11. Hopf proved that the Hopf map has Hopf invariant 1, and therefore is not null-homotopic . 12. Two mappings are "'homotopic "'if one can be continuously deformed into the other. 13. At any rate, we can see that there are gauge transformations U homotopic to the trivial gauge transformation. 14. The "'Borel conjecture "'states that the map f is homotopic to a homeomorphism. 15. A weak form of Tutte's homotopy theorem states that any closed path is homotopic to the trivial path. 16. The Hawaiian earring is not simply connected, since the loop parametrising any circle is not homotopic to a trivial loop. 17. Under what conditions is f homotopic to the identity \ text { id } _ T : T \ to T. 18. With Jack Snoeyink he extended the algorithms for simple polygons to compute homotopic shortest paths among polygonal obstacles in the plane. 19. The gluing map is homotopic to the identity map on the circle and so the complex is homotopy equivalent to the disc. 20. In this sense, the homotopy pushouts treats homotopic spaces as well as the ( ordinary ) pushout does with homeomorphic spaces.
