Question

Difference homoemorphic equivalence and homotopy equivalence

Answer 1

When you say X and Y are homotopic, I assume you mean that they are homotopy equivalent. Anyways, homotopy equivalence is weaker than homeomorphic. Counterexample to your claim: the 2-dimensional cylinder and a Mobius strip are both 2-dimensional maifolds and homotopy equivalent, but not homoemorphic.

Answer 2

Answer:

Possessing similarity of form, 2. Continuous, one-to-one, in surjection, and having a continuous inverse. The most common meaning is possessing intrinsic topological equivalence. Two objects are homeomorphic if they can be deformed into each other by a continuous, invertible mapping.

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.