The Hopf Fibration over S Admits No S-subfibration*
نویسندگان
چکیده
It is shown that there does not exist a PL-bundle over S8 with fibre and total space PL-manifolds homotopy equivalent to CP and CP respectively. Consequently, the Hopf fibration over S8 admits no subfibration by PL-circles.
منابع مشابه
The Hopf Fibration and its Applications
Among the most important topological constructions are Hopf fibrations. That they both play an important role in topology itself and have numerous physical applications lends credence to this statement. In this paper, I will discuss some of these applications as motivation for mathematicians to learn about the physical significance of the Hopf fibration in greater detail than that presented her...
متن کاملSymplectic Lefschetz Fibrations on S ×m
A remarkable theorem of Donaldson [D] says that a symplectic 4manifold admits a Lefschetz pencil by symplectic surfaces. Given a Lefschetz pencil on a 4-manifold X, one can blow up points at the base locus to get a Lefschetz fibration of X over S, which admits a nice handlebody decomposition and can be described by geometric monodromy representation into the mapping class group of a regular fib...
متن کاملGeometry of Entangled States , Bloch Spheres and Hopf Fibrations
We discuss a generalization to 2 qubits of the standard Bloch sphere representation for a single qubit, in the framework of Hopf fibrations of high dimensional spheres by lower dimensional spheres. The single qubit Hilbert space is the 3-dimensional sphere S. The S base space of a suitably oriented S Hopf fibration is nothing but the Bloch sphere, while the circular fibres represent the qubit o...
متن کاملLefschetz Fibrations and 3-fold Branched Covering Spaces
Let M be a smooth 4-manifold which admits a genus g C∞-Lefschetz fibration over S, and assume that all of the vanishing cycles of this fibration are nonseparating curves. We show that M can be obtained as an irregular simple 3-fold cover of an S-bundle over S, branched over an embedded surface. Moreover, for g = 2, we show that M is a cyclic 2-fold cover of an S-bundle over S, branched over an ...
متن کاملHyperelliptic Lefschetz Fibrations and Branched Covering Spaces
Let M be a smooth 4-manifold which admits a relatively minimal hyperelliptic genus h Lefschetz fibration over S. If all of the vanishing cycles for this fibration are nonseparating curves, then we show that M is a 2-fold cover of an S-bundle over S, branched over an embedded surface. If the collection of vanishing cycles for this fibration includes σ separating curves, we show that M is the rel...
متن کامل