Smoothing 3-manifolds in 5-manifolds

The paper proves that every locally flat topological embedding of a 3-manifold into a smooth 5-manifold can be approximated by a smooth embedding via a small homotopy, thereby establishing that topologically locally flat concordance implies smooth concordance for surfaces in 4-manifolds.

The Gist

Here is an explanation of the paper "Smoothing 3-Manifolds in 5-Manifolds" by Michelle Daher and Mark Powell, translated into everyday language with creative analogies.

The Big Picture: Straightening Out a Crumpled Sheet

Imagine you have a piece of fabric (a 3-manifold) that you want to lay perfectly flat on a large, smooth table (a 5-manifold).

In the world of mathematics, there are two ways to look at this fabric:

1. Topological: The fabric is made of a stretchy, rubbery material. It can be crumpled, twisted, and knotted, as long as it doesn't tear or pass through itself. It's "locally flat," meaning if you zoom in close enough, it looks like a flat sheet, even if the whole thing is a mess.
2. Smooth: The fabric is made of perfect silk. It has no kinks, no sharp corners, and flows perfectly.

The Problem: Mathematicians knew that you could usually turn a crumpled rubber sheet into a smooth silk one if the table was big enough (high dimensions) or the sheet was small enough (low dimensions). But there was a tricky middle ground: 3-dimensional sheets on 5-dimensional tables.

For a long time, mathematicians weren't sure if you could always take a crumpled rubber sheet and turn it into a smooth silk one without tearing it. The paper by Daher and Powell says: "Yes, you can!"

But there's a catch: You can't just push the crumpled sheet into a smooth shape (an "isotopy"). Sometimes, the knot is too tight. Instead, you have to gently stretch and wiggle the sheet (a "homotopy") to untangle it just enough to make it smooth.

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The Two-Step Dance

The authors prove this by breaking the problem down into two main steps, like a two-part dance routine.

Step 1: The "Magic Glue" Fix (Dealing with the Knots)

Imagine your rubber sheet is tangled in a way that creates a "knot" in the fabric of the universe around it. In math, this is called an obstruction. Sometimes, the way the sheet sits in the 5D space creates a "glitch" that prevents it from being smooth.

• The Analogy: Think of the 5D table as a giant room. Your rubber sheet is floating in the middle. Sometimes, the room itself has a weird "twist" in its geometry that makes it impossible for the sheet to be smooth.
• The Solution: The authors use a special tool called Lashof's Knot. Imagine this as a "magic knot" that, when you tie it into your sheet, cancels out the weird twist in the room.
• The Action: They take the crumpled sheet and perform a tiny, almost invisible surgery. They tie in these magic knots (which are actually just tiny, specific 3D shapes) to neutralize the geometric glitches.
• The Result: Suddenly, the sheet can be smooth, but only if you change the definition of "smooth" for the whole room slightly. The sheet is now smooth, but the room's "smoothness" is different from the original room.

Step 2: The "Local Patch" Fix (Matching the Original Room)

Now the sheet is smooth, but it's sitting in a "modified" room. We want it to be smooth in the original room (the standard 5-manifold).

• The Analogy: Imagine you fixed the sheet, but now the floor tiles under it are slightly different colors than the rest of the room. You need to make the sheet look perfect against the original floor.
• The Problem: The sheet and the original floor don't match perfectly in a few tiny spots. These spots are like tiny "bumps" where the smoothness clashes.
• The Solution: The authors use a technique inspired by Kervaire's Theorem (which proved that 2D knots in 4D space can always be untangled). They realize that these "bumps" are actually just small, isolated problems.
• The Action: They zoom in on each bump. They show that inside each tiny bump, the sheet is just a simple ball that can be reshaped. They replace the "bumpy" part of the sheet with a perfectly smooth "slice" that fits the original room's rules.
• The Result: The sheet is now perfectly smooth and fits perfectly into the original room.

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Why Does This Matter? (The "Concordance" Connection)

The paper has a cool side effect (Corollary 1.2) that applies to 4-dimensional spaces (like our universe, but with an extra dimension).

Imagine you have two smooth surfaces (like two different shapes of soap bubbles) floating in a 4D space.

• Topological Concordance: You can morph one bubble into the other if you allow the path between them to be crumpled and rubbery.
• Smooth Concordance: You can morph one into the other if the path must be perfectly smooth silk.

The Big Discovery: The authors prove that if you can morph them using the "rubbery" method, you can also morph them using the "silk" method.

• The Metaphor: If you can drive a car from City A to City B by driving off-road through mud and rocks (topological), you can also find a paved highway to do it (smooth). You don't need to invent a new car; you just need to find the right path.

Summary of the "Magic"

1. The Obstacle: Sometimes, a shape is so knotted in a 5D space that it can't be smoothed without changing the space itself.
2. The Trick: The authors use "Lashof's Knot" to untangle the space's geometry, allowing the shape to become smooth (Step 1).
3. The Cleanup: They then use local surgery to fix the few remaining spots where the new smoothness clashes with the old rules (Step 2).
4. The Payoff: This proves that for 3D shapes in 5D space, "rubbery" smoothness and "perfect" smoothness are essentially the same thing, provided you are willing to wiggle the shape a little bit.

In a nutshell: You can always smooth out a crumpled 3D sheet in a 5D room, as long as you are allowed to stretch it a tiny bit and tie in a few magic knots to fix the room's geometry first.

Technical Summary

Here is a detailed technical summary of the paper "Smoothing 3-Manifolds in 5-Manifolds" by Michelle Daher and Mark Powell.

1. Problem Statement

The paper addresses a fundamental question in geometric topology regarding the relationship between topological and smooth categories in codimension-two embeddings. Specifically, it investigates whether a locally flat topological embedding of a compact 3-manifold $Y$ into a smooth 5-manifold $N$ can be approximated by a smooth embedding.

• Context: In low dimensions (e.g., 3-manifolds in 3-manifolds), locally flat embeddings are isotopic to smooth ones. In dimension 4, there exist topologically slice knots that are not smoothly slice, implying that some locally flat embeddings of disks in $D^4$ cannot even be homotoped to smooth embeddings.
• The Gap: For codimension-two embeddings of 3-manifolds in 5-manifolds, it was unknown if every locally flat topological embedding is homotopic (via a small homotopy) to a smooth embedding.
• The Obstruction: The authors note that one cannot generally isotope a locally flat embedding to a smooth one. Lashof [Las71] constructed a locally flat 3-knot ($S^3 \subset S^5$) that is not even concordant to any smooth knot. Therefore, the authors aim to prove that a small homotopy (rather than an isotopy) suffices to achieve smoothness.

2. Methodology

The proof of the main theorem (Theorem A) is structured into two distinct steps, utilizing Smoothing Theory (Kirby-Siebenmann theory) and specific constructions involving Lashof's knot.

Step 1: Smoothing the Complement (The Global Obstruction)

The authors decompose the 5-manifold $N$ into a tubular neighborhood $\nu M$ of the image $M = f(Y)$ and its exterior $W_f = N \setminus \nu M$.

• Goal: Extend the smooth structure from the boundary $\partial N \cup \partial \nu M$ to the entire manifold $N$.
• Obstruction: Smoothing theory provides a Kirby-Siebenmann obstruction $ks(W_f, \partial W_f) \in H^4(W_f, \partial W_f; \mathbb{Z}/2)$. If this class vanishes, the smooth structure extends.
• Strategy: The authors show that this obstruction lies in a subgroup generated by meridians of the components of $M$. They utilize Lashof's non-smoothable 3-knot ($L \subset S^5$), which has a non-trivial Kirby-Siebenmann invariant in its exterior.
• Mechanism: By performing ambient connected sums of the components of $M$ with copies of Lashof's knot $L$, the authors can "cancel out" the obstruction. Specifically, if the obstruction is non-zero for a component, they connect sum that component with $L$. This modifies the embedding $f$ via a small homotopy to a new embedding $g$ such that the obstruction vanishes.
• Result: $f$ is homotopic to an embedding $g$ that is smooth with respect to some smooth structure $\sigma$ on $N$ (which agrees with the standard structure near $\partial N$).

Step 2: Comparing Smooth Structures (The Local Obstruction)

At this stage, $g(Y)$ is smooth in the structure $\sigma$, but $N$ is originally equipped with a standard structure $\text{std}$. The structures $\sigma$ and $\text{std}$ may differ.

• Obstruction: The difference between $\sigma$ and $\text{std}$ is measured by an element $ks(\sigma, \text{std}) \in H^3(N, \partial N; \mathbb{Z}/2)$. By Poincaré duality, this corresponds to a class in $H_2(N; \mathbb{Z}/2)$.
• Strategy: The authors represent this class by a closed surface $S \subset \text{Int}(N)$ that is transverse to $g(Y)$.
• Reduction: Using transversality, the intersection $g(Y) \cap S$ consists of finitely many points. The problem reduces to local neighborhoods around these points.
• Resolution: In each local neighborhood, the problem becomes finding a smooth "slice" for the intersection. The authors invoke a generalization of Kervaire's theorem (that every 2-knot is smoothly slice) and Sunukjian's results on concordance of surfaces in 4-manifolds. They construct a smooth 3-ball (slice disc) in the standard structure that replaces the local intersection, effectively removing the discrepancy between $\sigma$ and $\text{std}$.
• Result: The embedding $g$ is homotoped (via a small homotopy) to a final embedding $g'$ that is smooth with respect to the original standard structure $\text{std}$.

3. Key Contributions and Results

Main Theorem (Theorem A)

Let $f: Y \to N$ be a locally flat proper topological embedding of a compact 3-manifold into a smooth 5-manifold, smooth near the boundary. Then $f$ is homotopic rel. boundary, via an arbitrarily small homotopy, to a smooth embedding.

Corollary: Concordance of Surfaces in 4-Manifolds

The authors deduce a significant result for 4-manifolds:

Corollary 1.2: Topological concordance implies smooth concordance for surfaces in smooth 4-manifolds.
If two smooth surfaces $\Sigma_0, \Sigma_1$ in a 4-manifold $X$ are topologically concordant (connected by a locally flat submanifold $\Sigma \times I$), they are also smoothly concordant.

This follows by applying Theorem A to the concordance $C \cong \Sigma \times I$ embedded in $X \times I$ (a 5-manifold).

Conditions for Isotopy (Section 5)

The paper clarifies when smoothing can be achieved via isotopy rather than just homotopy:

1. If the Kirby-Siebenmann obstruction of the exterior vanishes ($ks(W_f, \partial W_f) = 0$), no Lashof knots are needed, and the embedding is smooth in some structure.
2. If, additionally, the relative obstruction $ks(\sigma, \text{std})$ has zero intersection with the image of the manifold, the embedding can be isotoped to a smooth one.
   This highlights that the necessity of a homotopy (rather than an isotopy) arises precisely when these obstructions are non-trivial.

4. Significance and Impact

• Bridging Categories: The paper resolves a long-standing gap in the classification of embeddings in dimension 5. It establishes that while topological and smooth categories differ (as shown by Lashof's knot), they are "close" enough that any topological embedding can be made smooth with a small homotopy.
• Concordance Theory: The result unifies topological and smooth concordance for surfaces in 4-manifolds. Previously, obstructions were known in the smooth category (e.g., in [Sto93], [Sch19]), but this result implies that any topological concordance obstruction is automatically a smooth one. This simplifies the study of surface concordance, as researchers need not distinguish between the two categories for existence questions.
• Methodological Innovation: The proof creatively combines:
  • Kirby-Siebenmann obstruction theory to handle global smoothing.
  • Lashof's knot as a "corrective tool" to cancel obstructions via connected sums.
  • Sunukjian's generalization of Kervaire's slice theorem to handle the local differences between smooth structures.
• Dimensional Context: The result is specific to dimension 5. The authors note that for $m \ge 5$, Schultz's theorems apply (where isotopy is often possible), but for $m=3$ in $n=5$, the situation is unique and requires the homotopy approach described here.

In summary, Daher and Powell demonstrate that the "roughness" of a topological 3-manifold in a smooth 5-manifold can be smoothed out by a small homotopy, effectively proving that the topological and smooth categories of such embeddings are equivalent up to homotopy.