The unknot is not equivalent to the Trefoil knot since one cannot be deformed into the other through a continuous path of embeddings.
Shape theory: Many shape invariants (Borsuk groups, Quigley inward and approaching groups) of a compact metric space can be obtained as exterior homotopy groups of the exterior space determined by the open neighborhoods of a compact metric space embedded in the Hilbert cube.
Henri Poincaré's 1895 paper Analysis Situs studied three-and-higher-dimensional manifolds(which he called "varieties"), giving rigorous definitions of homology, homotopy (which had originally been defined in the context of late nineteenth-century knot theory, developed by Maxwell and others), and Betti numbers and raised a question, today known as the Poincaré conjecture, based his new concept of the fundamental group.
Hiroshi Toda used the composition product and Toda brackets to label many of the elements of homotopy groups.
Infinity categories (in the form of Joyal's quasi-categories) are a convenient framework to do homotopy theory in abstract settings.
Stasheff's research contributions include the study of associativity in loop spaces and the construction of the associahedron (also called the Stasheff polytope), ideas leading to the theory of operads; homotopy theoretic approaches to Hilbert's fifth problem on the characterization of Lie groups; and the study of Poisson algebras in mathematical physics.
The attempt to classify the objects of this category (up to homeomorphism) by invariants has motivated areas of research, such as homotopy theory, homology theory, and K-theory etc.