Nodar Berikashvili
For a fibration with the fiber $K(π,n)$-space, the algebraic model as a twisted tensor product of chains of the base with standard chains of $K(π,n)$-complex is given which preserves multiplicative structure as well. In terms of this model the action of the $n$-cohomology of the base with coefficients in $π$ on the homology of fibration is described.
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We define $\mathcal{H}$-fibration sequences as fibrations where the holonomy action of the fundamental group of the base on the fiber lies in a given subgroup $\mathcal{H}$ of $\mathcal{E}(F)$, where $\mathcal{E}(F)$ is the homotopy automorphism group of the fiber. Furthermore, we classify these $\mathcal{H}$-fibration sequences via a universal $\mathcal{H}$-fibration sequence.
In this note we define fibrations of topological stacks and establish their main properties. We prove various standard results about fibrations (fiber homotopy exact sequence, Leray-Serre and Eilenberg-Moore spectral sequences, etc.). We prove various criteria for a morphism of topological stacks to be a fibration, and use these to produce examples of fibrations. We prove that every morphism of topological stacks factors through a fibration and construct the homotopy fiber of a morphism of topological stacks. When restricted to topological spaces our notion of fibration coincides with the clas…
Let $G$ be a finite group and $\mathcal{H}$ be a family of subgroups of $G$ which is closed under conjugation and taking subgroups. Let $B$ be a $G$-$CW$-complex whose isotropy subgroups are in $\mathcal{H}$ and let $\mathcal{F}= \{F_H\}_{H \in \mathcal{H}}$ be a compatible family of $H$-spaces. A $G$-fibration over $B$ with fiber $\mathcal{F}= \{F_H\}_{H \in \mathcal{H}}$ is a $G$-equivariant fibration $p:E \rightarrow B$ where $p^{-1}(b)$ is $G_b$-homotopy equivalent to $F_{G_b}$ for each $b \in B$. In this paper, we develop an obstruction theory for constructing $G$-fibrations with fiber $\…
Much work has been done on the existence and uniqueness of broken Lefschetz fibrations such as those by Auroux et al., Gay and Kirby, Lekili, Akbulut and Karakurt, Baykur, and Williams, but there has been a lack of explicit examples. A theorem of Gay and Kirby suggests the existence of a broken Lefschetz fibration of S^4 over S^2 with a 2-knot fiber. In the case of a spun or twist-spun torus knot, we present a procedure to construct such fibrations explicitly. The fibrations constructed have no cusps nor Lefschetz singularities.
We prove that for a fibration of simply-connected spaces of finite type $F\hookrightarrow E\to B$ with $F$ being positively elliptic and $H^*(F,\qq)$ not possessing non-trivial derivations of negative degree, the base $B$ is formal if and only if the total space $E$ is formal. Moreover, in this case the fibration map is a formal map. As a geometric application we show that positive quaternion Kähler manifolds are formal and so are their associated twistor fibration maps.
We propose a simplified definition of Quillen's fibration sequences in a pointed model category that fully captures the theory, although it is completely independent of the concept of action. This advantage arises from the understanding that the homotopy theory of the model category's arrow category contains all homotopical information about its long fibration sequences.
We describe pairs (p,n) such that n-dimensional affine space is fibered by pairwise skew p-dimensional affine subspaces. The problem is closely related with the theorem of Adams on vector fields on spheres and the Hurwitz-Radon theory of composition of quadratic forms.
The purpose of this paper is to give some solutions for the classification problem in fibration theory by using the homotopy sequences of fibrations (sequences of $n$-th homotopy groups $ π_{n}(S,s_{o}) $ of total spaces of fibrations). In particular, to show the role of homotopy sequence of $n$-th homotopy to get the required fiber map in Fadell-Dold theorem such that the restriction of this fiber map on some fiber spaces is a homotopy equivalence.
Consider a fibration \[p:E\to S^{p-1}\] with fiber $F$. We have the following natural question: Under what conditions does this fibration admit a cross-section? Our purpose is to discuss this problem when the fibration $p$ is the Milnor fibration $f_{\mid}:f^{-1}(S^{p-1}_δ)\cap D^{n}_ε\to S^{p-1}_δ$ with Milnor fiber $F_f$ and the Milnor fibration of arrangements $Q:\mathcal{M}(\mathcal{A})\to \mathbb{C}^\ast$ with fiber $F=F(\mathcal{A})$. Furthermore, we use our results to study the tasking planning problem for the Milnor fibration as a work map and we give the tasking algorithms.
Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent diagrams and their homotopy colimits. We also comment on the simple homotopy type of the total space and give an application to the fibering of Hilbert cube manifolds.
Let $f:E\longrightarrow O$ be a Hurewicz fibration with a fiber space $F_{r_{o}}$ and a lifting function $L_{f}$. The \emph{$Lf-$function} $Θ_{L_{f}}$ of $f$ is defined by the restriction map of $L_{f}$ on the space $Ω(O,r_{o})\times F_{r_{o}}\times \{1\}$. The purpose of this paper is to give some results which show the role of $Lf-$functions in finding a fiber homotopically equivalent relation between two fibrations, over a common polyhedron base. Furthermore we will prove the equivalently between our results and Dold's theorem in fiber bundles, over a common suspension base of polyhedron sp…
We investigate the bounded cohomology of Lefschetz fibrations. If a Lefschetz fibration has regular fiber of genus at least 2 and it has at least two distinct vanishing cycles, we show that its Euler class is not bounded. As a consequence, we exclude the existence of negatively curved metrics on Lefschetz fibrations with more than one singular fiber.
We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibers admits a reduction to a fiber bundle with compact topological manifold fibers. The criteria lead to a new and unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjective splitting of the assembly map for Waldhausen's functor A(X). We also give concrete examples of fibrations having a reduction to a fiber bundle with compact topological manifold fibers but which fail to admit a compact fiber smoothing. The examples are detected by algebraic K-theory invariant…
We construct characteristic classes of smooth (Hamiltonian) fibrations as as fiber integrals of products of Pontriagin (or Chern) classes of vertical vector bundles over the total space of the universal fibration. We give explicit formulae of these fiber integrals for toric manifolds and get estimates of the dimension of the cohomology groups of classifying spaces.
Let $M$ be a Liouville 6-manifold which is the smooth fiber of a Lefschetz fibration on $\mathbb{C}^4$ constructed by suspending a Lefschetz fibration on $\mathbb{C}^3$. We prove that for many examples including stabilizations of Milnor fibers of hypersurface cusp singularities, the compact Fukaya category $\mathcal{F}(M)$ and the wrapped Fukaya category $\mathcal{W}(M)$ are related through $A_\infty$-Koszul duality, by identifying them with cyclic and Calabi-Yau completions of the same quiver algebra. This implies the split-generation of the compact Fukaya category $\mathcal{F}(M)$ by vanishi…
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