변양현 교수

The unstable tangent fibration of a Poincaré complex is defined so that it is consistent with the manifold case. It exists uniquely for each Poincaré complex X up to fibrewise homotopy equivalence and, furthermore, if a Poincaré embedding structure exists on the diagonal X→X×X, its normal fibration is the tangent fibration.

There is a Poincaré embedding structure on the diagonal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X right-arrow upper X times upper X"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>X</mml:mi> <mml:mo> × </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X\rightarrow X\times X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> und

There is a homotopy equivalence φ:M→M′ between closed smooth manifolds of an odd dimension such that φ*TM′, TM are stably isomorphic but not isomorphic to each other.

We show that a flat principal bundle with compact connected structure group and its adjoint bundles of Lie groups have the same cohomology as the trivial bundle, which is done by proving they satisfy the condition for the Leray-Hirsch theorem. This information has been used to construct a cohomology class of the adjoint bundle of a flat bundle whose restriction to each fiber is the class of the Maurer-Cartan 3-form. Then we use this result to define an invariant of a gauge transformation of a fl

We investigate when the product of two smooth manifolds admits a weakly Lagrangian embedding. Prove that, if and are smooth manifolds such that M admits a weakly Lagrangian embedding into whose normal bundle has a nowhere vanishing section and N admits a weakly Lagrangian immersion into , then admits a weakly Lagrangian embedding into . As a corollary, we obtain that admits a weakly Lagrangian embedding into if n=1,3. We investigate the problem of whether in general admits a weakly Lagrangian em

We provide product formulae for self-intersection numbers in various coefficients.

Let W be a parallelizable compact oriented manifold of dimension <TEX>$n$</TEX> with boundary <TEX>${\partial}W=M$</TEX>. We define the so-called Gauss map <TEX>$f:M{\rightarrow}S^{n-1}$</TEX> using a framing of TW and show that the degree of <TEX>$f$</TEX> is equal to Euler-Poincar<TEX>$\acute{e}$</TEX> number <TEX>${\chi}(W)$</TEX>, regardless of the specific framing. As a special case, we get a Hopf theorem.

We develop strong lower bounds for the span of the projective Stiefel manifolds $X_{n,r}=O(n)/(O(n-r)\times \mathbb Z/2)$, which enable very accurate (in many cases exact) estimates of the span. The technique, for the most part, involves elementary stability properties of vector bundles. However, the case $X_{n,2}$ with $n$ odd presents extra difficulties, which are partially resolved using the Browder-Dupont invariant. In the process, we observe that the symmetric lift due to Sutherland does no

We consider a hypothetical bound state of two electrons even if we leave the binding mechanism unknown. The bound electron pairs should behave as bosons and we conclude that the paired electrons are 'apparently' in the highest energy states of all the occupied states. We call this multi-particle state an 'apparent' Fermi surface. Then the solid in concern must be a type 2 superconductor for a charged boson fluid is theoretically known to be as such. Moreover the Bose-Einstein condensation is kno

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript r Baseline left-parenthesis upper X right-parenthesis subset-of pi Subscript r Baseline left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⊂ </mml:mo> <mml:msub> <mml:mi> π </mml:mi> <mm

Abstract We invoke the classical fact that the algebra of bi-invariant forms on a compact connected Lie group G is naturally isomorphic to the de Rham cohomology H * dR ( G ) itself. Then, we show that when a flat connection A exists on a principal G -bundle P , we may construct a homomorphism E A : H * dR ( G )→ H * dR ( P ), which eventually shows that the bundle satisfies a condition for the Leray–Hirsch theorem. A similar argument is shown to apply to its adjoint bundle. As a corollary, we s