MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR S. P. NOVIKOV, B. Yu. STERNIN

TRACES OF ELLIPTIC OPERATORS ON SUBMANIFOLDS AND \(K\)-THEORY

The authors started from the following Dirichlet–Sobolev problem: let an elliptic operator \(A=A_{s,t}\) be given on a closed manifold \(X\):

\[ \Gamma^s(X,E_1)\to \Gamma^t(X,E_2), \]

where \(E_1, E_2\) are complex vector bundles; \(\Gamma^i\) is the Sobolev space of sections; \(s>0,\ t<0,\ |t|\le |s|\). For every submanifold \(Y^i\subset X\) of codimension \(k\), define the Dirichlet–Sobolev operator

\[ (A,SD):\Gamma^s(X,E_1)\to \Gamma^t(X,E_2)/R_Y^t\oplus \sum_{\substack{j\le l^k_{s,t}}}\Gamma^{s-k/2-j}(Y,i^*E_1\otimes S^j c\bar n), \]

putting

\[ (A,SD)[u]=[Au(\bmod R_Y^t),i^*u). \]

Here \(R_Y^{-t}\) is the subspace in \(\Gamma^t(X,E_2)\) consisting of sections concentrated on \(Y\); \(S^j\) is the symmetric power; \(n\) is the normal bundle to \(Y\) in \(X\); \(\bar n\) is its conjugate; \(c\) is complexification;

\[ i^*:\Gamma^s(X,E_1)\to \sum_{\substack{j\le l^k_{s,t}}}\Gamma^{s-k/2-j}(Y,i^*E_1\otimes S^j c\bar n) \]

is the restriction operator with normal derivatives up to order

\[ l^k_{s,t}=[-t-k/2]-1+\operatorname{sgn}(-t-k/2-[t-k/2])). \]

If the operator is Fredholm,* denote its (analytic) index by \(I_a(A,SD)\). Put

\[ I_a(A,Y)=I_a(A,SD)-I_a(A). \]

How is \(I_a(A,Y)\) to be computed? (For \(I_a(A)\) there is the Atiyah–Singer formula \((^1)\).)

For simplicity, we shall assume that \(\dim X\) is even, since the pair \(X,Y\) (with any operator) is easily reduced to the pair \((X\times S^1,Y\times S^1)\). Denote by \(\operatorname{ch}A\) the Atiyah–Singer character of the operator \(A\); \(T\) is the Todd genus; \(\eta\) is the tangent bundle.

Let \(R_c(SO_{2q})\) be the ring of virtual representations of the group \(SO_{2q}\), and let \(\delta\in R_c(SO_{2q})\) be such an element of this ring that its restriction to \(R_c(SO_{2q-1})\) is zero. Put

\[ \underline{\operatorname{ch}}\delta(V)=\operatorname{ch}\delta(V)/\chi_{2q}(V), \]

where \(\chi_{2q}\) is the Euler class. (This notion is correctly defined for the universal bundle and is therefore transferred to all the others.)

Theorem 1. a) If the codimension of the submanifold \(Y\) is odd, then

\[ I_a(A,Y)=0; \]

b) If the codimension of \(Y\) is even, then the formula holds

\[ I_a(A,V)=-i^*(\operatorname{ch}A\cdot T(c\eta_X))\underline{\operatorname{ch}}\left\{ \sum_{\substack{p\le k\\ j\le l^k_{s,t}}}(-1)^p S^j\otimes \Lambda^p\operatorname{ch} \right\}[Y], \]

where \(k=\operatorname{codim}Y\);

\[ l^k_{s,t}=[-t-k/2]-1+\operatorname{sgn}(-t-k/2-[-t-k/2]); \]

\(|s|\ge |t|\); \(\Lambda^p\) is the exterior-power-taking operator.

Corollary. a) If \(2k>\dim X\), then \(I_a(A,Y)=0\); b) if \(2k=\dim X\), then

\[ I_a(A,Y)=(\operatorname{ch}^0 A)\,(\dim_c\sum S^j c n)\,(Y\circ Y). \]

Here by \(\operatorname{ch}^0\) we have denoted the zero-dimensional component of the Atiyah–Singer character of the operator \(A\); \((Y\circ Y)\) is the self-intersection index of the manifold \(Y\).

Consider in the space \(\Gamma^t(X,E)\) \((t<0)\) the subspace

\[ R_Y^t\subset \Gamma^t(X,E). \]

* Necessary and sufficient conditions for the Fredholm property of such an operator were found in \((^4)\).

Lemma 1. There are defined operators
\[ \chi_j^t:\Gamma^{t+k/2+j}(Y,i^*E\otimes S^j cn)\to R_Y^t\subset \Gamma^t(X,E'), \qquad \chi=\sum \chi_j^{(t)}, \]
whose images are all homogeneous differential expressions of order \(j\) in \(\delta\)-functions concentrated on \(Y\), \(j\le l_{s,t}^k\), and \(t\) differentiations are taken in the normal directions; the image of
\[ \chi=\sum \chi_j^{(t)} \]
is exactly \(R_Y^t\).

Let
\[ F=\sum_{j\le l_{s,t}^k} F_j\subset \sum S^j cn. \]
Define the generalized Dirichlet–Sobolev operator
\[ (A,SD_F):\Gamma^s(X,E_1)\to \Gamma^t(X,E_2)/\operatorname{Im}\chi_F\oplus \sum_j \Gamma^{(s)}(Y,i^*E_1\otimes \bar F), \]
where \(\operatorname{Im}\chi_F\) is the image of the composition
\[ \sum \Gamma^{t+k/2+j}(Y,i^*E_2\otimes F_j) \to \sum \Gamma^{t+k/2+j}(Y,i^*E_2\otimes S^j cn) \to \Gamma^t(X,E_2) \]
and \(\bar F=\operatorname{Hom}_{\mathbb C}(F,\mathbb C)\).

Theorem 2. The index of the operator \((A,SD_F)\) is computed by the formula
\[ I_a(A,SD_F)=I_a(A)+I_a(A,Y;F), \]
where
\[ I_a(A,Y;F)=-\,i^*\bigl(\operatorname{ch} AT'(cn_X)\bigr)\frac{\operatorname{ch}F}{T(\operatorname{ch})}\chi(n)[Y]. \]

We indicate here one interesting case: let \(F=cF_k^q(n)\), \(l_{s,t}^k=\frac{k}{2}q\), with
\[ F_{k_1+k_2}^q(n_1\oplus n_2)=F_{k_1}^q(n_1)\oplus F_{k_2}^q(n_2) \]
and
\[ F_1^q(\xi)=\sum_{j\le q}\xi^j=\sum_{j\le q} S^j\xi \]
for a one-dimensional bundle \(\xi\).

For such an operator \((A,SD_F)\) we have

Corollary.
\[ I_a(A,Y;F_k^q)=-\,i^*\bigl(\operatorname{ch} AT(cn_X)\bigr) \frac{(q+1)^{2k}}{T(\psi^{q+1}cn)}\chi(n)[Y], \]
where \(\psi^p:K(X)\to K(X)\) are the Adams operations \((^2)\).

Next we shall need to consider “matrix” operators (not necessarily elliptic):
\[ A:\sum_j \Gamma^{s_j}(X,E_j)\to \sum_k \Gamma^{t_k}(X,E'_k), \]
where either a) all \(s_j\ge 0\), all \(t_k\le 0\), or b) all \(s_j\le 0\), all \(t_k\ge 0\). The operator \(A\) is a matrix with components
\[ A^{jk}:\Gamma^{s_j}(X,E_j)\to \Gamma^{t_k}(X,E'_k). \]

Let \(A:\Gamma^s(X,E_1)\to \Gamma^t(X,E_2)\) be an arbitrary operator of type b), and let an embedding \(i:Y\subset X\), \(\operatorname{codim}Y=2q\), be given. We have the following operators:
\[ i_n^t:\Gamma^t(X,E_2)\to \sum_{j\le l}\Gamma^{t-q-j}(Y,i^*E_2\otimes S^j cn) \quad\text{(restriction),} \]
\[ \chi:\sum_{j\le l}\Gamma^{s+q+j}(Y,S^j cn\otimes i^*E_1)\to \Gamma^s(X,E_1) \quad\text{(corestriction).} \]
Put \(i_{SD}(A)=i_n\circ A\circ \chi\).

Further, we shall consider only operators of type b), replacing an elliptic operator of type a) by \(A^{-1}(\bmod\ \mathrm{comp})\).

For a “matrix” operator
\[ A:\sum_j \Gamma^{s_j}(X,E'_j)\to \sum_k \Gamma^{t_k}(X,E'_k), \]
given by the matrix \(A=(A^{jk})\), \(s_j\le 0\), \(t_k\ge 0\), we put
\[ i_{SD}(A)=(i_{SD}A^{jk}). \]

Let now \(E\) be a real vector bundle over \(X\) of dimension \(2q\). Fix the numbers \(s,t\) and denote by
\[ \operatorname{Op}_{s}^{\,t}(X,E) \]
the set of operators of the form
\[ A_{s,t}(X,E): \sum_{j\le l_{s,t-q}}\Gamma^{s+q+j}(X,S^j cE\otimes E_1) \to \sum_{k\le l_{s,t-q}}\Gamma^{t-q-k}(X,S^j c\bar E\otimes E_2), \]
\[ l_{s,t}=[\min\{|s|,|t|\}]-1+\operatorname{sgn}(\min(|s|,|t|)-(\min(|s|,|t|)). \]

In the case where the bundle \(E=0\), these operators are simply the \(A_{s,t}\) that we considered at the beginning. We note that the Whitney sum \(\oplus\) turns the set of operators \(\operatorname{Op}^{s,t}(X,E)\) into a semigroup.

Lemma 2. The operator \(i_{SD}\) defines a mapping
\[ i_{SD}:\operatorname{Op}^{s,t}(X,E)\longrightarrow \operatorname{Op}^{s,t}(Y,\eta\oplus i^*E). \]

Moreover, if \(i_1:Z\subset Y,\ i_2:Y\subset X\), then
\[ (i_2\circ i_1)_{SD}=(i_1)_{SD}\circ (i^2)_{SD}. \]

Definition. We shall call an operator \(A\in \operatorname{Op}^{s,t}(X,E)\) normally elliptic if, for all \(j\le l_{s,t}-q\), the operators
\[ A_j:\sum_{k\le j}\Gamma^{s+q+k}(X,S^k\mathbb C E\otimes E_1)\longrightarrow \sum_{k\le j}\Gamma^{t-q-k}(X,S^k\mathbb C\overline E\otimes E_2), \]
defined by the principal minors of the matrix \(A\), are elliptic.

Normally elliptic operators define a subsemigroup
\[ \operatorname{Ell}^{s,t}(X,E)\subset \operatorname{Op}^{s,t}(X,E). \]
On this subsemigroup the Atiyah—Singer character is defined:
\[ \operatorname{ch}:\operatorname{Ell}^{s,t}(X,E)\longrightarrow H^*(X;Q). \]

Let us define the components
\[ \operatorname{ch}_k:\operatorname{Ell}^{s,t}(X,E)\longrightarrow H^*(X,Q),\quad k\le l_{s,t}-q. \]
Namely, let \(\operatorname{ch}_0=\operatorname{ch}A_0\). Represent \(A_j\) in the form of a matrix
\[ A_j= \begin{pmatrix} A_{j-1} & \beta_j\\ \gamma_j & \delta_j \end{pmatrix}, \]
where
\[ \delta_j:\Gamma^{s+q+j}(X,S^j\mathbb C E\otimes E_1)\longrightarrow \Gamma^{t-q-j}(X,S^j\mathbb C\overline E\otimes E_2), \]
is, in exactness, the operator \(A_j^j\), and \(\beta_j,\gamma_j\) are rectangular matrices. From normal ellipticity it follows that the operator
\[ \widetilde\delta_j=\delta_j-\beta_j\circ \widetilde A_{j-1}^{-1}\circ \gamma_j \]
is elliptic. We now put
\[ \operatorname{ch}_j A=\operatorname{ch}\widetilde\delta_j,\quad j\le l_{s,t}-q, \]
where \(\widetilde A_{j-1}^{-1}\) is such an operator that
\[ A_{j-1}\circ \widetilde A_{j-1}^{-1}=1\pmod{\operatorname{Comp}},\quad \widetilde A_{j-1}^{-1}\circ A_{j-1}=1\pmod{\operatorname{Comp}}, \]
where \(\operatorname{Comp}\) denotes compact operators.

The following simple fact holds.

Lemma 3. The operator \(A\in \operatorname{Ell}^{s,t}(X,E)\) is homotopic (by means of an absolutely universal homotopy) in the set \(\operatorname{Ell}^{s,t}(X,E)\) to the diagonal operator
\[ \sum_{j\le l-q}\delta_j. \]

Introduce the notation
\[ \operatorname{Ch}A[t]=\sum_{i\ge0}(\operatorname{ch}_i A)t^i\in H^*(X,Q)[t]/t^{\,l-q+1}, \]
\[ \operatorname{Ch}A[1]=\operatorname{ch}A;\qquad \operatorname{Ph}\xi=\sum_{i\ge0}(\operatorname{Ch}S^i\xi)t^i \]
for a bundle \(\xi\).

We note that
\[ \operatorname{Ph}(\xi\oplus\eta)=\operatorname{Ph}\xi\cdot \operatorname{Ph}\eta. \]

If \(Y\subset X,\ \operatorname{codim}Y=2q'\), we naturally put
\[ i^*:H^*(X,Q)[t]/t^{m+1}\longrightarrow H^*(Y,Q)[t]/t^{m-q'+1},\quad m=l-q. \]

Our main Riemann—Roch theorem holds:

Main Theorem. If \(A\in \operatorname{Ell}^{s,t}(X,E)\), \(\dim_R E=2q\), and an embedding \(i:Y\subset X\) of codimension \(2q'\) is given such that
\[ i_{SD}A\in \operatorname{Ell}^{s,t}(Y,\eta\oplus i^*E), \]
then the following relations hold in \(H^*(Y,Q)[t]/t^{\,l-q-q'+1}\):
\[ 1)\quad T(c\eta_Y)\operatorname{Ch}i_{SD}(A) = \left(\operatorname{ch}\sum_{p\le 2q'}(-1)^p\Lambda^p c\eta\right) \operatorname{Ph}(\eta)\circ i^*(T(c\eta_X)\operatorname{Ch}A); \]

for \(E=0\):
\[ 2)\quad I_a(A;Y)=I_a(i_{SD}A)= (\operatorname{Ch}i_{SD}A\circ T(c\eta_Y))[Y]\big|_{t=1}, \]
where \(\operatorname{Ph}(\xi)\), \(\operatorname{Ch}A\), \(i_{SD}A\), \(\operatorname{Ell}^{s,t}(X,E)\) are defined above, and by \(I_a(A,Y)\) is meant the difference of the indices of the operators
\[ I_a(A^{-1},SD)-I_a(A^{-1}), \]
where \(A^{-1}\) is an operator of type \(a\), inverse \((\bmod\,\operatorname{Comp})\) to \(A\). The operator \((A^{-1},SD)\) is Fredholm under the hypotheses of the theorem.

Concluding remark. In the immediately following work by the authors (3), the most interesting types of operators are indicated, a number of examples (of group type) are examined, the geometric meaning of the Fredholm conditions established in (4) is explained, a homomorphic variant of the Riemann–Roch type theorem given here is presented, and the connection with \(K\)-theory and algebraic geometry is clarified.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
22 IV 1966

REFERENCES

\(^{1}\) M. F. Atiyah, I. M. Singer, Bull. Am. Math. Soc., 69, No. 3, 422 (1963).
\(^{2}\) J. F. Adams, Bull. Am. Math. Soc., 68, No. 1, 38 (1961).
\(^{3}\) S. P. Novikov, B. Yu. Sternin, DAN, 171, No. 3 (1966).
\(^{4}\) Yu. B. Sternin, Tr. Mosk. matem. obshch., 15 (1966).