All Questions
Tagged with differential-topology spectral-sequences
8 questions
5
votes
1
answer
366
views
Is the space of real conics with a singular point an orientable manifold?
Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set
of such a polynomial gives a real curve ...
- 3,245
5
votes
1
answer
653
views
Spin bordism group of classifying space $BG$ with a finite Abelian $G$
The spin bordism group for the classifying space $BG$ of group $G$ can be denoted as $\Omega^{Spin}_d(BG)$.
For example, $\Omega^{Spin}_d(pt)$ are computed by Anderson-Brown-Peterson (D. W. Anderson, ...
- 755
4
votes
1
answer
598
views
Thom space, homotopy group and cohomology group
In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...
- 10.5k
3
votes
1
answer
437
views
First cohomology of the space of long knots in ℝ⁴
Let's consider the space of long knots in $\mathbb R^n$, $n>3$. I know that there are many results (Vassiliev, Turchin, Sinha, Kontsevich) about different expressions of cohomology of this space. I ...
- 5,063
3
votes
1
answer
205
views
Prove or disprove that there exists no $G$ structure with its bordism group $\Omega_1^{G} =\mathbb{Z}/N$ for $N>2$
It can be found that there are the following bordism group $\Omega_0^{G}$ at $d=0$ and 1 dimensions by requiring $G$ structure for $d$-manifolds:
$$
\Omega_0^{SO} = \mathbb{Z} , \quad \Omega_1^{SO} = ...
- 2,147
3
votes
1
answer
463
views
cohomology module of unit tangent vector bundles over spheres
Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration
$$
S^{m-1}\longrightarrow \tau(S^m)\...
- 2,223
2
votes
1
answer
148
views
An attempt at an alternative calculation of the rank of $\pi_n(MO)$
$\newcommand{\a}{\mathfrak a}\newcommand{\Z}{\mathbb Z}$Let $MO$ be the Thom Spectrum, then I am trying to come up with an alternative calculation that the rank of $\pi_n(MO)$ as a $\Z_2$ vector space ...
- 391
2
votes
0
answers
71
views
Connected topological/Lie group $H$ and $Q$, inflate $Q$-cocycle to coboundary in $H$
I am interested in finding mathematical examples and criteria of inflating $Q$-cocycle to coboundary in $H$, under the requirement:
(1) Both $H$ and $Q$ are connected topological groups or Lie groups (...
- 2,147