Analytic cycles on real analytic manifolds (Q1889443)

Let \(X\) be a compact real analytic manifold of dimension \(n\); the authors study homology classes in \(H_* (X, \mathbb{Z}/2)\) represented by analytic subsets of \(X\) (here analytic means the set of zeros of global analytic functions). If \(H_k^{an} (X, \mathbb{Z}/ 2)\) is the subgroup of classes of \(H_k (X, \mathbb{Z}/2)\) corresponding to analytic sets \(V \subset X\) of dimension \(k\), and if \(H_{an}^l (X, \mathbb{Z}/2) = D^{-1}_X (H_k^{an} (X,\mathbb{Z}/2)\) where \(D_X\) is the Poincaré duality isomorphism, it is well known that \(H^1 (X, \mathbb{Z}/ 2) = H^1_{an} (X, \mathbb{Z}/2)\). But for \(H^2_{an} (X, \mathbb{Z}/2)\) the situation is different. The authors obtain an elegant description of this group. Namely, let \(W(X) = \{ v \in H^2(X, \mathbb{Z}/ 2) \mid v = w_2 (\xi) \text{ for some }\; \xi \in\text{Vect} (X) \}\) where Vect\(X\) is the set of isomorphism classes of topological vector bundles over \(X\) and \(w_2 (\xi)\) is the Stiefel-Whitney class of \(\xi\). Then (theorem 1.1) they prove that \(H^2_{an} (X, \mathbb{Z}/2) = W(X)\), i.e. that \(v\) a cohomology class in \(H^2(X, \mathbb{Z}/2)\) is represented by an analytic subset \(V\) iff \(v \in W(X)\). For this the authors prove first a real analytic version of Grothendieck formula, i.e. that for \(X\) as above and \(v \in H^2_{an} (X, \mathbb{Z}/2)\) there exists an analytic vector bundle \(\xi\) over \(X\) such that \(w_1 (\xi) = 0\) and \(w_2(\xi) = v\) which proves that \(H^2_{an} (X, \mathbb{Z}/Z) \subseteq W(X)\). The opposite inclusion follows from the fact that if \(\xi\) is a vector bundle of rank \(p\) over \(X\) there is \(f : X \to G_{m, p} (\mathbb{R})\) a continuous map (for some \(m\)) such that \(\xi \simeq F^* \gamma\) with \(\gamma\) the universal bundle over the Grassmanian \(G_{m, p}(\mathbb{R})\) and the \(h^* (G_{m, p} (\mathbb{R}), \mathbb{Z}/2) = H^*_{an} (G_{m, p} (\mathbb{R}, \mathbb{Z}/ 2)\) [\textit{J. Bochnak, M. Coste} and \textit{M.-F. Roy}, Real Algebraic Geometry, Springer (1998; Zbl 0912.14023)] and that (prop. 31) if \(f : X \to Y\) is a continuous map between real analytic manifolds, then the induced map \(f^*\) satisfies \(f^* (H^k_{an} (Y, \mathbb{Z}/2)) \subseteq H^k_{an} (X, \mathbb{Z}/2)\) for each \(k \geq 0\) (this last result is obtained using a Whitney statification of the analytic subset \(W\) of \(Y\) representing a cohomology class \(W \in H^k_{an}(Y, \mathbb{Z}/ 2)\), verifying condition a) which ensures openness of the set of all smooth maps from \(X\) to \(Y\) transverse to the strates of \(W\), and a result of \textit{A. Borel} and \textit{A. Haeflinger} [Bull. Soc. Math. Fr. 89, 461--513 (1961; Zbl 0102.38502)] that the analytic map \(g : X \to Y\), homotopic to \(f\) and transverse to the states of \(W\) has the property that \(v = D_X^{-1} ([V_0]_X)\) satisfies \(v = g^*(w)\) where \(V= g^{-1}(W)\). A corollary of theorem 1.1 is that for each \(n \geq 6\) there exists a compact \(n\)-dimensional real analytic manifold such that \(H^2(X, \mathbb{Z}/2) \neq H^2_{an} (X, \mathbb{Z}/2)\).