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The researchers provide a unified skyrmion-hopfion homotopy classification and offer an insight into the diversity of topological solitons in three-dimensional chiral magnets.

Sullivan says that the result he is proudest of is one he obtained in 1977, which distils the crucial properties of a space using a tool called rational homotopy.

But contractability guarantees that any two composite paths are homotopic, any two homotopies relating two composite paths are connected by a higher homotopy, and so on.

This term refers to the notion of isomorphism in the more exotic homotopy category of spaces.

A homotopy equivalence is another type of continuous deformation, but in this case, you can identify distinct points.