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A new invariant and parametric connected sum of embeddings

A. Skopenkov (2007)

Fundamenta Mathematicae

We define an isotopy invariant of embeddings N → ℝ m of manifolds into Euclidean space. This invariant together with the α-invariant of Haefliger-Wu is complete in the dimension range where the α-invariant could be incomplete. We also define parametric connected sum of certain embeddings (analogous to surgery). This allows us to obtain new completeness results for the α-invariant and the following estimation of isotopy classes of embeddings. In the piecewise-linear category, for a (3n-2m+2)-connected...

A note on characteristic classes

Jianwei Zhou (2006)

Czechoslovak Mathematical Journal

This paper studies the relationship between the sections and the Chern or Pontrjagin classes of a vector bundle by the theory of connection. Our results are natural generalizations of the Gauss-Bonnet Theorem.

A note on generalized flag structures

Tomasz Rybicki (1998)

Annales Polonici Mathematici

Generalized flag structures occur naturally in modern geometry. By extending Stefan's well-known statement on generalized foliations we show that such structures admit distinguished charts. Several examples are included.

A note on rational surgeries on a Hopf link

Velibor Bojković, Jovana Nikolić, Mladen Zekić (2023)

Czechoslovak Mathematical Journal

It is clear that every rational surgery on a Hopf link in 3 -sphere is a lens space surgery. In this note we give an explicit computation which lens space is a resulting manifold. The main tool we use is the calculus of continued fractions. As a corollary, we recover the (well-known) result on the criterion for when rational surgery on a Hopf link gives the 3 -sphere.

A note on the cohomology ring of the oriented Grassmann manifolds G ˜ n , 4

Tomáš Rusin (2019)

Archivum Mathematicum

We use known results on the characteristic rank of the canonical 4 –plane bundle over the oriented Grassmann manifold G ˜ n , 4 to compute the generators of the ℤ 2 –cohomology groups H j ( G ˜ n , 4 ) for n = 8 , 9 , 10 , 11 . Drawing from the similarities of these examples with the general description of the cohomology rings of G ˜ n , 3 we conjecture some predictions.

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