nLab formal immersion of smooth manifolds

Definition

A formal immersion $F$ of one smooth manifold, $M$, into another, $N$, is an injective bundle morphism $T M \to T N$ between their tangent bundles.

That is, $F$ consists of a smooth function $f \;\colon\; M \to N$ and a homomorphism of vector bundles $F \;\colon\; T M\to T N$ covering $f$ such that the linear function $F|_{x} \;\colon\; T_x M \to T_{f(x)} N$ is an injective function for every point $x$ in $M$.

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

Note: Some authors define formal immersions in terms of continuous functions (e.g., Laudenbach17, p. 6). However, the space $Map^{sm}(M, N)$ is homotopy equivalent to the space of all continuous functions, $Map(M, N)$, 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 $d f$, there is a natural continuous function $Imm(M,N) \to Imm^f(M,N)$, sending an actual immersion $f$ to the formal immersion with injective bundle morphism $d f$.

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

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