formal immersion of smooth manifolds in nLab

Contents

Definition

A formal immersion FF of one smooth manifold, MM, into another, NN, is an injective bundle morphism TM→TNT M \to T N between their tangent bundles.

That is, FF consists of a smooth function f:M→Nf \;\colon\; M \to N and a homomorphism of vector bundles F:TM→TNF \;\colon\; T M\to T N covering ff such that the linear function F| x:T xM→T f(x)NF|_{x} \;\colon\; T_x M \to T_{f(x)} N is an injective function for every point xx in MM.

Write Imm f(M,N)Imm^f(M,N) for the space of such formal immersions. There is a fibration over the space of smooth functions from MM to NN, Map sm(M,N)Map^{sm}(M, N), forgetting the bundle homomorphism, whose fiber over ff is Hom Vect M inj(TM,f *TN)Hom^{inj}_{Vect_M}(T M, f^{\ast}T N).

Note: Some authors define formal immersions in terms of continuous functions (e.g., Laudenbach17, p. 6). However, the space Map sm(M,N)Map^{sm}(M, N) is homotopy equivalent to the space of all continuous functions, Map(M,N)Map(M, N), due to integrating against a smoothing kernel.

Relation to immersions

Since an actual immersion of smooth manifolds is a formal immersion where the bundle morphism in question is specifically taken to be the pointwise derivative dfd f, there is a natural continuous function Imm(M,N)→Imm f(M,N)Imm(M,N) \to Imm^f(M,N), sending an actual immersion ff to the formal immersion with injective bundle morphism dfd f.

Stephen Smale and Morris Hirsch established that when MM is compact, and also either MM is open (in the sense that the complement of the boundary has no compact component) or dim(M)<dim(N)dim(M) \lt dim(N), then the map Imm(M,N)→Imm f(M,N)Imm(M,N) \to Imm^f(M,N) is a weak homotopy equivalence. This is an instance of the h-principle.

When combined with the result that Imm f(S k,ℝ n+k)→Map(S k,V k(ℝ n+k))Imm^f(S^k,\mathbb{R}^{n+k}) \to Map(S^k, V_k(\mathbb{R}^{n+k})) can be shown to be a homotopy equivalence, where V k(ℝ n+k)V_k(\mathbb{R}^{n+k}) is the Stiefel manifold of kk-frames in ℝ n+k\mathbb{R}^{n+k}, the previous result establishes that isotopy classes of immersions of S kS^k into ℝ n+k\mathbb{R}^{n+k} are in bijection with π kV k(ℝ n+k)\pi_k V_k(\mathbb{R}^{n+k}). In the case where k=2k = 2 and n=1n = 1, we find that immersions of the 2-sphere into ℝ 3\mathbb{R}^3 are classified by π 2V 2(ℝ 3)=π 2(SO(3))=0\pi_2 V_2(\mathbb{R}^3)= \pi_2(SO(3))= 0, in other words, all such immersions are isotopic. In particular, S 2S^2 can be turned inside-out (sphere eversion) inside R 3R^3 by moving through a family of immersions.

References

• John Francis, The h-principle, lectures 1 and 2: overview, (pdf)

• Konrad Voelkel, Helene Sigloch, Homotopy sheaves and h-principles, (pdf)

• Francois Laudenbach, René Thom and an anticipated h-principle, (arXiv:1703.08108)