MATHEMATICS

A. S. SCHWARZ

HOMOTOPIC DUALITY FOR A SPACE WITH A GROUP OF OPERATORS

(Presented by Academician P. S. Aleksandrov, July 7, 1960)

The purpose of the present note is to generalize Spanier–Whitehead duality \(^{(1,2)}\) to spaces with a group of operators.

Let \(G\) be a topological group. We shall call a topological space \(X\) a \(G\)-space if the group \(G\) acts in \(X\) without fixed points, giving rise to a principal fibration \(\pi : X \to X_G\) (by \(X_G\) we denote the space of trajectories of the group \(G\) acting in \(X\)). A \(G\)-space \(X\) is called a \(G\)-polyhedron if the spaces \(X\) and \(X_G\) are finite polyhedra. The set of homotopy classes of admissible (i.e., commuting with the transformations of the group \(G\)) mappings of a \(G\)-space \(X\) into a \(G\)-space \(Y\), relative to admissible homotopies, will be denoted by \([X,Y]\).

The join \(X * Y\) of two \(G\)-spaces \(X\) and \(Y\) naturally becomes a \(G\)-space (the points of the join \(X * Y\) are represented by triples \((x,y,t)\) with the identifications \((x,y,0)\sim(x,y',0)\); \((x,y,1)\sim(x',y,1)\); the group \(G\) acts on \(X * Y\) by the formula \(g(x,y,t)=(g(x),g(y),t)\); here \(x,x'\in X\), \(y,y'\in Y\), \(g\in G\), \(0\le t\le 1\)). If \(\varphi:X\to X'\), \(\psi:Y\to Y'\) are admissible mappings of \(G\)-spaces, then their join is naturally defined: the admissible mapping \(\varphi * \psi:X*Y\to X'*Y'\).

By the homology groups \(H_i(X;A)\) (cohomology groups \(H^i(X;A)\)) we shall always mean the reduced singular homology (cohomology) groups. We note that the group \(G\) acts in the homology and cohomology groups of a \(G\)-space.

We fix some \(G\)-space \(R\). The join \(R * \cdots * R\) of \(k\) copies of the \(G\)-space \(R\) will be denoted by \(R_k\). By an \(R\)-mapping of a \(G\)-space \(X\) into a \(G\)-space \(Y\) we shall mean an admissible mapping \(\varphi:R_k*X\to R_k*Y\), where \(k=0,1,2,\ldots\). If \(\varphi\) is an admissible mapping of \(R_k*X\) into \(R_k*Y\), then there is naturally defined an admissible mapping \(r_l(\varphi):R_{k+l}*X\to R_{k+l}*Y\) as the join of the identity mapping of the space \(R_l\) and the mapping \(\varphi\) (here \(R_{k+l}*X\) and \(R_{k+l}*Y\) are considered respectively as \(R_l*(R_k*X)\) and \(R_l*(R_k*Y)\)). Two \(R\)-mappings \(\varphi:R_k*X\to R_k*Y\) and \(\psi:R_l*X\to R_l*Y\) will be called \(R\)-homotopic if, for some \(n\ge k,l\), the mappings \(r_{n-k}(\varphi)\) and \(r_{n-l}(\psi)\) are admissibly homotopic. The set of \(R\)-homotopy classes of \(R\)-mappings of \(X\) into \(Y\) will be denoted by \(\{X,Y\}\). We note that the sets \([R_k*X,R_k*Y]\), in particular the set \([X,Y]\), are naturally mapped into the set \(\{X,Y\}\); the set \(\{X,Y\}\) may be regarded as the limit of the direct spectrum of the sets \([R_k*X,R_k*Y]\) with respect to the mappings
\[ [R_k*X,R_k*Y]\to [R_{k+1}*X,R_{k+1}*Y]. \]
If \(\alpha\in\{X,Y\}\) and \(\beta\in\{Y,Z\}\), then, by means of composing the \(R\)-mappings defining the elements \(\alpha\) and \(\beta\), a composition of these elements \(\beta\cdot\alpha\in\{X,Z\}\) can be defined.

In what follows we shall assume that the fixed \(G\)-space \(R\) is an \(r\)-dimensional \(G\)-polyhedron, homotopy equivalent to the \(r\)-dimensional sphere.

Theorem 1. Let \(Y\) be a \(G\)-space, aspherical in dimensions \(< n\) (i.e., \(Y\) is connected and \(\pi_i(Y)=0\) for \(1 \leq i < n\)), and let \(X\) be a \(G\)-polyhedron with trajectory space \(X_G\) of dimension \(\leq 2n-2\). Then the natural mapping of the set \([X,Y]\) into the set \(\{X,Y\}\) is one-to-one.

Corollary. If \(X\) and \(Y\) are \(G\)-polyhedra, then for sufficiently large \(k\) the natural mapping of the set \([R_k * X, R_k * Y]\) into the set \(\{X,Y\}\) is one-to-one.

Every \(R\)-mapping of a \(G\)-space \(X\) into a \(G\)-space \(Y\), by virtue of the relation \(H_i(X;A)=H_{i+(r+1)k}(R_k * X;A)\), induces a homomorphism of the group \(H_i(X;A)\) into the group \(H_i(Y;A)\), commuting with the transformations of the group \(G\) (here \(A\) is any abelian group).

Let \(X,Y,Z\) be \(G\)-spaces, \(f\) an \(R\)-mapping of the \(G\)-space \(X * Y\) into the \(G\)-space \(Z\), and \(A\) some field. It is known \((^3)\) that the tensor product \(H_i(X;A)\otimes H_{s-i-1}(Y;A)\) is naturally embedded in \(H_s(X*Y,A)\); therefore the mapping \(f_*\) gives rise to a homomorphism of the group \(H_i(X;A)\otimes H_{s-i-1}(Y;A)\) into the group \(H_s(Z;A)\), or, in other words, a pairing of the groups \(H_i(X;A)\) and \(H_{s-i-1}(Y;A)\) in the group \(H_s(Z;A)\). If the space \(Z\) is homotopy equivalent to the \(s\)-dimensional sphere, then the pairing of the groups \(H_i(X;A)\) and \(H_{s-i-1}(Y;A)\) in the group \(H_s(Z;A)=A\) may be regarded as a scalar product (recall that the groups \(H_i(X;A)\) and \(H_{s-i-1}(Y;A)\) are endowed with the structure of vector spaces over the field \(A\)).

Definition. We shall say that the \(G\)-polyhedron \(Y\) is \(n\)-dual to the \(G\)-polyhedron \(X\), and write \(Y=D_nX\), if there exists such an \(R\)-mapping of the \(G\)-space \(X*Y\) into \(R_n\) that the scalar product of the vector spaces \(H_i(X;A)\) and \(H_{n(r+1)-i-2}(Y;A)\), defined by this \(R\)-mapping, is nondegenerate for every field \(A\) and every \(i \geq 0\).

We note the following properties of the duality operator \(D_n\):

1. If \(Y=D_nX\), then \(X=D_nY\) (i.e., \(D_n=D_n^{-1}\)).
2. \(R_k * D_nX=D_{n+k}X;\quad D_n(R_k * X)=D_{n+k}X\).
3. \(D_nX * D_mX'=D_{m+n}(X*X')\).
4. If \(Y=D_nX\), then there exists an isomorphism \(D_n^*: H_i(X)\to H^{n(r+1)-i-2}(Y)\), commuting with the transformations of the group \(G\).
5. If \(Y=D_nX\), then the polyhedron \(Y\) is weakly \((nr+n-1)\)-dual to the polyhedron \(X\) in the sense of Spanier—Whitehead \((^1)\).
6. The space \(D_nX\) is determined up to \(R\)-homotopy type (in other words, if \(Y=D_nX\) and \(Y'=D_nX\), then for some \(k \geq 0\) there is an admissible homotopy equivalence \(\varphi:R_k*Y\to R_k*Y'\)).

Let \(X\) and \(Y\) be \(G\)-polyhedra, \(Y'=D_nY\), and let
\(u:R_k*Y*Y'\to R_k*R_n=R_{k+n}\) be an \(R\)-mapping of \(Y*Y'\) into \(R_n\), giving the duality of \(Y\) and \(Y'\). Define a mapping \(P_u\) of the set \(\{X,Y\}\) into the set \(\{X*Y',R_n\}\) by means of the following convention. If an element \(\alpha\in\{X,Y\}\) is defined by a mapping \(\varphi:R_l*X\to R_l*Y\), then the element \(P_u(\alpha)\in\{X*Y',R_n\}\) is specified by the superposition \(\psi\) of the mappings \(\varphi*1:(R_l*X)*(R_k*Y')\to(R_l*Y)*(R_k*Y')\) and \(1*u:R_l*(R_k*Y*Y')\to R_l*(R_k*R_n)\) (by \(1\) we denote the identity mapping; one may speak of the superposition of the mappings \(\varphi*1\) and \(1*u\), since \((R_l*Y)*(R_k*Y')=R_l*(R_k*Y*Y')\); the mapping \(\psi:(R_l*X)*(R_k*Y')\to R_l*(R_k*R_n)\), by virtue of the relations \((R_k*X)*(R_k*Y')=R_{k+l}*(X*Y')\), \(R_l*(R_k*R_n)=R_{k+l}*R_n\), may be regarded as an \(R\)-mapping of \(X*Y'\) into \(R_n\)).

Main Lemma. The mapping \(P_u\) is a one-to-one correspondence between the sets \(\{X,Y\}\) and \(\{X*Y',R_n\}\).

Definition 2. Let \(X\) and \(Y\) be \(G\)-polyhedra, \(X'=D_nX\), \(Y'=D_nY\) their \(n\)-dual \(G\)-polyhedra; \(u\) and \(v\) be \(R\)-mappingsж

\(X * X'\) in \(R_n\) and \(Y * Y'\) in \(R_n\), generating these duality relations. We shall say that an element \(a \in \{X, Y\}\) is \(n\)-dual to an element \(\beta \in \{Y', X'\} = \{D_nY, D_nX\}\), and write \(\beta = D_n(u, v)(a)\) (or simply \(\beta = D_n(a)\)), if \(P_u(a) = P_v(\beta)\) (the element \(P_u(a)\) lies in the set \(\{X * Y', R_n\}\), the element \(\beta\) in the set \(\{Y' * X, R_n\}\), but \(X * Y' = Y' * X\), and therefore the equality \(P_u(a) = P_v(\beta)\) makes sense).

From the fundamental lemma the following follows.

Theorem 2. The mapping \(D_n(u, v)\) is a one-to-one correspondence between the sets \(\{X, Y\}\) and \(\{D_nY, D_nX\}\).

Let us indicate some properties of the mapping
\[ D_n : \{X, Y\} \to \{D_nY, D_nX\}: \]

1. The mapping \(D_n\) takes the composition \(\beta \cdot a\) of elements \(a \in \{X, Y\}\), \(\beta \in \{Y, Z\}\) to the composition \(D_n(a)\cdot D_n(\beta)\) of the elements \(D_n(\beta) \in \{D_nZ, D_nY\}\), \(D_n(a) \in \{D_nY, D_nX\}\).

2. The homomorphisms
   \[ H_i(X; A) \to H_i(Y; A) \]
   and
   \[ H_{n(r+1)-i-2}(D_nY; A) \to H_{n(r+1)-i-2}(D_nX; A), \]
   induced by the elements \(a \in \{X, Y\}\) and \(D_n(a) \in \{D_nY; D_nX\}\), are, for any field, adjoint homomorphisms with respect to the scalar products generated by the mappings defining the duality.

The previously indicated properties 1–5 of the duality operator \(D_n : X \to D_nX\) carry over (with the corresponding changes in the formulations) to the operator
\[ D_n : \{X, Y\} \to \{D_nY, D_nX\}. \]

The description of the operator \(D_n\) given above can be simplified in the case when \(R = S^r\) and the transformations of the group \(G\) on \(R = S^r\) are orthogonal transformations. Then the space \(R_n\) can be identified with the sphere \(S^{nr+n-1}\), and the transformations of the group \(G\) will also be orthogonal transformations on \(R_n = S^{nr+n-1}\).

Theorem 3. Let \(X, Y\) be disjoint subsets of the sphere \(R_n = S^{nr+n-1}\), invariant with respect to the group \(G\). Suppose that \(X, Y\) are \(G\)-polyhedra with respect to the action of the group \(G\) defined in them; \(Y\) is a deformation retract of the set \(R_n \setminus X\), and \(Y\) contains no points diametrically opposite to points of the set \(X\). Then the \(G\)-polyhedron \(Y\) is \(n\)-dual to the \(G\)-polyhedron \(X\) (i.e. \(Y = D_nX\)). If \(X', Y'\) are two other subsets of the sphere \(R_n\) satisfying the same conditions, and \(X \subset X'\), \(Y' \subset Y\), then the inclusion mappings \(X \to X'\), \(Y' \to Y\) generate \(n\)-dual elements of the sets \(\{X, X'\}\), \(\{Y', Y\}\).

Let us indicate how, under the conditions of this theorem, one constructs the duality-generating mapping \(u : X * Y \to R_n\). If a point \(a \in X * Y\) is determined by the triple \((x, y, t)\) \((x \in X, y \in Y, 0 \le t \le 1)\), then the point \(u(a) \in R_n\) is defined as the point that divides the smaller of the arcs of the great circle joining the points \(x\) and \(y\), in the ratio \(t : (1 - t)\).

The construction of duality described in Theorem 3 can be applied, in particular, to the cyclic group \(Z_n\) (for it one may take \(R = S^1\), and for \(n = 2\) also \(R = S^0\)) and to the groups \(S^1 = U(1)\) of complex numbers of modulus 1, and \(S^3 = \operatorname{Sp}(1)\) of quaternions of norm 1 (for these groups one may take for \(R\) the group space itself, in which the group acts by left translations).

Example. Let \(G = R = S^1\), and let \(X = Y_{n,2}\) be the Stiefel manifold of orthonormal 2-frames, in which an element \(e^{i\varphi} \in S^1\) acts by rotating each 2-frame through the angle \(\varphi\) in the plane of this frame. The \(S^1\)-space \(Y\), \(n\)-dual to the \(S^1\)-space \(X\), can be constructed as follows. The points of the space \(Y\) are pairs \((a, b)\), where \(a \in S^{n-1}\), \(b \in S^1\), with the identifications \((a, b) \sim (-a, -b)\); the group \(S^1\) acts in \(Y\) by the formula \(g(a, b) = (a, gb)\).

Remark 1. The \(G\)-space \(R\), with the aid of which the duality was constructed, was assumed here to be an \(r\)-dimensional polyhedron homotopy equivalent to the \(r\)-dimensional sphere. However, at the cost of some complication of the definitions, formulations, and proofs, almost all of the above-listed …

assertions can be carried over to the case where \(R\) is an arbitrary \(G\)-space homotopy equivalent to a sphere.

Remark 2. The above-indicated duality of \(G\)-spaces is closely connected with the duality of fiber spaces, the construction of which will be the subject of a separate note \({}^{4}\).

Voronezh State
University

Received
4 VII 1960

References

\({}^{1}\) E. Spanier, J. H. C. Whitehead, Mathematica, 2, 56 (1955).
\({}^{2}\) E. Spanier, Ann. Math., 70, No. 2, 338 (1959).
\({}^{3}\) J. Milnor, Ann. Math., 63, No. 3, 430 (1956).
\({}^{4}\) A. S. Shvarts, DAN, 136, No. 2 (1961).