Regular homotopy
In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another.
The homotopy must be a 1-parameter family of immersions.
Similar to homotopy classes, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them.
Regular homotopy for immersions is similar to isotopy of embeddings: they are both restricted types of homotopies.
Stated another way, two continuous functions
are homotopic if they represent points in the same path-components of the mapping space
, given the compact-open topology.
The space of immersions is the subspace of
consisting of immersions, denoted by
are regularly homotopic if they represent points in the same path-component of
Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy.The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if their Gauss maps have the same degree/winding number.
Stephen Smale classified the regular homotopy classes of a k-sphere immersed in
– they are classified by homotopy groups of Stiefel manifolds, which is a generalization of the Gauss map, with here k partial derivatives not vanishing.
More precisely, the set
of regular homotopy classes of embeddings of sphere
is in one-to-one correspondence with elements of group
π
is path connected,
and due to Bott periodicity theorem we have
{\displaystyle \pi _{6}(SO(6))\cong \pi _{6}(\operatorname {Spin} (6))\cong \pi _{6}(SU(4))\cong \pi _{6}(U(4))\cong 0}
Therefore all immersions of spheres
in euclidean spaces of one more dimension are regular homotopic.
admit eversion if
A corollary of his work is that there is only one regular homotopy class of a 2-sphere immersed in
In particular, this means that sphere eversions exist, i.e. one can turn the 2-sphere "inside-out".
Both of these examples consist of reducing regular homotopy to homotopy; this has subsequently been substantially generalized in the homotopy principle (or h-principle) approach.
For locally convex, closed space curves, one can also define non-degenerate homotopy.
Here, the 1-parameter family of immersions must be non-degenerate (i.e. the curvature may never vanish).
There are 2 distinct non-degenerate homotopy classes.
[1] Further restrictions of non-vanishing torsion lead to 4 distinct equivalence classes.
