Determine whether two quotient spaces derived from S³ by collapsing different subspaces are homeomorphic | Step-by-Step Solution

Understanding Quotient Spaces and Homeomorphisms

What We're Solving:

We need to determine if two quotient spaces are homeomorphic: S³/(S¹ × D²) and S³/S¹. This means we're "collapsing" different subspaces within the 3-sphere and asking if the resulting spaces have the same topological structure.

The Approach:

The key insight is to analyze what happens near the "collapsed points" in each quotient space. When we quotient a space X by a subspace A, we're essentially gluing all points in A together into a single point. The topology near this collapsed point tells us a lot about the overall structure. We'll:

• 1. Understand what each quotient space looks like
• 2. Examine neighborhoods around the collapsed points
• 3. Compare their local topological properties

Step-by-Step Solution:

Step 1: Visualize S³/(S¹ × D²)

• Think of S³ as sitting in ℝ⁴, and S¹ × D² as a "solid torus" inside it
• When we collapse the entire solid torus S¹ × D² to a point, we get S³/(S¹ × D²)
• This is actually homeomorphic to S² (the 2-sphere)!
• Why? The complement of S¹ × D² in S³ is another solid torus, and when we collapse the first solid torus to a point, we get a space that's topologically a 2-sphere

Step 2: Visualize S³/S¹

• Here we're only collapsing a circle S¹ (not a solid torus) to a point
• Think of S¹ as a simple loop inside S³
• The quotient S³/S¹ has a more complex structure

Step 3: Analyze neighborhoods of collapsed points

For S³/(S¹ × D²) ≅ S²:

• Near the collapsed point, neighborhoods look like neighborhoods in S²
• Every neighborhood of the collapsed point is homeomorphic to a disk D²
• This gives us the "round" structure of a 2-sphere

For S³/S¹:

• Near the collapsed point, we need to understand what the original neighborhoods of S¹ in S³ looked like
• A tubular neighborhood of S¹ in S³ looks like S¹ × D², but we're only collapsing the S¹ factor
• After quotient, neighborhoods of the collapsed point look like cone-like structures over S¹
• The space S³/S¹ is actually homeomorphic to a cone over the real projective plane!

Step 4: Compare the structures

• S³/(S¹ × D²) ≅ S² has neighborhoods that are always disk-like
• S³/S¹ has more complex neighborhoods around the collapsed point
• These different local structures mean the spaces cannot be homeomorphic

The Answer:

No, S³/(S¹ × D²) and S³/S¹ are not homeomorphic.

S³/(S¹ × D²) ≅ S² (2-sphere), while S³/S¹ has a more complex structure with cone-like neighborhoods around the collapsed point. The fundamental difference in their local topology near the collapsed points proves they're not homeomorphic.

Memory Tip:

Remember: "The more you collapse, the simpler it gets!" Collapsing the entire solid torus (S¹ × D²) gives you the nice, smooth S². Collapsing just the circle (S¹) leaves behind a more complex, cone-like structure. The size of what you're collapsing dramatically affects the resulting topology!

Great work tackling this advanced topology problem! Understanding quotient spaces takes practice, but analyzing local neighborhoods is always your best friend for determining homeomorphisms.