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Results tagged with gt.geometric-topology
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user 10366
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
2
votes
Accepted
A torus bundle whose vertical tangent bundle is indecomposable
Put $T=\{(z_0,z_1,z_2)\in(S^1)^3:z_0z_1z_2=1\}$, so $T$ is homeomorphic to $S^1\times S^1$ and has an obvious action of the symmetric group $\Sigma_3$. The action of $\Sigma_3$ on $H_1(T;\mathbb{R})$ …
- 56.9k
9
votes
Which homotopy classes $S^3 \to S^2$ lift to embeddings $S^3 \to S^2 \times D^3$?
This is not really an answer, but it provides some context. We can consider $S^3$ as the space of unit quaternions, $S^2$ as the subspace of unit pure-imaginary quaternions, and $D^3$ as the space of …
- 56.9k
1
vote
Approximate Jordan-Brouwer theorem (corrected)
Let $u$ be the generator of $H_n(S^n)$. It is given that $f_*(u)\neq 0$ in $H_n(\mathbb{R}^{n+1}\setminus\{x\})$. The assumptions ensure that the straight line from $f(p)$ to $g(p)$ never passes thr …
- 56.9k
4
votes
Hadamard-like product on orientable surfaces
You don't say what the morphisms in $C$ are supposed to be. I'll take them to be isotopy classes of orientation-preserving diffeomorphisms. Let $\mathbb{N}$ be the category with object set $\mathbb{ …
- 56.9k
13
votes
Classification of fibrations $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$
I'll assume that $1\leq k<d$, the other cases being easy. Then the long exact sequence of homotopy groups shows that $B$ is simply connected, so we have a Serre spectral sequence with untwisted coeff …
- 56.9k
17
votes
Accepted
Which cohomology classes are detected by tori?
I'd prefer to work in homology (with coefficients $\mathbb{Q}$ everywhere). I'll say that a class $x\in H_k(X)$ is basically toral if there is a map $f\colon T^k\to X$ sending the fundamental class o …
- 56.9k
16
votes
Accepted
Relationship between induced maps at homotopy groups level for maps $f:S^2\to S^2$
I'll write $n_d$ for the degree $n$ map on $S^d$, and $\eta$ for the Hopf map $S^3\to S^2$. It is well-known that $\eta_*\colon\pi_3(S^3)\to\pi_3(S^2)$ is an isomorphism, so that $\pi_3(S^2)=\{\eta\c …
- 56.9k
8
votes
Künneth formulas/theorem for bordism groups and cobordisms?
If you work with the unoriented bordism groups $\Omega^O_*(X)=MO_*(X)$ then there is a Künneth isomorphism. This is just because there is a natural isomorphism $MO_*(X)=H_*(X;\mathbb{Z}/2)\otimes MO_ …
- 56.9k
16
votes
Infinite loop space structure of $BU^+$
Firstly, no plus construction is needed here. The plus construction is used to kill a perfect subgroup of the fundamental group, but $\pi_1(BU)$ is already trivial.
Next, note that the space $\mathb …
- 56.9k
6
votes
Accepted
Eilenberg-Zilber-type theorem for good fiber products?
There are fibrations
\begin{align*}
F \to X & \to B \\
G \to Y & \to B \\
F\times G \to X\times_B Y & \to B \\
F\times G \to X\times Y & \to B\times B \\
F \to X\times_BY &\to Y \\
G \to X\times …
- 56.9k
8
votes
Accepted
Classifying spaces of finitely presented groups with torsion elements
This is not possible. Let $B$ be a space of type $K(\Gamma,1)$ and of dimension $d<\infty$, and let $E$ be the universal cover, which is contractible. Then for any $A\leq\Gamma$ we see that $E/A$ is …
- 56.9k
8
votes
Spin bordism group of classifying space $BG$ with a finite Abelian $G$
The paper of Anderson, Brown and Peterson also shows that the localisation of the spectrum $MSpin$ splits as a wedge of suspensions of the real connective $K$-theory spectrum $kO$ and various closely …
- 56.9k
3
votes
Accepted
Question about maps of $S^{3}$-bundles
Here is an argument that is essentially the same as Oscar's, but organized a little differently.
First, we have a fibration $S^3\to X\to\Sigma$, giving an exact sequence
$$ \pi_2(\Sigma) \to \pi_1(S …
- 56.9k
2
votes
Is the action of $G$ on $H_1(T^n, \mathbb{Z}) = \mathbb{Z}^n$ faithful?
We can expand the answer from user83633, following hints in the notes by Edmonds, as follows.
Let $G$ be a finite group acting faithfully on an $n$-dimensional torus $T$ and fixing a point $p$. We …
- 56.9k
10
votes
Accepted
Are two equivariant maps between aspherical topological spaces homotopic?
In the language of equivariant homotopy theory, your question is as follows. You have a group homomorphism $\phi\colon G\to H$, which you use to make $EH$ into a $G$-space, and then you ask whether $ …
- 56.9k