Determine if a specific topological quotient space is diffeomorphic to the original space under given geometric and group action conditions | Step-by-Step Solution

Understanding Quotient Spaces and Group Actions on Manifolds

What We're Solving:

We need to determine whether the quotient space $(S^2 \times [-1,1])/\mathbb{Z}_2$ is diffeomorphic to the original space $S^2 \times [-1,1]$, where the $\mathbb{Z}_2$-action (involution) has a fixed point set consisting of two strands connecting the boundary spheres.

The Approach:

This is a problem that combines group actions, quotient topology, and differential geometry! We'll analyze this systematically by:

• Understanding what the involution looks like geometrically
• Examining the fixed point set and its implications
• Using topological invariants to determine if the spaces can be diffeomorphic

Step-by-Step Solution:

Step 1: Visualize the Setup Think of $S^2 \times [-1,1]$ as a "thick sphere" - imagine taking a 2-sphere and thickening it into a 3-dimensional region. The boundary consists of two spheres: $S^2 \times \{-1\}$ and $S^2 \times \{1\}$.

Step 2: Understand the Fixed Point Set The fixed point set consists of "two strands connecting boundary spheres." This means we have two curves (1-dimensional) that each connect a point on $S^2 \times \{-1\}$ to a point on $S^2 \times \{1\}$. Let's call these strands $\gamma_1$ and $\gamma_2$.

Step 3: Analyze the Involution Since we have an involution $\tau: S^2 \times [-1,1] \to S^2 \times [-1,1]$ with $\tau^2 = \text{id}$, and the fixed point set has dimension 1, this tells us something crucial about how the involution acts on the complement of the fixed set.

Step 4: Apply Topological Analysis Here's the key insight: Let's examine the Euler characteristic and fundamental group.

For the original space $S^2 \times [-1,1]$:

• This space is homotopy equivalent to $S^2$
• $\pi_1(S^2 \times [-1,1]) = 0$ (simply connected)
• The space is orientable

Step 5: Examine the Quotient Space When we take the quotient by the $\mathbb{Z}_2$ action:

• Points not on the fixed set are identified in pairs
• The two strands remain as "singular" curves in some sense
• The quotient will have a different topology

Step 6: Use the Fixed Point Structure The crucial observation is that having exactly two separate fixed point strands creates a topological obstruction. The quotient space will have a fundamental group that's non-trivial because loops can wind around these strands in ways that cannot be continuously deformed away.

The Answer:

No, the quotient space is NOT diffeomorphic to $S^2 \times [-1,1]$.

The key reason is that the quotient space $(S^2 \times [-1,1])/\mathbb{Z}_2$ has a non-trivial fundamental group (specifically $\mathbb{Z}_2$), while $S^2 \times [-1,1]$ is simply connected. Since diffeomorphic spaces must have isomorphic fundamental groups, they cannot be diffeomorphic.

Additionally, the presence of two separate fixed point strands creates a topological structure that cannot be "undone" by any smooth deformation.

Memory Tip:

Remember the mantra: "Fixed points leave fingerprints!" When a group action has isolated fixed point sets (especially multiple components), the quotient space often has fundamentally different topology from the original. The number and arrangement of fixed point components directly affect the fundamental group of the quotient!

Great work tackling such an advanced topology problem! These quotient space problems really showcase how group actions can create fascinating new geometric objects.