LS-category and topological complexity of several families of fibre bundles (Q6621497)

This paper is concerned with estimating the Lusternik--Schnirelmann catgeory (LS-category) and the topological complexity (in the sense of M.~Farber) of several classes of spaces, including some fiber bundles, higher dimensional Klein bottles, projective product spaces, and the so-called Dold manifolds of Grassmann type.\N\NRecall that the LS-category of a space \(X\), denote \(\mathsf{cat}(X)\), is the smallest integer \(n\) such that \(X\) can be covered by \(n\) number of open sets that are contractible in \(X\). Also recall that the topological complexity of \(X\), denoted \(\mathsf{TC}(X)\), is the smallest integer \(m\) such that \(X\times X\) can be covered by \(m\) number of open sets \(V_i\) over each of which there exists a continuous map \(s_i:V_i\to P(X)\) with the property that \(s_i(x,y)(0)=x\) and \(s_i(x,y)(1)=y\) for each \((x,y)\in V_i\). Here, \(P(X)=X^{[0,1]}\) denotes the path space of \(X\).\N\NThe problem of estimating the topological complexity of fiber bundles of manifolds is classical. For a Hurewicz fibration \(p:E\to B\) with fiber \(F\), Farber and Grant showed that \(\mathsf{TC}(E)\le \mathsf{TC}(F)\cdot\mathsf{cat}(B\times B)\) and asked whether \(\mathsf{TC}(E)\le \mathsf{TC}(F)\cdot\mathsf{TC}(B)\) is true. For some special fibrations, the former upper bound was improved by Grant. In the current paper, the authors ask for what fiber bundles \(p:E\to B\) with fiber \(F\) does the inequality \(\mathsf{TC}(E)\le\mathsf{TC}(F)+\mathsf{TC}(B)-1\) holds, and they answer this question for a specific family of fiber bundles (see Theorem 2.2).\N\NThe authors then estimate the topological complexity of the \(n\)-dimensional Klein bottle \(K_n\). Recall that \(K_n:=(S^1)^n/\sim\), where \((z_1,\ldots, z_{n-1},z_n)\sim (\overline{z}_1,\ldots,\overline{z}_{n-1},-z_n)\). Using a fiber bundle \((S^1)^{n-1}\hookrightarrow K_n\to \mathbb{R} P^1\), it has been shown in this paper that \(n+3\le\mathsf{TC}(K_n)\le \tfrac{3n+4}{2}\). These bounds turn out to be sharp in the cases \(n=2,3\).\N\NIn 2010, \textit{D. M. Davis} [J. Topol. 3, No. 2, 265--279 (2010; Zbl 1198.55008)] introduced the projective product space\N\[\NP(n_1,\ldots,n_r):=\frac{S^{n_1}\times\cdots\times S^{n_r}}{(x_1,\ldots,x_r)\sim (-x_1,\ldots,-x_r)},\N\]\Nwhose LS-category was computed by Fişekci and Vandembroucq. In the current paper, the authors introduce a generalization of these projective product spaces by considering involutions on \(S^{n_i}\) and topological spaces with free involutions. Then by studying the cohomology rings of these spaces, they estimate the LS-category and topological complexity of generalized projective product spaces in Propositions 4.3 and 4.5, respectively. Furthermore, they consider a specific class of generalized projective product spaces which consists of the following spaces:\N\[\NX_g^{n-2}:=\frac{(S^1)^{n-2}\times\Sigma_g}{(z_1,\ldots,z_{n-2},x)\sim (\overline{z}_1,\ldots,\overline{z}_{n-2},-x)},\N\]\Nwhere \(\Sigma_g\) denotes the closed orientable surface of genus \(g\). Using a fiber bundle \((S^1)^{n-2}\hookrightarrow X^{n-2}_g\to N_{g+1}\), where \(N_{g+1}\) denotes the closed non-orientable surface of genus \(g+1\), the authors conclude that \(\mathsf{cat}(X^{n-2}_g)=n+1\) and that \(n+4\le\mathsf{TC}(X^{n-2}_g)\le \tfrac{3n+5}{2}\) for each \(n\) and \(g\).\N\NThe authors then turn towards another class of generalized projective product spaces, called the Dold manifolds of Grassmann type. Let \(\text{Gr}_d(\mathbb{C}^n)\) (resp. \(\text{Gr}_d(\mathbb{R}^n)\)) be the set of \(d\)-dimensional subspaces of \(\mathbb{C}^n\) (resp. \(\mathbb{R}^n\)), and let \(\tau\) be a complex conjugation involution on \(\text{Gr}_d(\mathbb{C}^n)\) with fixed point set \(\text{Gr}_d(\mathbb{R}^n)\). Then\N\[\NX(\text{Gr}_d(\mathbb{C}^n),n_1,\ldots,n_r):=\frac{\text{Gr}_d(\mathbb{C}^n)\times S^{n_1} \times\cdots\times S^{n_r}}{(y,x_1,\ldots,x_r)\sim(\tau(y),-x_1,\ldots,-x_r)}\N\]\Nis a Dold manifold of Grassmann type. By studying the cohomology ring of \(X(\text{Gr}_d(\mathbb{C}^n),n_1,\ldots,n_r)\) and a fiber bundle \(\text{Gr}_d(\mathbb{C}^n)\hookrightarrow X(\text{Gr}_d(\mathbb{C}^n),n_1,\ldots,n_r)\to P(n_1,\ldots,n_r)\), the authors determine that \(\mathsf{cat}(X(\text{Gr}_d(\mathbb{C}^n),n_1,\ldots,n_r))=d(n-d)+n_1+r\) and they also bound the topological complexity of \(X(\text{Gr}_d(\mathbb{C}^n),n_1,\ldots,n_r)\) from above and below in Proposition 5.2 and Theorem 5.5.\N\NThe authors conclude this paper by studying the equivariant version of LS-category and topological complexity and doing some interesting computations for a family of \(\mathbb{Z}_2\)-spaces related to the generalized projective product spaces and Dold manifolds.