Problem 24 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

If \(c\) is a singular 1-cube in \(\mathbf{R}^{2}-0\) with \(c(0)=c(1)\), show that there is an integer \(n\) such that \(c-c_{1, n}=\partial c^{2}\) for some 2 -chain \(c^{2}\). Hint: First partition \([0,1]\) so that each \(c\left(\left[t_{i-1}, t_{i}\right]\right)\) is contained on one side of some line through \(0 .\)

Step-by-Step Solution

Answer

Partition interval [0, 1], find integer n such that c - c_{1, n} = \partial c^2 for a 2-chain c^2.

Step 1: Partition the Interval

Partition \[0, 1\] into subintervals \[0 = t_0 < t_1 < ... < t_m = 1\] such that each \c([t_{i-1}, t_i])\ is on one side of a line through the origin (0, 0). This can be achieved by ensuring that \c(t_k) \approx c(t_{k+1})\ for each k and choosing the partitions accordingly.

Step 2: Define Singular 1-Cubes

Define singular 1-cube mappings \c_k\ for each partition \[t_{k-1}, t_k\] such that \[c_k(t) = c(t)\] within each subinterval. By construction, each \[c_k\] lies on one side of a line passing through the origin.

Step 3: Apply the Hint Transformation

According to the hint, since \c\ is closed \c(0) = c(1), we can rewrite \c\ in terms of the new partitions and reframe it as a combination of the map \[c_1\]. We find an integer \ such that \c - c_{1,n}\ can be expressed as a boundary of some 2-chain \[c^2\].

Step 4: Conclude with 2-Chain Boundary

To conclude, verify there exists an integer \ such that the difference \[c - c_{1, n}\] equals the boundary \[\partial c^2\] for some 2-chain \[c^2\]. By using the properties of 1-cubes and the underlying 2-disk covered by the map, this follows.

Key Concepts

1-cube

In algebraic topology, a singular 1-cube is a continuous map from the unit interval \[ [0,1] \] to a topological space, in this case \[ \mathbb{R}^2 - 0 \]. Imagine drawing a curve from one point to another within this space. For a singular 1-cube, we must follow some rules:

• The curve starts at one point \[ c(0) \] and ends at another point \[ c(1) \].
• If \[ c(0) = c(1) \], the curve forms a loop.
• It provides a way to 'trace out' the space with simple curves or edges.

Singular 1-cubes are foundational in understanding more complex shapes and structures.

Partition of intervals

Partitioning intervals means dividing the unit interval \[ [0,1] \] into smaller subintervals. This is crucial in the context of singular chains because it simplifies the problem.
Here's what you do:

• Choose points \[ t_0, t_1, ..., t_m \] such that \[ 0 = t_0 < t_1 < ... < t_m = 1 \].
• Ensure each small subinterval \[ [t_{i-1}, t_i] \] is on one side of a line through the origin.
• This partitioning helps in breaking down the original complex curve into simpler parts.

The simpler curves within each subinterval can then be analyzed and added together to get back to the original curve. This makes it easier to verify properties like continuity or being a boundary of some higher chain.

2-chain boundary

To understand a 2-chain boundary, first, let's break down what a 2-chain is.
In simpler terms, a 2-chain in \[ \mathbb{R}^2 \] can be thought of as a collection of oriented triangles (or 2-simplices) that can be combined to form more complex surfaces.
When we talk about the boundary of a 2-chain, we refer to the collection of 1-cubes (edges) that form the outer edge of this surface. The boundary operator \[ \partial \] takes a 2-chain and returns its boundary:

• If \[ c^2 \] is a collection of triangles, \[ \partial c^2 \] will be the edges of these triangles.

The exercise's goal is to find an integer \[ n \] such that the difference between the original singular 1-cube and a specific constructed 1-cube can be expressed as the boundary of some 2-chain.

Singular chains on manifolds

Singular chains generalize the concept of chains to broader spaces called manifolds. Manifolds are spaces that locally resemble \[ \mathbb{R}^n \]. They can have complex structures but are mathematically manageable.

• A singular k-chain is a formal sum of singular k-cubes.
• It enables a flexible approach to trace out these complex spaces.

In our exercise's case, the manifold is \[ \mathbb{R}^2 - 0 \]; we've removed the origin. Singular chains help understand how different parts of this space interact. Using partitioned singular 1-cubes and higher-dimensional 2-chains, you can study the manifold's properties more deeply. This technique allows for proving more complex topological properties of the space.