Problem 11 - A First Course in Algebraic Topology

Question

(i) Prove that if \(\mathrm{h}:(0,1) \rightarrow(0.1)\) is a homeomorphism then there exists a homeomorphism \(\mathrm{f}:[0,1]\) such that \(\mathrm{f} \mid(0,1)=\mathrm{h}\). Moreover, prove that \(f\) is the unique such homeomorphism. (Hint: Look at the closed interval \((0, a]\) (closed in \(I\) ) and show that \(h((0, a])\) is of the form \((0, b]\) or \([c .1)\) for some \(b\) or \(c\).) (ii) Prove that if \(h: 1 \rightarrow 1\) is a homeomorphism then \(h(\partial I)=\partial I\). (Ilint: Use connectedness.) (iii) Suppose \(f, g: I \rightarrow X\) are paths in \(X\) so that \(f: 1 \rightarrow f(1)\) and \(g\) : \(I \rightarrow g(I)\) are homeomorphisms. Prove that if \(f(1)=g(I)\) then cither \(f \sim g\) or \(f \sim \bar{g}\). (Hint: Use (ii) above.) (iv) Suppose \(\mathrm{f}, \mathrm{g}: \mathrm{I} \rightarrow \mathrm{X}\) are closed paths in \(\mathrm{X}\) so that \(\mathrm{f}: \mathrm{I} \rightarrow \mathrm{f}(\mathrm{I})\) and \(\mathrm{g}: \mathrm{I} \rightarrow \mathrm{g}(\mathrm{I})\) are homeomorphisns. Prove that if \(\mathrm{f}(1)=\mathrm{g}(\mathrm{I})\) and \(f(\partial I)=g(\partial I)\) then cither \(f \sim g\) or \(f \sim \bar{g}\).

Step-by-Step Solution

Answer

The unique homeomorphism extends by mapping endpoints directly, ensuring boundary mapping in paths for homotopy.

Step 1: Understanding Homeomorphisms

A homeomorphism is a continuous, bijective function with a continuous inverse, acting as an isomorphism between topological spaces. The task requires extending a homeomorphism, \( h: (0,1) \rightarrow (0,1) \), to the boundary points of the interval [0,1].

Step 2: Extending the Homeomorphism

Given \( h: (0,1) \rightarrow (0,1) \) as a homeomorphism, to extend to \( f: [0,1] \rightarrow [0,1] \), we need to define values at 0 and 1. If \( h((0,a]) = (0,b]\), then map \( 0 \rightarrow 0 \). Similarly, \( h([a,1)) = [a,1)\) lets us map \( 1 \rightarrow 1 \). The resulting function \( f \) must be continuous.

Step 3: Uniqueness of f

The extension \( f \) is unique because for any other potential map differing at \( 0 \) or \( 1 \), continuity would be violated. Thus, any function with these properties also satisfies homeomorphism criteria throughout the extended domain.

Step 4: Applying to Part (ii)

For a homeomorphism \( h: I \rightarrow I \), since \( I \) is connected and \( \partial I = \{0, 1\}\), this boundary must map to itself due to connectedness; hence, \( h(\partial I)=\partial I \).

Step 5: Considering Paths in Part (iii)

Given \( f, g: I \rightarrow X \) are path homeomorphisms and \( f(I) = g(I) \) or inverse, use the result of Step 4 and connectedness to establish either homotopic to the other or to reverse.

Step 6: Extending to Closed Paths in Part (iv)

For closed paths \( f, g \) with similar initial and final point mapping, show with connectedness and endpoint equivalence, \( f \) must be homotopic to \( g \) or its reverse by continuity preserving mappings of boundaries as dictated by homeomorphism.

Key Concepts

Homeomorphism

A Homeomorphism is a special type of function that establishes an equivalence between two topological spaces. Simply put, it is a continuous, bijective function with a continuous inverse. This means that both the function and its inverse need to smoothly map all points between the two spaces without any breaks or interruptions.

The concept of homeomorphism is akin to creating a perfect stretchable and compressible model between two shapes. If you can distort one shape into another without tearing or gluing points together, you have a homeomorphism. The main aspects to confirm include:

• Continuity: Both the mapping from space A to B and its inverse from B to A must be continuous.
• Bijection: Each point in space A must match one and only one point in space B, and vice versa.
• Isomorphic Relation: The topological properties like connectedness and compactness must be preserved.

In the exercise, we extend a function defined on an open interval to the closed interval while maintaining homeomorphism. This involves carefully matching boundary points of the interval seamlessly.

Connectedness

Connectedness is a fundamental concept in topology, describing a space that cannot be split into two disjoint open sets. In essence, a connected space remains in one piece when you view it as a whole.

When dealing with a topological space like the closed interval [0,1], any continuous function mapping within it must respect its connected nature. This specifically means mapping the endpoints, or the boundary of the interval, must correspondingly map to themselves or remain intact.

• Continuous Mapping: Ensures no breaks, adhering to the space's innate connected character.
• Boundary Adherence: In our exercise, the transformation used this property to prove that boundary points will always map onto boundary points maintaining connectedness.

Applying this property in homeomorphisms, it becomes evident that transformations maintain the whole structure while extending functions naturally or dealing with enclosed paths.

Path Homotopy

Path Homotopy involves morphing one path into another continuously within a space. It’s like guiding a thread along the space without lifting it, managing to transform its shape smoothly into another path.

This concept is significant when dealing with paths that adhere to homeomorphic transformations, especially when the initial and final point mappings are the same. The operation helps determine if two paths can be considered equivalent by transformation or their reverse.

• Homotopic Paths: Paths that can be transformed into each other without leaving the hosting space.
• Reverse Homotopy: Sometimes reversing a path is also needed to maintain that each mapping considers continuity and boundary adherence, ensuring smooth transformation.

In exercises, path homotopies use this smooth continuity to prove or negate the equivalence of different mappings accordingly. This allows a deep dive into relationships, proving fundamental assertions on paths' existence and equivalency.

Continuous Function

A Continuous Function seamlessly maps elements from one space to another, preserving the locality of points without breaking or creating new connections. Simply put, if you move a point slightly in the domain, its image also moves slightly in the codomain.

This property is crucial for many mathematical and topological operations, including homeomorphisms. In a continuous function, no matter how small a change you make to the input, the change in output is just as small, meaning there are no sudden jumps.

• Preservation of Limits: Ensures each cluster of nearby points remains clustered, critical in topological spaces.
• Smooth Transitions: Essential in mapping between intervals, maintaining the same level of fidelity towards endpoints and interiors.
• Consistent at Boundaries: For interval operations, this means the function would naturally extend itself over boundaries without change in property.

The exercise applies the continuous property to guarantee smooth inclusivity across extensions and assure the mapping’s integrity both internally and at boundaries.