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Register a new userFibrations, unique path lifting, and continuous monodromy
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Given a path-connected space $X$ and $Hleqpi_1(X,x_0)$, there is essentially only one construction of a map $p_H:(widetilde{X}_H,widetilde{x}_0)rightarrow(X,x_0)$ with connected and locally path-connected domain that can possibly have the following two properties: $(p_{H})_{#}pi_1(widetilde{X}_H,widetilde{x}_0)=H$ and $p_H$ has the unique lifting property. $widetilde{X}_H$ consists of equivalence classes of paths starting at $x_0$, appropriately topologized, and $p_H$ is the endpoint projection. For $p_H$ to have these two properties, $T_1$ fibers are necessary and unique path lifting is sufficient. However, $p_H$ always admits the standard lifts of paths. We show that $p_H$ has unique path lifting if it has continuous (standard) monodromies toward a $T_1$ fiber over $x_0$. Assuming, in addition, that $H$ is locally quasinormal (e.g., if $H$ is normal) we show that $X$ is homotopically path Hausdorff relative to $H$. We show that $p_H$ is a fibration if $X$ is locally path connected, $H$ is locally quasinormal, and all (standard) monodromies are continuous.
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We show that in a fibration the coformality of the base space implies the coformality of the total space under reasonable conditions, and these conditions can not be weakened. The result is partially dual to the classical work of Lupton cite{Lup} on the formality within a fibration. Our result has two applications. First, we show that for certain cofibrations, the coformality of the cofiber implies the coformality of the base. Secondly, we show that the total spaces of certain spherical fibrations are Koszul in the sense of Berglund cite{Ber}.
A fibration of ${mathbb R}^n$ by oriented copies of ${mathbb R}^p$ is called skew if no two fibers intersect nor contain parallel directions. Conditions on $p$ and $n$ for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of ${mathbb R}^3$ by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of $S^3$ by Gluck and Warner. We show that Salvais classification has a topological variation which generalizes to characterize all continuous fibrations of ${mathbb R}^n$ by skew oriented copies of ${mathbb R}^p$. We show that the space of fibrations of ${mathbb R}^3$ by skew oriented lines deformation retracts to the subspace of Hopf fibrations, and therefore has the homotopy type of a pair of disjoint copies of $S^2$. We discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of ${mathbb C}^n$ (${mathbb H}^n$) by skew oriented copies of ${mathbb C}^p$ (${mathbb H}^p$).
We prove that the sectional category of the universal fibration with fibre X, for X any space that satisfies a well-known conjecture of Halperin, equals one after rationalization.
Let G be a profinite group, {X_alpha}_alpha a cofiltered diagram of discrete G-spectra, and Z a spectrum with trivial G-action. We show how to define the homotopy fixed point spectrum F(Z, holim_alpha X_alpha)^{hG} and that when G has finite virtual cohomological dimension (vcd), it is equivalent to F(Z, holim_alpha (X_alpha)^{hG}). With these tools, we show that the K(n)-local Spanier-Whitehead dual is always a homotopy fixed point spectrum, a well-known Adams-type spectral sequence is actually a descent spectral sequence, and, for a sufficiently nice k-local profinite G-Galois extension E, with K a closed normal subgroup of G, the equivalence (E^{h_kK})^{h_kG/K} simeq E^{h_kG} (due to Behrens and the author), where (-)^{h_k(-)} denotes k-local homotopy fixed points, can be upgraded to an equivalence that just uses ordinary (non-local) homotopy fixed points, when G/K has finite vcd.
We discuss the history of the monodromy theorem, starting from Weierstrass, and the concept of monodromy group. From this viewpoint we compare then the Weierstrass , the Legendre and other normal forms for elliptic curves, explaining their geometric meaning and distinguishing them by their stabilizer in P SL(2,Z) and their monodromy. Then we focus on the birth of the concept of the Jacobian variety, and the geometrization of the theory of Abelian functions and integrals. We end illustrating the methods of complex analysis in the simplest issue, the difference equation $f(z) = g(z+1) - g(z)$ on $mathbb C$.
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