Surgery Theory

Surgery theory is a fundamental tool and framework in geometric topology and differential topology that enables the systematic construction and analysis of high-dimensional manifolds by cutting and pasting operations. It provides a method for modifying manifolds in a controlled manner, with the aim of simplifying their structure or transforming them into other well-understood manifolds. This approach is instrumental in the classification of manifolds, especially in dimensions greater than four.

Basic Idea

The essence of surgery involves the following steps:

1. Removing a Submanifold: Start with a manifold $M$ and select a submanifold $N$ of lower dimension. Typically, $N$ is chosen to be homeomorphic to $S^k\times D^{n-k}$, where $S^k$ is a $k$-dimensional sphere and $D^{n-k}$ is an $(n-k)$-dimensional disk, with $n$ being the dimension of the original manifold $M$.

2. Gluing Back a Different Piece: Remove $N$ from $M$ and replace it with another manifold $P$ that has the same boundary as $N$. Often, $P$ is homeomorphic to $D^{k+1}\times S^{n-k-1}$. This step is akin to cutting out a hole and filling it with a “plug” that fits the hole’s boundary but may have a different interior structure.

3. Resulting Manifold: The result of this process is a new manifold that has been altered through this cutting and pasting operation. The aim is to change specific topological or geometric features of the original manifold without affecting others.

Objectives and Applications

• Classification of Manifolds: One of the primary goals of surgery theory is to classify manifolds up to homeomorphism or diffeomorphism. It particularly shines in dimensions 5 and higher, where it contributes to the classification and understanding of smooth structures on manifolds.

• Study of High-Dimensional Topology: Surgery theory provides insights into the structure and properties of manifolds in higher dimensions, helping to resolve questions about which manifolds can exist in certain dimensions and how they relate to each other.

• Understanding Manifold Invariants: Through surgery, mathematicians can analyze how various invariants of manifolds (such as homotopy groups, homology groups, and others) change. This is crucial for the development of algebraic and geometric topology.

Significance

Surgery theory has had significant impacts on mathematics:

• h-Cobordism Theorem: Surgery theory is instrumental in the proof of the h-cobordism theorem, which states that two simply connected smooth manifolds of dimension greater than 4 that are h-cobordant are actually diffeomorphic. This result, proven by Smale, was groundbreaking in the study of high-dimensional manifolds.

• Poincaré Conjecture in Higher Dimensions: Surgery techniques were pivotal in the proof of the generalized Poincaré conjecture in dimensions greater than four, accomplished by Stephen Smale, and later extended to dimension 4 by Michael Freedman.

• Exotic Spheres: The discovery and classification of exotic spheres, or smooth manifolds that are homeomorphic but not diffeomorphic to standard spheres, rely heavily on surgery theory.

Surgery theory highlights the deep interconnections between algebra, geometry, and topology, offering a powerful framework for understanding the complex world of manifolds.