About this video

• Video Title: What is...homotopy?
• Channel: VisualMath
• Speakers: Not specified
• Duration: 18:20

Overview

This video introduces the concept of homotopy equivalence as a notion of equivalence in algebraic topology. It contrasts homotopy equivalence with homeomorphism, explaining that homotopy equivalence is a weaker, more flexible concept that is particularly useful for algebraic topology as many algebraic constructions are invariant under it. The video uses examples like the letters of the alphabet and the Mobius strip to illustrate how homotopy equivalence relates to the "number of holes" in a topological space.

Key takeaways

• Homeomorphism vs. Homotopy Equivalence: Homeomorphism is a strict equivalence where two spaces are topologically identical. Homotopy equivalence is a weaker notion, allowing for continuous deformations (like stretching or squeezing) that preserve certain topological features, such as the number of holes.
• Deformation Retraction: A key concept for understanding homotopy equivalence is deformation retraction. A space X has a deformation retract onto a subspace A if A can be continuously shrunk down from X, while staying within X, and A itself remains fixed.
• Counting Holes: A useful intuition for homotopy equivalence is that it essentially classifies spaces based on their "holes." Spaces with the same number of holes are often homotopy equivalent.
• Application in Algebraic Topology: Homotopy equivalence is the preferred notion of equivalence in algebraic topology because many algebraic invariants (like homology or homotopy groups) are preserved under this type of equivalence, whereas they might not be preserved under stricter equivalences like homeomorphism.
• Weakness of Homotopy Equivalence: While useful, homotopy equivalence is a weak notion, meaning that many spaces that seem topologically different can be homotopy equivalent (e.g., a house with two rooms is homotopy equivalent to a point).

Imagine you have two shapes made of play-doh.

A homeomorphism is like saying these two shapes are exactly the same, just maybe turned or moved around a bit. You can't stretch, tear, or cut them. If one has a hole, the other must have a hole in the exact same spot and shape.

Homotopy equivalence is more like saying these shapes are the "same type" of thing, even if they aren't perfectly identical. You can stretch and squish them, but you can't create or destroy holes.

Think of it this way:

• A ball and a cube are homeomorphic. You can imagine smoothly deforming one into the other without tearing or creating holes.
• A ball and a donut (torus) are not homeomorphic because the donut has a hole, and the ball doesn't.
• However, a donut and a coffee mug are homotopy equivalent. Even though they look different, you can continuously deform the donut into a coffee mug shape without tearing, and importantly, you can do it in a way that preserves the single hole. The "hole" is the key topological feature here.

So, homotopy equivalence is a looser way of saying two shapes are fundamentally the same in terms of their basic structure (like how many holes they have), even if their exact form can be continuously changed.