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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).
31
votes
Is there a classification of open subsets of euclidean space up to homeomorphism?
If you have a presentation $P$ of a group $G$ with finitely many generators and relations then you can construct a $2$-dimensional simplicial complex $K(P)$ with $\pi_1(K(P))=G$. You can then embed $ …
- 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
19
votes
Accepted
On the fundamental group of a finite CW complex
Let $X$ be a CW-complex, and write $X_k$ for the $k$-skeleton. The cellular approximation theorem says that any based map $S^1\to X$ is homotopic to a cellular map, and that any two cellular maps tha …
- 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
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
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
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
15
votes
Accepted
contractible configuration spaces
The space $S^\infty$ is actually homeomorphic to $\mathbb{R}^\infty$. To see this, put
\begin{align*}
B_n &= \{x\in\mathbb{R}^\infty\::\: \|x\|\leq n, x_k = 0\text{ for } k \geq n\} \\
C_n &= \{x\ …
- 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
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
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
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
9
votes
Is there a good definition of (topological) K-Theory over arbitrary spaces?
Some comments:
You might want to look at 'Vector bundles over classifying spaces of compact Lie groups' by Jackowski and Oliver. They discuss a situation in which you can understand $K(V(X))$ quite …
- 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