The degree of equivariant maps. (English) Zbl 1064.57038
The \(G\)-index of a \(G\)-space \(X\) with coefficients in \(K\), \(Ind^G(X,K)\), is the kernel of the cohomology map \(c^*_G: H^*_G(*;K) \to H^*_G(X;K)\), where \(c: X \to *\) is the constant map into the one-point space \(*\). If \(G\) acts freely on \(X\) then \(H^*_G(*;K) \cong H^*(BG;K)\), where \(BG\) is the classifying space for \(G\); if \(\phi: X \to BG\) is a classifying map for \(X\), then under that isomorphism ker \(c^*_G \cong ker \phi*\). If \(X\) and \(Y\) are \(G\)-spaces and \(f: X \to Y\) is an equivariant map then \(Ind^G(Y,K) \subset Ind^G(X,K)\). The component of \(Ind^G(X,K)\) in dimension \(q\) is denoted by \(Ind^G_q(X,K)\). Let \(G\) be a compact Lie group acting freely on compact, connected, smooth manifolds \(M\) and \(N\), both of dimension \(n\). The author studies the degree of \(G\)-maps \(f: M \to N\). In this case \(M/G\) and \(N/G\) are manifolds of dimension \(n-dimG\) and \(H^{n-dimG} (M/G;{\mathbb Z}_2) \cong H^{n-dimG}(N/G;{\mathbb Z}_2) \cong {\mathbb Z}_2\).
The main results of the paper are contained in the following theorem. Theorem. Let \(f: M \to N\) be a \(G\)-map. (1) If \(Ind^G_{n-dimG}(N,{\mathbb Z}_2) = Ind^G_{n-dimG}(M,{\mathbb Z}_2) \neq H^{n-dimG}(BG;{\mathbb Z}_2)\) then \(f^*: H^*(N;{\mathbb Z}_2) \to H^*(M;{\mathbb Z}_2)\) is an isomorphism; if both \(M\) and \(N\) are oriented then the degree of \(f\) is odd. (2) If \(Ind^G_{n-dimG}(N,{\mathbb Z}_2) \subset Ind^G_{n-dimG}(M,{\mathbb Z}_2) = H^*(BG;{\mathbb Z}_2)\) (but \(Ind^G_{n-dimG}(N,{\mathbb Z}_2) \neq Ind^G_{n-dimG}(M,{\mathbb Z}_2)\)) then \(f^*: H^*(N/G;{\mathbb Z}_2) \to H^*(M/G;{\mathbb Z}_2)\) is zero; if both \(M\) and \(N\) are oriented then the degree of \(f\) is even. The transfer map (or “integration along the fibre” construction) plays a crucial role in the proofs. Analogous results for cohomology mod \(p\), where \(p\) is an odd prime, are also proved. These results are applied to maps of real and complex Stiefel manifolds. For instance, it is shown that if \(f: V_k({\mathbb R}^m) \to V_k({\mathbb R}^m)\) is an \(O(k)\)-map, then the degree of \(f\) is odd; if \(f: V_k({\mathbb C}^m) \to V_k({\mathbb C}^m)\) is an \(U(k)\)-map, then the degree of \(f\) is \(1\) or \(-1\).
The main results of the paper are contained in the following theorem. Theorem. Let \(f: M \to N\) be a \(G\)-map. (1) If \(Ind^G_{n-dimG}(N,{\mathbb Z}_2) = Ind^G_{n-dimG}(M,{\mathbb Z}_2) \neq H^{n-dimG}(BG;{\mathbb Z}_2)\) then \(f^*: H^*(N;{\mathbb Z}_2) \to H^*(M;{\mathbb Z}_2)\) is an isomorphism; if both \(M\) and \(N\) are oriented then the degree of \(f\) is odd. (2) If \(Ind^G_{n-dimG}(N,{\mathbb Z}_2) \subset Ind^G_{n-dimG}(M,{\mathbb Z}_2) = H^*(BG;{\mathbb Z}_2)\) (but \(Ind^G_{n-dimG}(N,{\mathbb Z}_2) \neq Ind^G_{n-dimG}(M,{\mathbb Z}_2)\)) then \(f^*: H^*(N/G;{\mathbb Z}_2) \to H^*(M/G;{\mathbb Z}_2)\) is zero; if both \(M\) and \(N\) are oriented then the degree of \(f\) is even. The transfer map (or “integration along the fibre” construction) plays a crucial role in the proofs. Analogous results for cohomology mod \(p\), where \(p\) is an odd prime, are also proved. These results are applied to maps of real and complex Stiefel manifolds. For instance, it is shown that if \(f: V_k({\mathbb R}^m) \to V_k({\mathbb R}^m)\) is an \(O(k)\)-map, then the degree of \(f\) is odd; if \(f: V_k({\mathbb C}^m) \to V_k({\mathbb C}^m)\) is an \(U(k)\)-map, then the degree of \(f\) is \(1\) or \(-1\).
Reviewer: Jan Jaworowski (Bloomington)
MSC:
| 57R91 | Equivariant algebraic topology of manifolds |
| 58E40 | Variational aspects of group actions in infinite-dimensional spaces |
| 57S15 | Compact Lie groups of differentiable transformations |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 55M25 | Degree, winding number |
| 57R19 | Algebraic topology on manifolds and differential topology |
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