Computing the Hopf invariant
We consider Whitehead's integral formula and propose an algorithm for computing the Hopf invariant for simplicial mappings.
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arXiv subjects
Timur Shamazov
Publications and source records attributed to Timur Shamazov.
A new proof of Milnor-Wood inequality
The Milnor-Wood inequality states that if a (topological) oriented circle bundle over an orientable surface of genus $g$ has a smooth transverse foliation, then the Euler class of the bundle satisfies $$|\mathcal{E}|\leq 2g-2.$$ We give a new proof of the inequality based on a (previously proven by the authors) local formula which computes $\mathcal{E}$ from the singularities of a quasisection. We also sketch two other proofs: one based on Poincar\`{e} rotation number theory, and the other of topological nature.
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Quasisections of circle bundles and Euler class
Let $ E \xrightarrow[\text{}]{\pi} B$ be an oriented circle bundle over an oriented closed surface $B$. A quasisection is a smooth surface ${Q}$ (either closed or bordered) mapped by a generic smooth mapping $q$ to $E$ such that $\pi\circ q({Q})=B$. In the paper we derive a local formula for the Euler number, that is, we show that Euler number (Euler class) of the bundle equals the sum of weights of (some of) singularities of a quasisection.We also prove the uniqueness of such a formula. The local formula is a close relative of M. Kazarian's formula which relates the Euler number and Morse bifurcations of a generic function defined on the total space $E$.
math.GT↗