \cong C$ is a diffeomorphism.

We wanted to understand the space of smooth structures on $M \amalg_{B,\varphi} N$ that is compatible with the smooth structures on $M$ and $N$. First let’s think about what we can get from having a smooth structure.

Lemma 18. Assume $X = M \amalg_{B,\varphi} N$ has a smooth structure compatible with $M$ and $N$. Then the inclusion $M \hookrightarrow X$ induces an isomorphism $$ M_B^{k\wedge} \xrightarrow{\cong} X_B^{k\wedge} $$ for all $k \ge 0$.

So from a smooth structure on $X = M \amalg_{B,\varphi} N$, we get isomorphisms $$ M_B^{k\wedge} \cong X_B^{k\wedge} \cong N_C^{k\wedge} $$ of real ringed spaces with evaluation.

Definition 19. A formal gluing datum is a collection of isomorphisms $$ \varphi_k \colon M_B^{k\wedge} \cong N_C^{k\wedge} $$ of real ringed spaces with evaluations for each $k \ge 0$, fitting in the commutative diagram $$ \begin{CD} B = M_B^{0\wedge} @>>> M_B^{1\wedge} @>>> M_B^{2\wedge} @>>> \cdots \br @V{\varphi = \varphi_0}VV @V{\varphi_1}VV @V{\varphi_2}VV \br C = N_C^{0\wedge} @>>> N_C^{1\wedge} @>>> N_C^{2\wedge} @>>> \cdots \end{CD} $$ such that $\varphi_1$ reverses the orientation of the formal direction.

The last orientation condition is just to make sure that we don’t glue $M$ and $N$ along $B$ in the same direction. The observation we made above was that a smooth structure on $M \amalg_{B,\varphi} N$ induces a formal gluing datum. Now we come to the main theorem.

Theorem 20. There is a canonical bijection between

• the set of smooth structures on $M \amalg_{B,\varphi} N$ compatible with those on $M$ and $N$, and
• the set of formal gluing data.

So how can we recover a smooth structure from a formal gluing datum? The idea is that if two smooth functions defined on $M$ and $N$ agree on the intersection, and moreover all derivatives agree on the intersection, then it must be smooth.

Definition 21. Let $\varphi_\bullet$ be a formal gluing datum. For each open subset $U \subseteq X = M \amalg_{B,\varphi} N$, define a subring $$ \mathscr{O}_ {X,\varphi_\bullet}(U) \subseteq C^0(U, \mathbb{R}) $$ consisting of those functions $f \colon U \to \mathbb{R}$ satisfying the following properties:

• $f \vert_{M \cap U}$ and $f \vert_{N \cap U}$ are smooth,
• for each $k \ge 0$, the image of $f \vert_{M \cap U}$ under $C^\infty(M \cap U, \mathbb{R}) \to \mathscr{O}_ {M_B^{k\wedge}}(B \cap U)$ and the image of $f \vert_{N \cap U}$ under $C^\infty(N \cap U, \mathbb{R}) \to \mathscr{O}_ {N_C^{k\wedge}}(C \cap U)$ are identified via $\varphi_k$.

The following lemma tells us that if $\varphi_\bullet$ came from a smooth structure on $X$, then $\mathscr{O}_ {X,\varphi_\bullet}$ precisely picks out the smooth functions on $X$ with respect to the original smooth structure.

Lemma 22. Let $f \colon \mathbb{R}^n \to \mathbb{R}$ be a continuous function. Let $f_+$ and $f_-$ be the restrictions of $f$ to $\mathbb{R}^{n-1} \times [0, \infty)$ and $\mathbb{R}^{n-1} \times (-\infty, 0]$, respectively. Assume that

• $f_+$ and $f_-$ are smooth,
• for every map $\varphi \colon D_k \to \mathbb{R}^n$ of real ringed spaces with evaluation, with image lying in $\mathbb{R}^{n-1} \times \lbrace 0 \rbrace$, we have $\varphi^\ast f_+ = \varphi^\ast f_-$.

Then $f$ is smooth.

What remains to prove is that this sheaf $\mathscr{O}_ {X,\varphi_\bullet}$ always defines a smooth manifold. We will need the following lemma.

Lemma 23 (Borel). Let $f_0, f_1, \dotsc \in C^\infty(\mathbb{R}^{n-1}, \mathbb{R})$ be smooth functions. Then there exists a smooth function $f \in C^\infty(\mathbb{R}^n, \mathbb{R})$ whose Taylor expansion along $\mathbb{R}^{n-1} \times \lbrace 0 \rbrace$ is given by $f_0 + f_1 x_n + f_2 x_n^2 + \dotsb$.

Proposition 24. For every formal gluing datum $\varphi_\bullet$, the space $X$ together with the subsheaf $\mathscr{O}_ {X,\varphi_\bullet} \subseteq C^0(-, \mathbb{R})$ is a smooth manifold.

This finishes the proof!

The space of gluings#

So we have finally reduced the study of smooth structures on the glued manifolds to the study of formal gluing data. But how many formal gluing data are there? Do they even exist? To understand the space of formal gluing data, we will start from a particular $\varphi = \varphi_0$ and try to lift the isomorphism one by one. The first step is basically something we have seen.

Lemma 25. There canonically exists a fiber bundle over $B$ where the fiber over $p \in B$ parametrizes orientation reversing isomorphisms $(T_p M, T_p B) \cong (T_{\varphi(p)} N, T_{\varphi(p)} C)$.

This can also be thought of as an open subset of an affine bundle for the vector bundle $\Hom_B(TM \vert_B / TB, \varphi^\ast (TN \vert_C))$.

Proposition 26. Let $(M, B)$ and $(N, C)$ be two smooth manifolds and let $\varphi \colon B \cong C$ be an diffeomorphism. Then a lift $\varphi_1 \colon M_B^{1\wedge} \cong N_C^{1\wedge}$ of $\varphi$ that is orientation reversing corresponds to a smooth global section of of this fiber bundle.

Going to the next step is a bit tricky.

Proposition 27. Given a $\varphi_k \colon M_B^{k\wedge} \cong N_C^{k\wedge}$, a lift to $\varphi_{k+1} \colon M_B^{(k+1)\wedge} \cong N_C^{(k+1)\wedge}$ corresponds to a smooth section of a certain smooth affine bundle (depending on $\varphi_k$) over $X$, which is affine with respect to the vector bundle $\Hom((TM \vert_B / TB)^{\otimes (k+1)}, \varphi^\ast (TN \vert_C))$.

Remark 28. We really need to do this in steps. The crucial thing we need is $M_B^{k\wedge} \hookrightarrow M_B^{(k+1)\wedge}$ is a “square zero thickening.” If try to lift too much at once, we won’t have a fiber bundles whose sections correspond to extensions.

The rough idea is that given a Taylor expansion $\gamma(t) = v_0 + v_1 t + \dotsb + v_k t^k + O(t^{k+1})$, the number of ways we can lift it to a Taylor expansion of higher order roughly resembles the tangent space.

Proposition 29. Let $M$ be a smooth manifold (even possibly with boundary), and let $F \to M$ be an affine bundle. Then there exists a smooth global section.

Once there is a smooth global section, we can use that to trivialize the affine bundle and find an isomorphism $F \cong V$ over $M$, where $V$ is the vector bundle corresponding to $F$.

Corollary 30. Given a $\varphi_k$, there are infinitely many ways to extend it to $\varphi_{k+1}$ (as long as $B \neq \emptyset$).

So there indeed are lots and lots of ways to glue two smooth manifolds along their boundary. What we have now achieved are canonical bijections between the following:

• smooth structures on $M \amalg_{B,\varphi} N$ compatible with those on $M$ and $N$,
• formal gluing data $\varphi_\bullet \colon M_B^{\bullet\wedge} \cong N_C^{\bullet\wedge}$ extending $\varphi = \varphi_0$,
• iterative smooth global sections of certain contractible smooth fiber bundles.

Families of gluings#

So far we have tried to understand the space of gluings as a set. This is good to know, but we also want to understand it as a space. Giving it a structure of a topological space seems to be quite difficult, but we can still try to define it as a “smooth” space. That is, as we have done with the tangent bundle, we can define what a “smooth family” of gluings is.

Definition 31. Let $X$ be a “test” manifold (possibly with boundary). A family of gluings over $X$ is a smooth structure on $(M \times X) \amalg_{B \times X} (N \times X)$ such that the natural projection map $$ (M \times X) \amalg_{B \times X} (N \times X) \to X $$ is smooth.

In a similar way, we will have an interpretation in terms of a family of formal gluing data.

Definition 32. Let $X$ be a “test” manifold (possibly with boundary). A family of formal gluing data over $X$ is a compatible family of isomorphisms $$ \begin{CD} M_B^{k\wedge} \dot{\times} X @>{\varphi_k}>> N_C^{k\wedge} \dot{\times} X \br @VVV @VVV \br X @= X, \end{CD} $$ where we define $M_B^{k\wedge} \dot{\times} X = (M \times X)_ {B \times X}^{k\wedge}$.

Theorem 33. There is a canonical bijection between the set of families of gluings over $X$ and the set of families of formal gluing data over $X$.

These can be interpreted in terms of iterative sections of smooth affine bundles.

Proposition 34. Given a $\varphi_k \colon M_B^{k\wedge} \dot{\times} X \cong N_C^{k\wedge} \dot{\times} X$, the ways it lifts to $\varphi_{k+1}$ correspond to smooth sections of a certain affine bundle on $B \times X$.

Theorem 35. Let $(X, \partial X)$ be a smooth manifold with boundary. Then any family of gluings over $\partial X$ extends to a family of gluings over $X$.

What this is saying is that if we somehow think of the “space of gluings” as a space $\mathcal{M}$, then any smooth map $\partial X \to \mathcal{M}$ extends to a smooth map $X \to \mathcal{M}$. So essentially it is saying that $\mathcal{M}$ is weakly contractible!

Corollary 36. Assume $B \cong C$ is compact. Let $G_1, G_2$ be two smooth structures on $M \amalg_B N$. Then there exists a diffeomorphism $G_1 \cong G_2$.

Note that this cannot be the identity map on topological spaces!