All Questions
22
questions
5
votes
0
answers
147
views
Representing some odd multiples of integral homology classes by embedded submanifolds
Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
- 467
13
votes
1
answer
490
views
Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces
Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
- 467
13
votes
1
answer
353
views
Realizing integral homology classes on non-orientable manifolds by embedded orientable submanifolds
Let $M^m$ denote a compact, non-orientable smooth manifold and $\nu$ an integral homology class of dimension $n$. I am interested in understanding the representability of $\nu$ by embedded, orientable ...
- 467
2
votes
0
answers
83
views
Turning cocycles in cobordism into an inclusion or a fibering
By the classical Pontryagin-Thom construction, we know that the cobordism group $\Omega^n_U(X)$ is represented by cocyles
$$ M\hookrightarrow X\times \mathbb{R}^{2k}\rightarrow X,$$
where $M$ is a ...
- 171
20
votes
2
answers
803
views
Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds
Let $M$ be an $m$-dimensional compact closed smooth manifold and $z\in H_n(M,\mathbb{Z})$ an $n$-dimensional integral homology class, with $m>n.$ Does there exist a pair of $M$ and $z$ so that $z$ ...
- 467
6
votes
0
answers
271
views
Regarding homology of fiber bundle
Let $f: X\to Y$ be a smooth map between smooth manifolds, both connected. Let $Y=\cup_{i=1}^k Y_i$ be a finite union of disjoint locally closed submanifold $Y_i$ such that $f^{-1}(Y_i)\to Y_i$ is ...
- 585
3
votes
1
answer
249
views
Relation between cohomological dimensions of manifolds
$\DeclareMathOperator\Ch{Ch}$Let $M$ be a connected manifold of finite type. We denote $\Ch_{\mathbb{Q}}(M),$ $\Ch_{\mathbb{Z}}(M)$ and $\Ch_{\mathbb{\pm}\mathbb{Z}}(M)$ by cohomological dimensions of ...
- 173
3
votes
0
answers
106
views
Reference request: $K$-theoretic wrong-way map for a boundary inclusion
Let $W$ be a compact manifold with boundary. Let $i:\partial W\hookrightarrow W$ be the natural inclusion. We have a long exact sequence in complex $K$-theory:
$$\ldots\to K^*(\partial W)\xrightarrow{...
- 1,901
4
votes
1
answer
221
views
A differential form whose support is in a tubular neighborhood of $T^k\times \{0\}^{n-k}\subset T^n$
Let $\alpha$ be a differential form on the torus $T^n$ whose support $\mathrm{supp}(\alpha)$ is contained in a small neighborhood of the subtorus $T^k\equiv T^k\times \{0\}^{n-k}$.
Question:
Suppose $...
- 2,739
5
votes
0
answers
175
views
Clarify formula for Steifel-Whitney (Poincaré dual) homology classes in a barycentric subdivision?
Let $X$ be a triangulated manifold of dimension $n$. Let $[W_{n-p}] \in H_{n-p}(X,\mathbb{Z}_2)$, be the homology class that's Poincaré dual to the $p$-th Stiefel-Whitney class $[w_p] \in H^p(X,\...
- 535
2
votes
0
answers
95
views
Compact $G$-ENR's and Euler characteristic computed with Alexander-Spanier cohomology with compact support
Let $(Z,A)$ a compact ENR pair, then
$$\chi(Z)=\chi_c(Z-A)+\chi(A)$$
where $\chi_c$ is the Euler characteristic taken in Alexander-Spanier cohomology with compact support (ENR means euclidean ...
- 795
7
votes
1
answer
1k
views
What is the scope of validity of Kunneth formula for de Rham?
In books like Bott-Tu or all pdf texts I have found on internet, the Kunneth formula for manifolds $M$ and $N$ and their de Rham cohomology
$$ H^{\bullet}_{dR}(M \times N) \simeq H^{\bullet}_{dR}(M) \...
- 1,326
11
votes
0
answers
532
views
Third cohomology of symplectic $6$-manifolds
Suppose that $A$ is a finitely generated abelian group with even rank and without $2$-torsion. Does there exist a compact symplectic $6$-manifold $(M,\omega)$ such that $A$ Is isormorphic to $H^{3}(M,\...
- 6,975
9
votes
2
answers
530
views
non-triviality of the underlying real vector bundle of the complexification of a real vector bundle
Let $M$ be a given manifold and $\xi$ be a given $k$-dimensional vector bundle over $M$. How to determine whether the underlying real vector bundle of $\xi\otimes\mathbb{C}$, i.e. the Whitney sum $\xi\...
- 93
11
votes
1
answer
738
views
Primary obstruction to the existence of a cross-section of $V_{n - q}(\omega)$ is a cohomology class in $H^{2q+2}(B, \pi_{2q+1} V_{n - q}(F))$?
Let $V_{n - q}(\mathbb{C}^n)$ denote the complex Stiefel manifold consisting of all complex $(n - q)$-frames in $\mathbb{C}^n$, where $0 \le q < n$. This manifold is $2q$-connected, and$$\pi_{2q + ...
user82367