Dupin hypersurfaces

For me, one of the great appeals of algebraic topology is its power to answer questions that don't appear to be all that related to homotopy theoretic data. There are two particular instances that appeared in some master's courses I followed. The first is 

Theorem 1 (Adams): If there are \( l \) linearly independent vector fields on the sphere \( S^m \) then \( m + 1 \) is a multiple of \( 2^{\phi(l)} \).

Here's a little bit about the proof. Let \( \mathbb{RP}^m \) be the real projective space of dimension \( m \), and for \( 0 < k \leq m \) let \( \mathbb{RP}^m_k \) be the 'stunted' projective space obtained from \( \mathbb{RP}^m \) by collapsing the subspace \( \mathbb{RP}^{k-1} \subset \mathbb{RP}^m \) to a point. We recall that the standard CW-structure of \( \mathbb{RP}^m \) has one cell of dimension \( s \) for \( 0 < s \leq m \) (working with spaces with base points we don't count the 0-cell which gives the base point). Since \( \mathbb{RP}^{k-1} \) is the \( (k-1) \)-skeleton of \( \mathbb{RP}^m \), the quotient space \( \mathbb{RP}^m_k \) is a CW-complex with one \( s \)-cell for every \( k \leq s \leq m \). In particular, the quotient of \( \mathbb{RP}^m_k \) obtained by collapsing the \( (m-1) \)-skeleton \( \mathbb{RP}_k^{m-1} \) to a point can be identified with the sphere \( S^m \). We denote the projection map onto this quotient by \( p: \mathbb{RP}_k^m \to S^m \). Recall that a \textbf{stable map} from a pointed space \( X \) to a pointed space \( Y \) is a map \( f: \Sigma^k X \to \Sigma^k Y \) between suitable suspensions of \( X \) and \( Y \).

Theorem 2 (Adams): Assume that the top cell of \( \mathbb{RP}_{m-l}^m \) splits off stably, i.e., there is a stable map \( s: S^m \to \mathbb{RP}_{m-l}^m \) such that the composition

\[
S^m \xrightarrow{s} \mathbb{RP}_{m-l}^m \xrightarrow{p} S^m
\]

has mapping degree one. Then \( m+1 \) is a multiple of \( 2^{\phi(l)} \).

 (This is the "Spanier-Whitehead dual" of what Adams actually proved.)

Here is one possible argument deducing Adams' theorem on vector fields from his homotopy theoretic result on mapping degrees.

1. If \( S^m \) admits \( l \) linearly independent vector fields, then the Thom space \( T(\tau S^m) \) of the tangent bundle of \( S^m \) (i.e., the quotient space obtained from the disc bundle of \( \tau S^m \) by collapsing the sphere bundle to a point) desuspends \( l \) times (i.e., \( T(\tau S^m) \) is homotopy equivalent to \( \Sigma^l X \) for some space \( X \)).
2. If \( T(\tau S^m) \) desuspends \( l \) times then the top cell of \( \mathbb{RP}_{m-l}^m \) splits off stably.

The Hurwitz-Radon-Eckmann construction produces \( l \) linearly independent vector fields on the sphere \( S^m \) from a module \( V \) of real dimension \( m + 1 \) over the Clifford algebra \( C_l \). Adams' result shows that the maximal possible number of linearly independent vector fields is achieved by this construction. 

The second result is the classification of real division algebras, and the closely related question of whether there exists a map with Hopf-invariant 1. For a map $f:S^{4n-1} \to S^{2n}$, the Hopf invariant is defined as follows: take $X_f=S^{2n} \cup_fD^{4n}$ and consider the cohomology class $a \in H^{2n}(X_f; \mathbb{Z})$ representing a generator of $H^{2n}(S^{2n}; \mathbb{Z})$. Let $b$ denote the image of a generator of $H^{4n}(S^{4n}; \mathbb{Z)}$ under the map $c$ defined by collapsing $S^{2n}$. $a \cup a =h(f) b$ for some integer $h(f)$, well-defined up to a sign, that is called the Hopf invariant. This turns out to be a homomorphism $\pi_{4n-1}(S^{2n}) \to \mathbb{Z}$, responsible for the only non-trivial classes in the rational homotopy groups of spheres (above the diagonal). Adams' work on $K$-theory led to Adams operations, which can be used to give a short proof that if there is a map $f$ with Hopf invariant 1, then $2n=2,4,8$. All these cases occur and correspond to the complex numbers, quaternions, and octonions.

There is a notion of a generalised Hopf invariant, due to Stolz, which will take a little bit of work to define. 

Definition: Given an integer \( k \geq 1 \), define \( D_{k,2}(X) = \left( S_+^{k-1} \wedge X \wedge X \right) / \Sigma_2 \), where \( S_+^{k-1} \) is the sphere \( S^{k-1} \) equipped with a disjoint base point and the two element group \( \Sigma_2 \) acts on \( S_+^{k-1} \) by the antipodal map and on \( X \wedge X \) by permuting factors. 

 It is well-known that \( D_{k,2}(S^n) \) is homeomorphic to  
\[
\Sigma^n \mathbb{RP}_n^{n+k-1}.
\]  
To see this, we note that  
\[
D_{k,2}(S^n) = (S_+^{k-1} \wedge S^n \wedge S^n) / \Sigma_2
\]  
is homeomorphic to the Thom space of the vector bundle  
\[
(S^{k-1} \times (\mathbb{R}^n \oplus \mathbb{R}^n)) / \Sigma_2
\]  
over \( S^{k-1} / \Sigma_2 = \mathbb{RP}^{k-1} \). Here \( \Sigma_2 \) acts on \( S^{k-1} \) by the antipodal map and on \( \mathbb{R}^n \oplus \mathbb{R}^n \) by permuting the summands. As a representation of \( \Sigma_2 \), \( \mathbb{R}^n \oplus \mathbb{R}^n \) is isomorphic to the direct sum of \( n \) copies of the non-trivial one dimensional representation \( \mathbb{R}^- \) and \( n \) copies of the trivial representation. Hence the above vector bundle is isomorphic to \( nH_{k-1} \oplus n\mathbb{R} \), where \( H_{k-1} = (S^{k-1} \times \mathbb{R}^-) / \Sigma_2 \) is the Hopf line bundle over \( \mathbb{RP}^{k-1} \), and \( \mathbb{R} \) is the one dimensional trivial real line bundle. It follows that the Thom space of this vector bundle is homeomorphic to  
\[
T(nH_{k-1} \oplus n\mathbb{R}) \approx \Sigma^n T(nH_{k-1})
\]  
which in turn is homeomorphic to  
\[
\Sigma^n \mathbb{RP}_n^{n+k-1}
\]  


Stolz first deduces the following from work of May, Milgram, and Segal.

Theorem 3: Let \( X \) be a \((r-1)\)-connected space, \( r \geq 1 \). There is a map  
\[
\beta_k : D_{k,2}(X) \to \Omega^k \Sigma^k X / X
\]  
which is a \((3r-1)\)-equivalence (i.e., the induced homomorphism on \( \pi_q \) is an isomorphism for \( q < 3r-1 \), and a surjection for \( q = 3r-1 \)). 

Let \( X \) be a \((r-1)\)-connected space, \( r \geq 2 \). We note that Freudenthal's suspension Theorem is equivalent to the statement that the pair \((\Omega^k \Sigma^k X, X)\) is \((2r-1)\)-connected (i.e., the homotopy groups \(\pi_q(\Omega^k \Sigma^k X, X)\) are zero for \( q < 2r\)). Then the Blakers-Massey Theorem implies that the map  
\[
\pi_q(\Omega^k \Sigma^k X, X) \to \pi_q(\Omega^k \Sigma^k X / X)
\]  
is an isomorphism for \( q < 3r-1 \) and surjective for \( q = 3r-1 \). Together with the isomorphism from Theorem 3 we obtain for \( q < 3r - 1 \) an isomorphism  
\[
\pi_q(\Omega^k \Sigma^k X, X) \cong \pi_q(D_{k,2}(X)).
\]  
Hence the long exact homotopy sequence of the pair \((\Omega^k \Sigma^k X, X)\) implies the following result.


Corollary 4: If \( X \) is an \((r-1)\)-connected space, \( r \geq 2 \), then for \( q < 3r - 1 \) there is an exact sequence  

\[
\pi_q(X) \xrightarrow{\Sigma^k} \pi_{k+q}(\Sigma^k X) \xrightarrow{H^k} \pi_q(D_{k,2}(X)) \xrightarrow{\partial} \pi_{q-1}(X) \xrightarrow{\Sigma^k} \pi_{k+q-1}(\Sigma^k X) \xrightarrow{\dots}
\]

The homomorphism \( H^k \) is called the (generalized) Hopf invariant. The letters EHP stand for ``Einhängung'' (the German word for suspension), ``Hopf invariant'' and ``product'', respectively (if \( X \) is a sphere and \( k = 1 \), the boundary map \( \partial \) can be interpreted as a Whitehead product).


This is a nice formalism, but it really packs a punch in Stolz's hands when he uses this to make significant progress on a classical topic of differential geometry. 

Say that a hypersurface \(M^n \subset S^{n+1} \) is isoparametric if its principal curvatures (the eigenvalues of the shape operator) are constant, and Dupin if, more generally, the number of distinct eigenvalues of the shape operator at \( x \in M \) is required to be independent of \( x \) and the \( i \)-th  eigenvalue \( \lambda_i(x) \) is constant on the leaves of the foliation defined by the eigenspaces corresponding to \( \lambda_i(x) \). Let $g$ denote the number of distinct principal curvatures. Assume that the eigenvalues are ordered according to size: \( \lambda_1 < \lambda_2 < \ldots < \lambda_g \)).

Elie Cartan classified those isoparametric hypersurfaces with \( g = 1, 2,\) or $3$. Moreover, it was shown that all Dupin hypersurfaces with \( g = 1, 2, 3 \) are Lie equivalent to isoparametric surfaces.

When $g>3$, more insight is needed.

Theorem 5: Let \( M^n \subset S^{n+1} \) be a Dupin hypersurface with \( g \) distinct principal curvatures and multiplicities denoted by \( m_1, \ldots, m_g \). There is a decomposition of the ambient sphere

\[ \quad S^{n+1} = D\gamma_1 \cup_M D\gamma_2, \tag{1}
\]

as the union of two linear disc bundles \( D\gamma_i \to M_i \) over smooth manifolds \( M_1, M_2 \) with common boundary \( \partial D\gamma_1 = \partial D\gamma_2 = M \), where the fibres of the fibre bundle \( p_i : M = \partial D\gamma_i \to M_i \) are spheres of dimension \( m_i \). Geometrically the manifolds \( M_i \) (considered as submanifolds of \( S^{n+1} \subset M \) via the identification (2.1)) are focal manifolds of \( M \), and \( \gamma_i \) is the normal bundle of \( M_i \). 

For isoparametric hypersurfaces, this breakthrough is due to Münzner, and the general case of Dupin hypersurfaces follows from work of Thorbergsson and Grove--Halperin. Here is some indication of the things they compute, in analogy with the computations on the Hopf invariant we mentioned at the start:

Lemma 6: If \( m_1 \) and \( m_2 \) are both larger than one, then \( M, M_1 \) and \( M_2 \) are simply connected; in particular, they are orientable.

Proof: The exact sequence of homotopy groups associated to the fibre bundle \( p_i : M \to M_i \) with fibre \( S^{m_i} \) shows that the inclusion map \( M \hookrightarrow D(\gamma_i) \sim M_i \) induces an isomorphism of fundamental groups. Applying van Kampen's Theorem to equation 1 then proves the lemma. $\blacksquare$

Applying the Mayer-Vietoris sequence to equation 1 we conclude that

\[
H^q(M_1) \oplus H^q(M_2) \xrightarrow{p_1^* \oplus p_2^*} H^q(M)
\]

is an isomorphism for \( 0 < q < n \). This shows in particular that the homomorphisms \( p_1^* \) and \( p_2^* \) are injective; we will use these maps to identify \( H^*(M_i) \) with a subring of \( H^*(M) \). The cohomology of the manifolds \( M_i \) is (additively) given as follows.

Proposition 7: Assume that \( S^{n+1} \) decomposes in the form of equation 1, and let \( m = m_1 + m_2 \). Then the cohomology groups of \( M_i \), \( i = 1, 2 \), are given by

\[
H^q(M_i) =
\begin{cases} 
R & q \equiv 0, m_{i+1} \mod m, \quad 0 \leq q < n \\
0 & \text{otherwise}
\end{cases}
\]

This implies in particular that \( \dim H^*(M_i) = \frac{1}{2} \dim H^*(M) = g \), where \( g \) is defined by \( n = gm/2 \).

Next we discuss some of the multiplicative structure of \( H^*(M) \). Let \( 1 \in H^0(M) \) the unit of the cohomology ring \( H^*(M) \). For \( g \geq 2 \), let \( a_i \in H^{m_i+1}(M_i) \cong R \) be a generator. The Leray–Hirsch Theorem applied to the fibre bundle \( M \to M_i \) with fibre \( S^{m_i} \) leads to the following result.

Proposition 8: For \( g \geq 2 \) the cohomology ring \( H^*(M) \) is a free module over \( H^*(M_i) \) with basis \( \{1, a_{i+1}\}, i = 1, 2 \).

Corollary 9: For \( g \geq 3 \) there is a choice of generators \( b_i \in H^m(M_i) \cong R \) such that \( a_1a_2 = b_1 + b_2 \).

Proof: The elements \( b_1, b_2 \) form a basis of \( H^m(M) \) due to the isomorphism from Mayer-Vietoris. Hence \( a_1a_2 = r_1b_1 + r_2b_2 \) with \( r_i \in R \). Proposition 8 implies that also the elements \( b_1, a_1a_2 \) form a basis and hence \( r_2 = \pm 1 \). By the same argument \( r_1 = \pm 1 \). $\blacksquare$

Note that the dimension of the Dupin hypersurface \( M^n \subset S^{n+1} \) is the sum of its multiplicities \( m_1, \ldots, m_g \). Using this, one proves further that, for any Dupin hypersurfaces, \( g = 1, 2, 3, 4 \), or \( 6 \) and that \( m_{i+2} = m_i \), \( i \in \mathbb{Z}/g \); in particular, all multiplicities \( m_i \) are determined by \( m_1 \) and \( m_2 \), and \( m_1 = m_2 \) for \( g \) odd. Hence

\[ \quad n = \frac{gm}{2} \quad \text{for} \quad m = m_1 + m_2. \tag{2}
\]

Theorem (Münzner, Grove--Halperin): Assume that \( S^{n+1} \) decomposes in the form of equation 1, and let \( g \) be given by equation 2. Then

1.  \( g = 1, 2, 3, 4 \), or 6.
2. If \( g = 6 \), then \( m_1 = m_2 = 1 \) or \( m_1 = m_2 = 2 \).
3. If \( g = 4 \) and \( m_1 = m_2 \), then \( m_1 = 1 \) or \( 2 \); if \( g = 4 \), and \( 2 \leq m_1 < m_2 \), then \( m_1 + m_2 \) is odd.

Note that the shape operator \( A \) of the embedding \( M \subset S^{n+1} \) involves the choice of a normal direction; making the opposite choice replaces \( A \) by \(-A\) and hence switches \( m_1 \) and \( m_2 \). Hence we may assume \( m_1 \leq m_2 \) without loss of generality. Stolz's contribution is the following:

Theorem 10: Let \( M^n \subset S^{n+1} \) be a Dupin hypersurface with \( g = 4 \) distinct principal curvatures and multiplicities \( m_1 \leq m_2 \). Then \((m_1, m_2) = (2, 2)\) or \((4, 5)\), or \( m_1 + m_2 + 1 \) is a multiple of \( 2^{\phi(m_1-1)} \). Here \( \phi(l) \) is the number of integers \( s \) with \( 1 \leq s \leq l \) and \( s \equiv 0, 1, 2, 4 \mod 8 \).

The above restrictions on the multiplicities are sharp. To show this we recall that the known examples of isoparametric hypersurfaces are of the following two types.

1.  Hsiang and Lawson noticed that every homogeneous isoparametric hypersurface occurs as a principal orbit of the isotropy representation of some symmetric space of rank 2. These were studied in detail by Takagi and Takahashi and they show in particular that for \( g = 6 \) the multiplicities \((1, 1)\) and \((2, 2)\) occur, and for \( g = 4 \), among others, the multiplicities \((2, 2)\) and \((4, 5)\).
2. Extending work of Ozeki-Takeuchi,  Ferus, Karcher, and M\"unzner introduced (and classified) a class of isoparametric hypersurfaces with four distinct principal curvatures in spheres defined by means of real representations of Clifford algebras or, equivalently, Clifford systems. A \emph{Clifford system} consists of \( m + 1 \) symmetric matrices \( P_0, \ldots, P_m \) with \( m \geq 1 \) such that \( P_i^2 = E \) and \( P_iP_j + P_jP_i = 0 \) for \( i, j = 0, \ldots, m \) with \( i \neq j \), where \( E \) denotes the identity matrix. Isoparametric hypersurfaces of \emph{Clifford type} in the unit sphere \( S^{2l-1} \) of the Euclidean vector space \( \mathbb{R}^{2l} \) have the property that there exists a Clifford system \( P_0, \ldots, P_m \) of symmetric \( (2l \times 2l) \)-matrices with \( l - m - 1 > 0 \) such that one of their two focal manifolds is given as \[\{x \in S^{2l-1} \mid \langle P_i x, x \rangle = 0 \text{ for } i = 0, \ldots, m\},\] where \( \langle \cdot, \cdot \rangle \) denotes the standard scalar product. Families of isoparametric hypersurfaces in spheres are completely determined by one of their focal manifolds, hence the above description of one of the focal manifolds by means of a Clifford system characterizes precisely the isoparametric hypersurfaces of Clifford type. 

We will discuss some aspects of the proof below, but not the full proof since it uses some heavy machinery involving homotopy groups of spheres and structure of modules over the Steenrod algebra. First I want say a bit about developments since Stolz's beautiful result. Cecil--Chi--Jensen proved that when an isoparametric hypersurface $M^n$ satisfies $g=4$ and $m_2 \geq 2m_1-1$, then M is of Clifford type. A much simpler proof of this was given by Immervoll. This proof is really remarkable in that it appears to have very little analysis at all, and is mostly linear algebra. 

M\"unzner proved there is a homogeneous polynomial function \( F \) of degree 4 such that \( M = F^{-1}(c) \cap \mathbb{S}^{2l-1} \) for some \( c \in (-1, 1) \). This \textbf{Cartan-M\"unzner polynomial} \( F \) satisfies the two partial differential equations

\[
\langle \operatorname{grad} F(x), \operatorname{grad} F(x) \rangle = 16\langle x, x \rangle^3,
\]

\[
\Delta F(x) = 8(m_2 - m_1)\langle x, x \rangle.
\]

By interchanging the multiplicities \( m_1 \) and \( m_2 \) we see that the polynomial \(-F\) is also a Cartan-M\"unzner polynomial. The polynomial \( F \) takes its maximum 1 (minimum \(-1\)) on \( \mathbb{S}^{2l-1} \) on the two focal manifolds. For a fixed Cartan-M\"unzner polynomial \( F \), let \( M_+ \) always denote the focal manifold on which \( F \) takes its maximum 1. Then we have \( M_+ = F^{-1}(1) \cap \mathbb{S}^{2l-1} \) and \( M_- = F^{-1}(-1) \cap \mathbb{S}^{2l-1} \), where \( \dim M_+ = m_1 + 2m_2 \) and \( \dim M_- = 2m_1 + m_2 \). One key step in Immervoll's proof is giving a linear-algebraic characterisation of $M_+$:

Lemma 11: Let \( M \) be an isoparametric hypersurface with four distinct principal curvatures in the unit sphere \( \mathbb{S}^{2l-1} \) of the Euclidean vector space \( V = \mathbb{R}^{2l} \). Let  \( S_{2l}(\mathbb{R}) \) be the space of real, symmetric (2l × 2l)-matrices. Set \( A(M_+) = \{A \in S_{2l}(\mathbb{R}) \mid \langle x, Ax \rangle = 0 \text{ for every } x \in M_+ \} \)  
and assume that \( m_2 \geq 2m_1 - 1 \). Then we have  

\[
M_+ = \{x \in S^{2l-1} \mid \langle x, Ax \rangle = 0 \text{ for every } A \in A(M_+)\}.
\]

Combined with later work of Chi, this shows that isoparametric hypersurfaces with $g=4$ must be either homogeneous or one of the known inhomogeneous examples, with the possible exception of $(m_1,m_2)=(7,8)$. The case $(g,m)=(6,1)$ had been settled as early as 1985 by Dorfmeister--Neher by a lengthy algebraic proof. After this, Miyaoka (related to but not the Miyaoka of the Bogomolov--Miyaoka--Yau inequality) wrote a paper claiming that isoparametric hypersurfaces with $(g,m)=(6,2)$ are homogeneous, but the reader should note that there is an erratum. 

Stolz's proof

Stolz's headline theorem is a consequence of the Münzner--Grove--Halperin decomposition result and the following theorem:

Theorem 12: Assume that \( S^{n+1} \) decomposes in the form (2.1), and let \( g \) be given by equation 2. Assume that \( g = 4 \) and \( m_1 \leq m_2 \). Then \( (m_1, m_2) = (2, 2) \) or \( (4, 5) \), or \( m_1 + m_2 + 1 \) is a multiple of \( 2^{\phi(m_1-1)} \).

Stolz notes that, both in philosophy and in technique, his proof of this theorem is analogous to Adams' proof. Compare the following two steps in Stolz's proof to those from Adams' proof that I mentioned at the start.

Proposition 13: Assume that \( S^{n+1} \) decomposes as a union of disc bundles with \( 2 \leq m_1 < m_2 \), \( n = 2m \), \( m = m_1 + m_2 \). If \( (m_1, m_2) \neq (2, 2), (4, 5) \), then \( (M_1 \wedge M_2)^{(2m)} \) (the \( 2m \)-skeleton of \( M_1 \wedge M_2 \)) desuspends \( l \) times for some \( l \) with \( \phi(l) \geq \phi(m_1 - 1) \).

Proposition 14: Assume that \( S^{n+1} \) decomposes as a union of disc bundles with \( 2 \leq m_1 < m_2 \), \( n = 2m \), \( m = m_1 + m_2 \). If \( (M_1 \wedge M_2)^{(2m)} \) desuspends \( l \) times, \( 0 < l \leq m_1 - 1 \), then the top cell of \( \mathbb{RP}_{m-l}^m \) splits off stably.

To prove the latter proposition, Stolz proves a generalisation of the following property of the generalized Hopf invariant. To simplify notation we write \( H \) for \( H^\infty \) and \( D_2(X) \) for \( D_{\infty,2}(X) \).

Lemma 15: If \( X \) is an \((r-1)\)-connected space, \( r \geq 2 \), then in the metastable range \( q < 3r - 1 \) the following diagram commutes

\[
\begin{array}{ccc}
\pi_q^s(X) & \xrightarrow{H} & \pi_q(D_2(X)) \\
{\scriptstyle\cong}\downarrow & & \downarrow{\scriptstyle\Delta_*} \\
\pi_{l+q}^s(\Sigma^l X) & \xrightarrow{H} & \pi_{l+q}(D_2(\Sigma^l X))
\end{array}
\]

Here \( H \) in the top row is the Hopf invariant for the space \( X \); in the lower row it is the Hopf invariant for the suspension \( \Sigma^l X \). The map \( \Delta : \Sigma^l D_2(X) \to D_2(\Sigma^l X) \) sends \( u \wedge [e \wedge x \wedge y] \) to \([e \wedge (u \wedge x) \wedge (u \wedge y)]\) for \( u \in S^l, e \in S_+^\infty, x, y \in X \).

For the proof of the former proposition Stolz uses the following corollary of Proposition 14 and Adams' result.

Corollary 16: If \( (M_1 \wedge M_2) \) desuspends \( l \) times, \( 0 < l \leq m_1 - 1 \), then \( m + 1 \) is a multiple of \( 2^{\phi(l)} \).