Mathematics

S. I. AL′BER

HOMOLOGIES OF THE SPACE OF UNORIENTED LOOPS AND THEIR APPLICATION TO THE CALCULUS OF VARIATIONS IN THE LARGE

(Presented by Academician P. S. Aleksandrov on 6 XI 1963)

The problem of estimating the number of geodesic loops on a Riemannian manifold was first considered in \((^1)\), where it was shown that on a manifold homeomorphic to an \(n\)-dimensional sphere there exists a countable sequence of geodesic loops with beginning and end (base point) at an arbitrarily chosen point of the manifold. In \((^2)\) A. I. Fet proved that on an arbitrary Riemannian manifold there exists at least one geodesic loop. From Serre’s results \((^3)\) it follows that on an arbitrary closed manifold there exists a countable sequence of geodesic loops; moreover \((^4)\), from this sequence one can extract a subsequence of loops whose lengths increase monotonically, but not faster than the terms of a certain arithmetic progression. (For the case of a spherical manifold this result was obtained by L. A. Lyusternik \((^5)\).) Estimates of the number of geodesic loops were obtained by studying the homology of the space of oriented loops \(\Omega(M)\) of the manifold \(M\), but in solving problems of the calculus of variations “in the large” significantly better estimates can be obtained if the variational functional is considered on spaces of unoriented loops, paths, and curves (cf. \((^6)\)).

In the present article* the homology of the space of unoriented loops of the \(n\)-dimensional sphere is completely computed, and a final estimate is obtained for the number of geodesic loops on a Riemannian manifold homeomorphic to the \(n\)-dimensional sphere. A countable sequence of series of geodesics is found; each series consists of \(n\) loops, of which only two enter the previously known sequence. The computations are carried out by the method of spectral sequences \((^{6-8})\).

1. The space of unoriented loops.

Let \(M^n\) be a Riemannian manifold homeomorphic to an \(n\)-dimensional sphere. We take an arbitrary point \(m_0 \in M^n\) as the base point. In the space \(\Omega(M^n)\) of oriented loops one can define an involution

\[ \Omega(M^n) \xrightarrow{\Theta} \Omega(M^n), \tag{1} \]

by setting

\[ f_2(t) = (\Theta f_1)(t) = f_1(1-t) \qquad (0 \leq t \leq 1). \tag{2} \]

Identifying in \(\Omega(M^n)\) the loops satisfying condition (2), we obtain the space \(\hat{\Omega}(M^n)\) of unoriented loops of the manifold \(M^n\). The natural mapping will be called the \(p\)-projection. A one-point loop will be denoted by \(O\). It can be shown that the spaces \(\hat{\Omega}(M^n)\) and \(\hat{\Omega}(M^n)/O\) are homotopically non-equivalent (cf. \((^9)\)), and in the space \(\hat{\Omega}(M^n)\) one cannot introduce an operation of multiplication of loops.

In the subspace of rectifiable loops we define a weak metric

* The results of the article were reported on 11 IV 1963 at the Topological Seminar of Moscow University and on 27 IX 1963 at the IV All-Union Topological Conference in Tashkent.

by the formula

\[ r_F(v_1,v_2)=\min\left[\max_t \rho\bigl(f_1(t),f_2(t)\bigr),\ \max_t \rho\bigl(f_1(t),f_2(1-t)\bigr)\right], \tag{3} \]

where \(v_j=pf_j(t)\), and \(\rho(x_1,x_2)\) is the distance between the points \(x_1,x_2\) on the manifold, and a strong metric by the formula

\[ r_M(v_1,v_2)=r_F(v_1,v_2)+\left|I\bigl(f_1(t)\bigr)-I\bigl(f_2(t)\bigr)\right|. \tag{4} \]

Here \(I(f(t))\) is the Riemannian length of the curve \(f(t)\). Contracting deformations in the resulting spaces \(\widehat{\Omega}(M^n)\) and \(\widehat{\Omega}_M(M^n)\) can be constructed either by the method of steepest descent or by the known method based on the local uniqueness of geodesics.

2. Cycles modulo 2 of the loop space. Since the spaces \(\widehat{\Omega}(M^n)\) and \(\widehat{\Omega}(S^n)\) are homeomorphic, in order to study the homology of \(\widehat{\Omega}(M^n)\) it is enough to compute the homology of the loop space on the sphere. We take the point \(m_0=(1,0,\ldots,0)\) on the sphere \(S^n:\ x_0^2+\cdots+x_n^2=1\) as the base point. Every great circle of the sphere \(S^n\) passing through the point \(m_0\) is completely determined by a tangent unit vector \(e\) at the point \(m_0\). We denote it by \(S(e)\). Let \(S(e)\) be an arbitrary oriented circle. Divide it into \((2k-1)\) equal parts by the points \(m_0,m_1,\ldots,m_{2k-2}\). In what follows we shall call the circle \(S(e)\) the supporting circle of the loop.

Connect successively the points \(m_0,m_k,m_1,m_{k+1},\ldots,m_{2k-2},m_{k-1},m_0\) by arcs of circles \(\sigma_1,\ldots,\sigma_{2k-1}\), so that the arc \(\sigma_i\) connects those points \(m_p\) and \(m_q\) for which

\[ p\equiv(i-1)k \pmod{(2k-1)};\qquad q\equiv ik \pmod{(2k-1)}. \tag{5} \]

Introducing on the constructed curve the reduced parameter, we obtain an oriented loop. The arcs \(\sigma_i(e)\) will be called the links of the loop. If \(\sigma_i\) coincides with the shortest geodesic connecting \(m_p\) with \(m_q\), we denote it by \(\sigma_i^{-}\). In the case when \(\sigma_i\) coincides with the longer arc of the circle \(S(e)\) connecting \(m_p\) and \(m_q\), we denote it by \(\sigma_i^{+}\). Under the \(p\)-projection an oriented loop passes to a \((2k-1)\)-link nonoriented loop with the same supporting circle.

Definition 1. The set of \((2k-1)\)-link nonoriented loops whose supporting circles lie on the sphere \(S^{j+1}:\ x_0^2+\cdots+x_{j+1}^2=1\) forms a cycle \([j,2k-1]\) modulo 2 of the space \(\widehat{\Omega}^{1}(S^n)\). We note that \(\dim [j,2k-1]=(2k-1)(n-1)+j\).

3. Homology groups modulo 2 of the space of nonoriented loops.

Theorem 1. The cycles \([j,2k-1]\) \((j=0,1,\ldots,n-1;\ k=1,2,\ldots)\) form a complete basic system of cycles modulo 2 of the space of nonoriented loops \(\widehat{\Omega}(S^n)\).

The Betti numbers modulo 2 of the space \(\widehat{\Omega}(S^n)\) are equal to:

\[ \pi_2^s[\widehat{\Omega}(S^n)]= \begin{cases} 0, & \text{for }(2k-2)(n-1)<s<(2k-1)(n-1),\\ 1, & \text{for }(2k-1)(n-1)\le s\le 2k(n-1). \end{cases} \tag{6} \]

Proof. Since on \(S^n\) geodesic loops coincide with multiply repeated great circles, the set of extremal points in the space \(\widehat{\Omega}(S^n)\) consists of a series of manifolds \(P_1^{\,n-1},P_2^{\,n-1},\ldots,\ldots,P_k^{\,n-1},\ldots\), homeomorphic to projective space, and the length of a loop \(v\in P_k^{\,n-1}\) is equal to \(I(v)=2\pi k\). Let \(A_l=\{I(v)\le 2\pi l\}\) and \(\varepsilon_k\ll1\). The groups \(H_s(A_{k+\varepsilon_k})\) stabilize for large \(k\), and therefore \(H_s(\widehat{\Omega}(S^n))=\)

\[ = \lim_{k\to\infty} H_s(A_{k+\varepsilon_k}). \]
Thus, for the proof it is enough to compute, by induction, the homology groups of the domains \(A_{k+\varepsilon_k}\).

Consider the exact sequence of the pair:
\[ \cdots \xrightarrow{\Delta_*} H_s(A_{k-1+\varepsilon_{k-1}}) \xrightarrow{i_*} H_s(A_{k+\varepsilon_k}) \xrightarrow{j_*} H_s(A_{k+\varepsilon_k}, A_{k-1+\varepsilon_{k-1}}) \xrightarrow{\Delta_*} \cdots \tag{7} \]

Using contracting deformations, one can show that \(A_{k-1+\varepsilon_{k-1}}\) is homotopy equivalent to \(A_{k-\varepsilon_k}\), and \(A_{k+\varepsilon_k}\) is homotopy equivalent to
\[ A_{k-\varepsilon_k}\cup S(P_k^{\,n-1},S_k). \]
By a continuous deformation in \(\hat{\Omega}(S^n)\), the spherical neighborhood \(S(P_k^{\,n-1},\delta_k)\) can be deformed into the skew product \(B_k\) with base \(P_k^{\,n-1}\) and fiber the direct product
\[ U_1^{\,n-1}\times \cdots \times U_{4k-1}^{\,n-1} \]
of balls of dimension \((n-1)\). Hence it follows that the groups
\[ H_s(A_{k+\varepsilon_k},A_{k-1+\varepsilon_{k-1}}) \]
are isomorphic to the groups
\[ H_s(A_{k-\varepsilon_k}\cup B_k,A_{k-\varepsilon_k}). \]
Computing the index of the extremal manifold \(P_k^{\,n-1}\) in the open manifold \(B_k\) and applying the dual homology sequences of the triple (8), we obtain that the cycles
\[ [j,2k-1]\quad (j=0,1,\ldots,n-1) \]
form a complete basis system of the groups
\[ H_s(A_{k+\varepsilon_k},A_{k-1+\varepsilon_{k-1}}). \]
Since \([j,2k-1]\) are absolute cycles, the boundary homomorphism \(\Delta_*\) in sequence (7) is trivial, whence it follows that the inclusion homomorphism
\[ H_s(A_{k-1+\varepsilon_{k-1}})\xrightarrow{i_*}H_s(A_{k+\varepsilon_k}) \tag{8} \]
is an isomorphism. The theorem is proved.

Let us note that for
\[ k>\frac{s}{2(n-1)}+\frac12 \]
the \(s\)-dimensional homology groups of the domains \(A_{k+\varepsilon_k}\) stabilize.

By the duality theorem there exists a basis system of cocycles
\[ \{j,2k-1\}\quad (j=0,1,\ldots,n-1;\ k=1,2,\ldots), \]
for which the scalar product
\[ ([i,2l-1],\{j,2k-1\})=\delta_j^i\delta_k^l . \]

We turn to the computation of the cohomology ring \(H(\hat{\Omega}(S^n))\). The natural embedding of the manifolds
\[ [n-1,2q-1]\to \hat{\Omega}(S^n) \]
induces inclusion homomorphisms of the cohomology rings
\[ H([n-1,2q-1])\xleftarrow{p_*} H(\hat{\Omega}(S^n)); \tag{9} \]
therefore the computation of the ring \(H(\hat{\Omega}(S^n))\) reduces to the study of the inclusion homomorphisms of homology groups
\[ H_s([n-1,2q-1])\xrightarrow{i_*} H_s(\hat{\Omega}(S^n)) \tag{10} \]
and to the computation of the rings \(H([n-1,2q-1])\).

4. Homology of the manifolds \([n-1,2q-1]\). Let the cycle
\[ \langle j,a_1,\ldots,a_{2q-1}\rangle \quad (0\le j\le n-1;\ a_i=1,+0,-0) \]
consist of \(p\)-parameter families of oriented loops for which: 1) when \(j>0\) the supporting circles lie on the sphere \(S^{j+1}\), and when \(j=0\) the oriented circle
\[ x_0^2+x_1^2=1 \]
serves as the supporting circle; 2) when \(a_i=1\), the arc \(\sigma_i(e)\) is an arbitrary arc of the circle joining the points \(m_p\) and \(m_q\); 3) when \(a_i=+0\), the arc \(\sigma_i(e)=\sigma_i^+(e)\); 4) when \(a_i=-0\), the arc \(\sigma_i(e)=\sigma_i^-(e)\). The dimension of the cycle is
\[ \dim \langle j,a_1,\ldots,a_{2q-1}\rangle =(a_1+\cdots+a_{2q-1})(n-1)+j. \]

We shall call a cycle symmetric if
\[ a_i=a_{2q-i}\quad \text{for } i=1,2,\ldots,q. \]
We shall call a symmetric cycle marked under the condition that, in the sequence \(a_1,\ldots,a_q\), the signs of the zeros alternate and the sign of the first zero in order is negative. We preliminarily divide nonsymmetric cycles into classes. The cycles
\[ \langle j,a_1,\ldots,a_{2q-1}\rangle,\quad \langle k,b_1,\ldots,b_{2q-1}\rangle \]
belong to the same class if \(k=j\) and either \(a_i=\pm b_i\) for \(1\le i\le 2q-1\), or
\[ a_i=\pm b_{2q-i} \]
for the same \(i\). Then from each class we choose the cycle, called marked, for which the sign of the first zero is negative and the signs of the zeros alternate.

By the method of (⁷) one proves

Theorem 2. The marked nonsymmetric cycles
\(\langle j,a_1,\ldots,a_{2q-1}\rangle\) \((j=0,n-1;\ \sum a_i>0)\) and the marked symmetric cycles \(\langle j,a_1,\ldots,a_{2q-1}\rangle\) \((j=0,1,\ldots,n-1)\) form a complete basic system of cycles modulo 2 of the manifold \([n-1,2q-1]\).

5. The cohomology ring of the space of nonoriented loops.

Theorem 3. Under the homomorphism of the embedding (10) the basic cycles are mapped as follows:

1) Nonsymmetric cycles
\[ i_*\langle n-1,a_1,\ldots,a_{2q-1}\rangle=0; \]
\[ i_*\langle 0,a_1,\ldots,a_{2q-1}\rangle=0, \qquad \text{if the sum } \sum a_i \text{ is even;} \]
\[ i_*\langle 0,a_1,\ldots,a_{2q-1}\rangle=[0,\sum a_i], \qquad \text{if the sum } \sum a_i \text{ is odd.} \]

2) Symmetric cycles \((0\le j\le n-1;\ \sum a_i>0)\)
\[ i_*\langle j,a_1,\ldots,a_{2q-1}\rangle=0, \qquad \text{if } \sum a_i \text{ is even;} \]
\[ i_*\langle j,a_1,\ldots,a_{2q-1}\rangle=[j,\sum a_i], \qquad \text{if } \sum a_i \text{ is odd.} \]

3)
\[ i_*\langle n-1,-0,+0,\ldots,+0,-0\rangle=0. \]

Having computed the cohomology rings \(H([n-1,2q-1])\) and studied the associated homomorphisms of cohomology groups, we obtain the theorem:

Theorem 4. In the cohomology ring \(H(\hat{\Omega}(S^n))\);
\[ \{0,2l-1\}*\{0,2k-1\} = \left[ \frac{(k+l-2)!}{(k-1)!(l-1)!} \right]\{n-1,2(k+l)-3\} \]
\[ (\text{for } k=1,2,\ldots;\ l=1,2,\ldots), \]
\[ \{j,2k-1\}*\{i,2l-1\}=0 \quad \text{in all other cases.} \]

Analogously to the way in which in § 17 of (⁶) the basic cocycles were constructed, in the space \(\hat{\Omega}(S^n)/O\) one can construct a one-dimensional cocycle \(\{1\}\), dual to the cycle \(\langle 1,-0,+0,\ldots,+0,-0\rangle\).

Theorem 5. The product of the cocycles \(\{j,2k-1\}\) of the basic system and the cocycle \(\{1\}\) is determined by the multiplication table:
\[ \{j,2k-1\}*\{1\}=\{j+1,2k-1\} \quad \text{for } j<n-1; \]
\[ \{n-1,2k-1\}*\{1\}=0. \]

6. Estimates of the number of geodesic loops on a Riemannian manifold.

From Theorems 1 and 6, with the aid of (⁶), it follows:

Theorem 6. Let \(M^n\) be a Riemannian manifold of class \(C^3\), homeomorphic to the \(n\)-dimensional sphere. Then: 1) on \(M^n\) there exists a countable sequence of series of geodesic loops with base point at \(m_0\in M^n\); 2) each series consists of \(n\) geodesic loops; 3) if the lengths \(C_p\) and \(C_q\) of two geodesic loops from one series coincide, then there exists a \((q-p)\)-dimensional set of geodesic loops of equal length \(C=C_p=C_q\).

Theorem 7. If \(M^n\) satisfies, in addition, the metric restriction of Morse (⁵, ⁶)
\[ 0<m\le dl/ds\le M<2m, \]
then the lengths of the geodesic loops of the first series satisfy the inequalities
\[ 2\pi m\le C_j\le 2\pi M, \]
and among them there is no pair in which one geodesic is a multiple traversal of the other.

The example of the \(n\)-dimensional ellipsoid shows that the estimate found cannot be improved.

Remark. By the same method one can investigate the homology of a number of other functional spaces.

Gorky State University
named after N. I. Lobachevsky

Received
30 X 1963

References

¹ G. Seifert, B. Threlfall, Variational Calculus in the Large, IL, 1947.
² A. I. Fet, Mat. sborn., 30 (72), 2, 270 (1952).
³ J.-P. Serre, Ann. Math., 54, 3, 425 (1951).
⁴ A. S. Schwarz, UMN, 13, 6 (84), 181 (1958).
⁵ L. A. Lyusternik, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 19 (1947).
⁶ S. I. Al’ber, UMN, 12, 4 (76), 57 (1957).
⁷ S. I. Al’ber, DAN, 108, No. 5, 763 (1956).
⁸ S. I. Al’ber, Tr. III matem. congress, 2, Moscow, 1956, p. 51.
⁹ A. S. Schwarz, Tr. Moskovsk. matem. obshch., 9, 3 (1960).