Cohomological relation between the tangent and normal Gauss maps of a submanifold of Euclidean space

For a closed orientable surface $M$ differentiably imbedded in $E^4$, the tangent Gauss map $t$ into the Grassmann manifold of oriented 2-planes satisfies, under the Plücker identification of that Grassmann manifold with a product of two 2-spheres, $t_*[M]=u_1S_1+u_2S_2$ with $u_1+u_2=\chi(M)$ and, after suitably orienting the spheres and $M$, $u_1=u_2=\chi(M)/2$; this refinement, pointed out by H. Hopf, was proved topologically by Chern and Spanier in 1951 using the orthogonal-complement involution on the Grassmann manifold and a Seifert–Whitney vanishing theorem for the normal characteristic class. The same paper poses the general question for a closed oriented $k$-manifold imbedded in $E^{k+l}$: what is the relation, for every cohomological degree $p$, between the pullback homomorphisms induced on cohomology by the tangent Gauss map $t:M\to G(k,l)$ and the normal Gauss map $n:M\to G(l,k)$? No treatment of that general question is known in the literature: the later work stays in the surface case or develops adjacent threads (complex points and totally real submanifolds, inverse Gauss-map problems, Grassmann-image geometry). The problem asks for the general theorem — which Grassmann cohomology classes are forced to pull back to zero, and which pullbacks are forced to be linked, by the imbedding into contractible Euclidean space.

Background

Let $E^{k+l}$ denote Euclidean $(k+l)$-space and let $G(k,l)$ denote the Grassmann manifold of oriented $k$-dimensional linear subspaces (planes through a fixed origin $O$) of $E^{k+l}$. An immersion of a $k$-dimensional manifold $M$ into $E^{k+l}$ is a differentiable map whose derivative is injective at every point, and an imbedding is an immersion that is a homeomorphism onto its image. A differentiable imbedding determines two maps: the tangent Gauss map $t:M\to G(k,l)$, sending each point $x$ of $M$ to the oriented tangent plane at $x$ translated to $O$, and the normal Gauss map $n:M\to G(l,k)$, sending $x$ to its oriented normal $l$-plane. The two are related by $n=\alpha\circ t$, where $\alpha:G(k,l)\to G(l,k)$ is the diffeomorphism taking a plane to its oriented orthogonal complement; hence the pullback homomorphisms induced on cohomology, $t^*:H^p(G(k,l))\to H^p(M)$ and $n^*:H^p(G(l,k))\to H^p(M)$, satisfy the formal identity $n^*=t^*\circ\alpha^*$ in every degree $p$ (cohomology with integer coefficients is understood throughout; for a continuous map $f$, the symbols $f^*$ and $f_*$ denote the homomorphisms induced by $f$ on cohomology and on homology). The content of the question is therefore not this identity but the constraints that the imbedding imposes: since $E^{k+l}$ is contractible, a closed imbedded $M$ is homologous to zero in it, and this forces some Grassmann cohomology classes to pull back to zero and some pairs of pullbacks to agree or be opposite. Which classes, and which pairs, for every $k$, $l$, and $p$, is the open question.

The classical instance is due to Chern and Spanier (1951), settling a question pointed out by H. Hopf. For a closed orientable surface ($k=l=2$), the Plücker coordinate map — which represents an oriented 2-plane by its six Plücker coordinates $a_{ij}$ (the $2\times 2$ minors of a matrix whose columns span the plane) and then passes to their self-dual and anti-self-dual combinations $x_i=a_{ij}+a_{kl}$ and $y_i=a_{ij}-a_{kl}$ on complementary index pairs, each of unit norm — identifies $G(2,2)$ with a product $S_1\times S_2$ of two 2-spheres. Writing $[M]$ for the fundamental class of $M$ (the generator of its top-dimensional homology) and using the same symbols $S_1,S_2$ for the fundamental classes of the two sphere factors, the image of the tangent map on homology is $t_*[M]=u_1S_1+u_2S_2$ for integers $u_1,u_2$; the sum $u_1+u_2$ equals the Euler characteristic $\chi(M)$, a formula due to W. Blaschke as credited in the paper. Chern and Spanier proved that the spheres and $M$ can be oriented so that $u_1=u_2=\chi(M)/2$. Their proof is topological: the unit circle bundle over $G(2,2)$, whose fiber over a plane is the unit circle in that plane, has Euler class $\Omega$ (the Euler class of an oriented circle bundle being the primary obstruction to a continuous choice of a unit vector in each fiber), and $\Omega$ generates the kernel of the bundle projection on cohomology; the complement involution $\alpha$ acts on the second sphere factor by $y\mapsto -y$, so it changes the sign of one of the two generating cohomology classes; and the Seifert–Whitney theorem — the normal characteristic class of an imbedding vanishes when the submanifold is homologous to zero in a Riemannian manifold, here the contractible $E^4$ — forces $n^*\Omega=0$, which yields $t^*S_1^*=t^*S_2^*$ (where $S_i^*$ is the cohomology class dual to $S_i$) and hence $u_1=u_2$.

The same paper poses the general problem: for $M^k$ imbedded in $E^{k+l}$, to study the relation between $t^*$ and $n^*$ on cohomology in all degrees. The later literature has developed adjacent directions rather than this question. The complex-point thread studies, for real submanifolds of complex space, the locus where the tangent space contains a complex subspace: Lai (1972) obtained formulas linking the Euler class of $M$, the Euler class of the normal bundle, and the algebraic number of such RC-singular points for immersions into complex manifolds, and showed that the Pontryagin classes (characteristic classes of a real vector bundle) and the Euler class are carried by the singular set; Webster (1985) computed the Euler and Pontryagin numbers of real $n$-manifolds in $\mathbb{C}^n$; Banchoff and Farris (1993) proved that for a compact oriented surface immersed in $\mathbb{R}^4=\mathbb{C}^2$ the sum of the tangential and normal Euler numbers equals the algebraic number of complex points, crediting this formula as implicit in the Chern–Spanier paper. Weiner (1984) studied the inverse Gauss-map problem for surfaces in $E^4$ — determining when a given smooth map from an oriented surface into the Grassmann manifold of oriented 2-planes arises as the Gauss map of an immersion into four-space, by partial-differential-equation methods of hyperbolic and elliptic type — and Weiner (1986) treated the analogous inverse problem for conformal immersions of surfaces into Euclidean space. The survey of Borisenko and Nikolaevskii (1991) on Grassmann images restates the surface theorem but does not treat the general cohomological question. In modern terms, $t$ and $n$ classify the tangent and normal bundles (each is the pullback of the tautological bundle over the corresponding Grassmann manifold, the bundle whose fiber over a plane is that plane itself), and the two bundles have trivial Whitney sum (direct sum as bundles), so their total characteristic classes are mutually inverse; this plausibly determines much of the characteristic-class level of the relation, but no explicit general treatment of the question as posed is known.

Problem Statement

Let $M$ be a closed oriented differentiable $k$-dimensional manifold differentiably imbedded (that is, immersed by a map that is a homeomorphism onto its image) in Euclidean space $E^{k+l}$, and let $t:M\to G(k,l)$ and $n:M\to G(l,k)$ be its tangent and normal Gauss maps into the Grassmann manifolds of oriented $k$-planes and $l$-planes through a fixed origin of $E^{k+l}$, so that $n=\alpha\circ t$ where $\alpha$ maps a plane to its oriented orthogonal complement. Determine, for every degree $p\ge 0$, the relation between the induced homomorphisms $t^*:H^p(G(k,l);\mathbb{Z})\to H^p(M;\mathbb{Z})$ and $n^*:H^p(G(l,k);\mathbb{Z})\to H^p(M;\mathbb{Z})$ that is forced by the existence of such an imbedding into the contractible space $E^{k+l}$, in which $M$ is necessarily homologous to zero. In particular: (i) characterize the classes $x\in H^*(G(k,l);\mathbb{Z})$ whose pullback $t^*x$ is necessarily zero — as happens when $k=l=2$: by the Seifert–Whitney theorem the normal characteristic class of an imbedding homologous to zero in a contractible Riemannian ambient space vanishes, $n^*\Omega=0$, and via $n^*=t^*\circ\alpha^*$ this is the vanishing of $t^*(\alpha^*\Omega)$, where under the identification $G(2,2)\cong S_1\times S_2$ one has $\Omega=S_1^*+S_2^*$ and $\alpha^*\Omega=S_1^*-S_2^*$; and (ii) characterize the pairs of classes whose pullbacks are necessarily equal, opposite, or otherwise linked — as $t^*S_1^*=t^*S_2^*$ when $k=l=2$, where $S_1^*,S_2^*$ are the cohomology classes dual to the fundamental classes of the two sphere factors under the Plücker-coordinate identification $G(2,2)\cong S^2\times S^2$. The case of closed orientable surfaces in $E^4$ ($k=l=2$, $p=2$) is settled by the source theorem: if $t_*[M]=u_1S_1+u_2S_2$ in $H_2(S_1\times S_2;\mathbb{Z})$, then the spheres and $M$ can be oriented so that $u_1=u_2=\chi(M)/2$, where $\chi(M)$ is the Euler characteristic of $M$. The requested result is the corresponding statement for general $k$ and $l$, at all degrees.

References

1. S.-S. Chern and E. H. Spanier, A theorem on orientable surfaces in four-dimensional space, Commentarii Mathematici Helvetici 25 (1951), 205–209. DOI: 10.1007/BF02566455, https://doi.org/10.1007/BF02566455.
2. Thomas Francis Banchoff and Frank Albert Farris, Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space, Pacific Journal of Mathematics 161 (1993), no. 1, 1–24. DOI: 10.2140/pjm.1993.161.1, https://doi.org/10.2140/pjm.1993.161.1.
3. A. A. Borisenko and Yu. A. Nikolaevskii, Grassmann manifolds and the Grassmann image of submanifolds, Russian Mathematical Surveys 46 (1991), no. 2, 45–94. DOI: 10.1070/RM1991v046n02ABEH002742, https://doi.org/10.1070/RM1991v046n02ABEH002742.
4. Joel L. Weiner, The Gauss map for surfaces in 4-space, Mathematische Annalen 269 (1984), 541–560. DOI: 10.1007/BF01450764, https://doi.org/10.1007/BF01450764.
5. Joel L. Weiner, The Gauss map for surfaces. II. The Euclidean case, Transactions of the American Mathematical Society 293 (1986), no. 2, 447–466. DOI: 10.1090/S0002-9947-1986-0816303-X, https://doi.org/10.1090/S0002-9947-1986-0816303-X. (The AMS-registered title is "The Gauss map for surfaces. II. The Euclidean case"; the JSTOR deposit of the same work, DOI 10.2307/2000015, lists it as "The Gauss Map for Surfaces: Part 2. The Euclidean Case".)
6. Hon-Fei Lai, Characteristic classes of real manifolds immersed in complex manifolds, Transactions of the American Mathematical Society 172 (1972), 1–33. DOI: 10.1090/S0002-9947-1972-0314066-8, https://doi.org/10.1090/S0002-9947-1972-0314066-8.
7. S. M. Webster, The Euler and Pontrjagin numbers of an n-manifold in C^n, Commentarii Mathematici Helvetici 60 (1985), 193–216. DOI: 10.1007/BF02567410, https://doi.org/10.1007/BF02567410.