Formality of certain CW complexes and applications to Schubert varieties and torus manifolds (Q2870993)

Any path connected topological space \(X\) has a functorial differential graded commutative algebra (abbrievated dgca) \(A_{PL}(X)\) over \(\mathbb Q\), a \textit{minimal model} \(({\mathcal{M}}_X, d)\) (which is a dgca) and a dgc algebra morphism \({\rho}_X : {\mathcal{M}}_X \rightarrow A_{PL}(X)\) such that \({\rho}_X\) induces an isomorphism in the cohomology. The minimal model is unique up to isomorphism. The space \(X\) is called \textit{formal} if there exists a dgca morphism \(({\mathcal{M}}_X, d)\rightarrow (H^* (X, \mathbb Q),0)\) which induces an isomorphism in the cohomology.NEWLINENEWLINEIf \(X\) is a simply-connected space, its rational homotopy type is determined by \({\mathcal{M}}_X\). In case \(X\) is formal, \({\mathcal{M}}_X\) is determined by \(H^* (X, \mathbb Q)\) and so the rational homotopy type of \(X\) is a formal consequence of its cohomology algebra.NEWLINENEWLINELet \(X\) be a simply-connected formal \(CW\) complex and let \(\alpha : S^{n-1}\rightarrow X\) represent an element in the kernel of the Hurewicz homomorphism \(\eta : {\pi}_{n-1}(X) \to H_{n-1}(X; \mathbb Q)\). Let \(Y = X {\cup}_{\alpha} e^n\). Then there are the inclusion map \(j : Y \rightarrow (Y, X)\) and the characteristic map \((D^n, S^{n-1})\rightarrow (Y, X)\). Hence \( j^* : H^n (Y, X; \mathbb Z) \rightarrow H^n (Y ; \mathbb Z)\) maps the positive generator \(\tilde{u} \in H^n (Y, X; \mathbb Z) \cong H^n (D, S^{n-1}) \cong \mathbb Z\) to a non-zero element \(u\) in \(H^n(Y ; \mathbb Z)\). If \(\alpha\) represents a torsion element in \({\pi}_{n-1}(X)\), then \(u\) is a non-zero indecomposable element in \(H^n (Y ; \mathbb Q)\). In this case, denoting the rationalization of \(X\) by \(X_0\), we see that \(X_0 {\cup}_{\alpha} e^n\) is homotopy equivalent to \(X_0 \vee S^n\). It follows that \(Y\) is rationally equivalent to \(X \vee S^n\) which is a formal space.NEWLINENEWLINEWe know that a minimal model of a simply-connected space is isomorphic as a graded algebra to \(\Lambda V\) where \(V\) is a graded \(\mathbb Q\)-vector space \(V= {\bigoplus}_{k\geq 2} V^k\) and \(\Lambda V\) stands for the free graded-commutative algebra over \(V\). Thus, \(\Lambda V\) is isomorphic to the tensor product over \(\mathbb Q\) of the symmetric algebra of \(V^{even} = {\bigoplus} V^{2k}\) and the exterior algebra of \(V^{odd} = {\bigoplus} V^{2k-1}\).NEWLINENEWLINENow we can discuss the present paper. I should indicate that there is a serious error in the proof of Theorem \(1.4\) of that paper (which has been brought to notice by the authors). That theorem is used to derive other results in the paper. The authors have corrected the error and salvaged the theorem in a weaker form in the new paper [``Formality of certain CW complexes'', \url{arXiv:1301.5421}]. This is the corrected version of the earlier version which contained the error in Theorem 1.4. This theorem, which now Theorem 1.1 of in the new version, has now been corrected. The corrected version is as follows:NEWLINENEWLINESuppose that \(X\) is a simply connected \(CW\) complex and is formal. Let \({\mathcal{M}}_X = \Lambda (V )\) and suppose that \(V= {\bigoplus}_{k\geq 0} V^k\) is a standard lower gradation of \(V\). Let \(Y = X {\cup}_{\alpha} e^n\). Suppose that \(\eta ([\alpha]) = 0\) so that \(j^* (\tilde{u}) =: u \neq 0\). (i) If \([\alpha] \in {\pi}_{n-1}(X)\) is a torsion element then \(u\) is indecomposable and \(Y\) is formal. (ii) Let \([\alpha] \neq 0\) in \({\pi}_{n-1}(X) \otimes \mathbb Q\). Suppose that \(\langle v, [\alpha]\rangle = 0\) for all \(v \in V_k, k \neq 1\), and that \(u\) is decomposable in \( H^* (Y ; \mathbb Q)\). Then \(Y\) is formal. (iii) If \([\alpha] \in {\pi}_{n-1}(X)\) is not a torsion element and \(u\) is not decomposable, then \(Y\) is not formal.NEWLINENEWLINEThe proofs of Theorems 1.1, 1.2, and 1.3 of the present version are not valid as they used the erroneous result. In fact, the authors in the new version of the present paper provide a counterexample to the assertion of Theorem 1.1 of the present paper there. Another thing is that the statement of Theorem 1.2, which asserted the formality of Schubert varieties in a generalized flag variety \(G/B\), is not valid. Theorem 1.3 is correct as stated as it had been proved previously by \textit{T. E. Panov} and \textit{N. Ray} in [Contemp. Math. 460, 293--322 (2008; Zbl 1149.55014)] using entirely different techniques.