Four-manifold invariants from higher-rank bundles

A note on singular points of bundle homomorphisms from a tangent distribution into a vector bundle of the same rank

We consider bundle homomorphisms between tangent distributions and vector bundles of the same rank. We study the conditions for fundamental singularities when the bundle homomorphism is induced from a Morin map. When the tangent distribution is the contact structure, we characterize singularities of the bundle homomorphism by using the Hamilton vector fields.

Minimal genus in circle bundles over 3-manifolds

An estimate for the genus function in circle bundles over irreducible 3-manifolds is proven. This estimate is in many cases an equality and it relates the minimal genus of the surfaces representing a given homology class with the self-intersection of the class and the Thurston norm of the underlying 3-manifold.

On $Q$-manifolds bundles

We prove a homological characterization of $Q$-manifolds bundles over $C$-spaces. This provides a partial answer to Question QM22 from \cite{w}.

$4$-流形上的球丛在环路化后是平凡的

Sphere bundles over $4$-manifolds are trivial after looping

本文证明了除两种特殊情况外，单连通$4$-流形上向量丛的球丛在环路化后分裂，揭示了无限多不等价球丛的全流形环路空间均同伦等价。

Involutory Hopf group-coalgebras and flat bundles over 3-manifolds

Given a group G, we use involutary Hopf G-coalgebras to define a scalar invariant of flat G-bundles over 3-manifolds. When G=1, this invariant equals to the one of 3-manifolds constructed by Kuperberg from involutary Hopf algebras. We give examples which show that this invariant is not trivial.

Equivariant principal bundles over spheres and cohomogeneity one manifolds

We classify SO(n)-equivariant principal bundles over $S^n$ in terms of their isotropy representations over the north and south poles. This is an example of a general result classifying equivariant $(Π, G)$-bundles over cohomogeneity one manifolds.