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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).
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
$$ \pi_2(\Sigma) \to \pi_1(S …
- 56.9k
3
votes
Accepted
Simple connectedness of $\mathbb{C}P^2$ intersected with an affine subspace
Put $V=\{A\in H_3(\mathbb{C}):a_{11}=a_{33}=1/3\}$. I claim that $V\cap\mathbb{C}P^2$ is the set of matrices of the form
$$ P = \frac{1}{3}\left[\begin{array}{ccc} 1 & z & zw \\ \overline{z} & 1 & w …
- 56.9k
11
votes
Accepted
Degree of maps on the sphere with a property of symmetry
First, for any $x\in S^2$ we have an endomorphism $A(x)$ of $\mathbb{R}^3$ given by $A(x)(w)=x\times w$. More generally, we have an orthogonal matrix $B(t,x)=\exp(t A(x))$, which is a rotation throug …
- 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
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
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
12
votes
Cup products of connected sum
The natural description is like this. There are augmentations $\epsilon_X:H^*(X)\to\mathbb{Z}$ (for $X\in\{M,N\}$) and orientation classes $u_X\in H^d(X)$. Put
$$R'=\{(a,b)\in H^*(M)\times H^*(N):\ …
- 56.9k
2
votes
Accepted
A torus bundle whose vertical tangent bundle is indecomposable
Put $T=\{(z_0,z_1,z_2)\in(S^1)^3:z_0z_1z_2=1\}$, so $T$ is homeomorphic to $S^1\times S^1$ and has an obvious action of the symmetric group $\Sigma_3$. The action of $\Sigma_3$ on $H_1(T;\mathbb{R})$ …
- 56.9k
7
votes
Accepted
Pseudo-manifolds and homology
You should be aware of the book "A geometric approach to homology theory" by Buoncristiano, Rourke and Sanderson. However, I am not willing to claim that it is modern (1976) or easy to read.
- 56.9k
2
votes
Is the action of $G$ on $H_1(T^n, \mathbb{Z}) = \mathbb{Z}^n$ faithful?
We can expand the answer from user83633, following hints in the notes by Edmonds, as follows.
Let $G$ be a finite group acting faithfully on an $n$-dimensional torus $T$ and fixing a point $p$. We …
- 56.9k
23
votes
$S^n \to S^m \to B$ bundle: possible?
More generally, there are bundles $S^0\to S^n\to \mathbb{R}P^n$ and $S^1\to S^{2n+1}\to\mathbb{C}P^n$ and $S^3\to S^{4n+3}\to\mathbb{H}P^n$. There is also an "octonionic projective plane" $\mathbb{O} …
- 56.9k
16
votes
Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?
Note that any continuous surjection from a compact space
to a Hausdorff space is automatically a quotient map.
Also, there are 'space-filling curves', which are continuous
surjections from $[0,1]$ to …
- 56.9k
3
votes
Accepted
space of homotopy equivalences of $S^1$
Put $$HE^+_1(S^1)=\{f\in HE^+(S^1):f(1)=1\}. $$
There is an evident homeomorphism $m:S^1\times HE_1^+(S^1)\to HE^+(S^1)$ given by $m(z,f)(x)=z f(x)$, and this restricts to give a homeomorphism $S^1\ti …
- 56.9k
7
votes
Accepted
Cohomology of the infinite loop space of the affine grassmanian (as in the generalized Mumfo...
Most of this is not special to the case of $AG_{\infty,2}^+$. For any spectrum $X$, we have a Hurewicz map $h:\pi_{\ast}(X)\to H_{\ast}(X)$, which induces a map $h':\mathbb{Q}\otimes\pi_{\ast}(X)\to\ …
- 56.9k
17
votes
Does $S^2$ have a trivial normal bundle in any closed orientable manifold?
The normal bundle to $\mathbb{C}P^1\simeq S^2$ in $\mathbb{C}P^2$ is the dual of the tautological bundle. This is nontrivial (even as a real bundle rather than a complex bundle); indeed, we have $H^* …
- 56.9k