Motivations for Homotopy Theory

Question

Guys we have motivations and significance behind every concept of Mathematics. As we know if two spaces are homeamorphic then they share lots of common properties - Euler's characteristics for instance. I know if two spaces are homotopically equivalent then they have same fundamental group. Apart from this, can anybody let me know which properties two homotopically equivalent spaces share? Or whey we even study this theory?

Answer 1

I'd argue that most topological invariants are actually homotopy invariants. Obviously "homeomorphism type" is not, but here is a small list of things that are homotopy invariants:

• Euler characteristic
• Fundamental group
• Higher homotopy groups
• All homology groups
• The cohomology ring
• ...

The idea behind all these invariants is to differentiate or even classify topological spaces. Hence, calculating these invariants is an important task and since they are homotopy invariants it is often easier to exchange your space $X$ for a homotopy equivalent space $Y$ and then calculate the invariants.