Problem 13 - A First Course in Algebraic Topology

Question

Prove that if \(\mathrm{A}\) is a strong deformation retract of \(\mathrm{X}\) then the inclusion map i: \(\mathrm{A} \rightarrow \mathrm{X}\) induces an isomorphism $$ \mathrm{i}_{*}: \pi(\mathrm{A}, \mathrm{a}) \rightarrow \pi(\mathrm{X}, \mathrm{a}) $$ for any point \(\mathrm{a} \in \mathrm{A}\).

Step-by-Step Solution

Answer

The inclusion map induces an isomorphism because the loop's properties in both spaces are preserved.

Step 1: Understanding the Concept

A strong deformation retract of a space \( \mathrm{X} \) onto a subspace \( \mathrm{A} \) means there exists a homotopy \( F: \mathrm{X} \times [0, 1] \to \mathrm{X} \) such that \( F(x, 0) = x \), \( F(x, 1) \in \mathrm{A} \) for all \( x \in \mathrm{X} \), \( F(a, t) = a \) for all \( a \in \mathrm{A} \) and \( t \in [0, 1] \). This essentially suggests that \( \mathrm{A} \) and \( \mathrm{X} \) have the same global topological properties up to homotopy.

Step 2: Setup the Map i

The inclusion map \( i: \mathrm{A} \rightarrow \mathrm{X} \) is defined simply as including each element of \( \mathrm{A} \) as part of \( \mathrm{X} \). This will maintain the position, features, and properties of \( \mathrm{A} \) as a subspace in \( \mathrm{X} \).

Step 3: Showing the Map i* Induces an Isomorphism

We need to show that the map \( i_*: \pi_1(\mathrm{A}, \mathrm{a}) \rightarrow \pi_1(\mathrm{X}, \mathrm{a}) \) is an isomorphism. A homotopy \( H \) shows \( i_* \) is both injective and surjective. Injectivity follows because any loop in \( \mathrm{A} \) that is homotopically trivial in \( \mathrm{X} \) will remain trivial when mapped back to \( \mathrm{A} \) via the retraction. Surjectivity follows because any loop in \( \mathrm{X} \) based at \( \mathrm{a} \) can be deformed to lie entirely within \( \mathrm{A} \), thanks to the properties of the deformation retract.

Step 4: Verify Homotopy Equivalence

The result holds since the homotopy equivalence between \( \mathrm{A} \) and \( \mathrm{X} \) implies there exists a retraction map \( r: \mathrm{X} \rightarrow \mathrm{A} \) such that \( r \circ i \) is the identity map on \( \mathrm{A} \). Therefore, for any loop in \( \mathrm{X} \), \( r \) maps it back into loops in \( \mathrm{A} \), showing that \( i_* \) is bijective, hence an isomorphism.

Key Concepts

Strong Deformation Retract

In algebraic topology, a strong deformation retract provides insight into how a subspace can mirror the structure of the larger space it is embedded in. Specifically, when a space \( \mathrm{X} \) can be transformed into a subspace \( \mathrm{A} \) through a continuous process, \( \mathrm{A} \) acts as a strong deformation retract of \( \mathrm{X} \). This is formalized using a homotopy \( F: \mathrm{X} \times [0, 1] \to \mathrm{X} \), which satisfies:

• \( F(x, 0) = x \) for all \( x \in \mathrm{X} \)
• \( F(x, 1) \in \mathrm{A} \) for all \( x \in \mathrm{X} \)
• \( F(a, t) = a \) for all \( a \in \mathrm{A}, \) and \( t \in [0, 1] \)

This essentially means one can "retract" \( \mathrm{X} \) onto \( \mathrm{A} \) in such a way that for points already on \( \mathrm{A} \), nothing changes. Hence, the subspace \( \mathrm{A} \) retains all the essential topological features of \( \mathrm{X} \), similar to how a sculpture can represent a larger model. Understanding this concept helps in simplifying complex spaces by focusing on their central characteristics.

Fundamental Group

The fundamental group is a powerful tool in algebraic topology used to study topological spaces. Denoted as \( \pi_1(X, a) \), it captures the essence of a space's shape and properties based on loops emanating from a point \( a \). This group consists of equivalence classes of loops, where two loops are considered equivalent if one can be continuously transformed into the other without breaking or tearing the space.

• The power of this group comes from its ability to distinguish between different topological spaces.
• If two spaces have different fundamental groups, they are not topologically equivalent.
• The operation to combine two loops, called the loop product, forms the group structure.

For instance, the fundamental group of a circle is infinite, reflecting the infinite number of ways a loop may wrap around it. When a strong deformation retract \( A \) exists from a space \( X \), it is particularly interesting that both share the same fundamental group, signifying a deep connection between the two structures.

Homotopy

Homotopy is a central concept in algebraic topology that explores the idea of continuously transforming one function or shape into another. When discussing topological spaces, a homotopy establishes a continuous deformation between two maps \( f, g: X \to Y \) over a parameter \( t \) in the interval \([0, 1]\). This continuous transformation is represented as \( H(x, t) \) such that:

• \( H(x, 0) = f(x) \) and \( H(x, 1) = g(x) \)
• For each fixed \( t \), \( H(x, t) \) itself is a function from \( X \) to \( Y \)

Homotopy is more than a tool for studying shapes. It reveals deeper relationships between spaces. Two spaces are deemed homotopy equivalent if they can be continuously transformed into each other, showcasing that they share core properties even if they appear different at first glance. This equivalence supports the results about deformation retracts and fundamental groups in our problem.

Isomorphism

In algebraic topology, an isomorphism implies a strong form of equality between mathematical structures. When we say that the inclusion map \( i_{*}: \pi_1(A, a) \to \pi_1(X, a) \) induces an isomorphism, it indicates that the fundamental groups of \( A \) and \( X \) are structurally the same. This means each element in \( \pi_1(A, a) \) corresponds uniquely to an element in \( \pi_1(X, a) \) and vice-versa.

• An isomorphism is bijective, covering both injectiveness (one-to-one) and surjectiveness (onto).
• It preserves the group operation, meaning if two loops are combined in \( A \), their image in \( X \) under the inclusion map behaves identically.
• In essence, it shows that two groups are indistinguishably equivalent in structure and function, thus serving as an algebraic manifestation of topological similarity.

Understanding this concept is crucial when analyzing spaces reduced by deformation retracts, ensuring that the core attributes are preserved across reductions and simplifications.