# Tagged Questions

**15**

votes

**2**answers

439 views

### Distinct manifolds with the same configuration spaces?

For a space $X$, let $C_k X$ denote the space of configurations of $k$ distinct unordered points in $X$.
What is an example of a pair of smooth manifolds $M$ and $N$ that are not homeomorphic but ...

**2**

votes

**1**answer

143 views

### isotopy classes of embeddings of the torus

Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$?
For each free homotopy classes $\gamma$ of mappings of the circle ...

**3**

votes

**0**answers

175 views

### Are there CW structures on homotopy limits of CW maps?

Consider the diagram of finite CW complexes $X \stackrel{f}\leftarrow Y$ where $f$ is a cellular map and note that its homotopy colimit is precisely the mapping cylinder
$$C_H = \frac{X \sqcup (Y ...

**6**

votes

**1**answer

221 views

### Detecting homotopy nontriviality of an element in a torsion homotopy group

I have a map, constructed geometrically, $S^4 \to S^3$. I suspect that it is a representative for the generator $\eta_3\in \pi_4(S^3) \simeq \mathbb{Z}_2$, but I am not 100% sure ($\eta_3$ is defined ...

**20**

votes

**4**answers

845 views

### Is the space of diffeomorphisms homotopy equivalent to a CW-complex?

Clarification: My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the compact-open $C^k$ topology, where $0< k \leq\infty$. I have stated my question below with $k=1$ ...

**21**

votes

**1**answer

702 views

### Is there a general theory of fiber theorems?

Here are three vague theorems rolled up in one.
Let $X$ and $Y$ be sufficiently nice topological spaces and $f:X \to Y$ a sufficiently nice surjection. If for each $y \in Y$, the fiber $f^{-1}(y) ...

**6**

votes

**0**answers

182 views

### What is the state of the art in 4-manfold 2-types?

In an old answer to an old question of mine, Peter Teichner commented that it is an open problem to determine which homotopy 2-types arise from 4-manifolds. In some instances we know that a 4-manifold ...

**10**

votes

**2**answers

890 views

### homotopy type of embeddings versus diffeomorphisms

Previously, I asked a question on mathoverflow comparing smooth embeddings and diffeomorphisms, which received a very interesting and somewhat unexpected answer by Agol. I now ask a further question ...

**7**

votes

**1**answer

526 views

### What is the homotopy type of the space of the homeomorphisms of the n-ball such that the homeomorphism restricted to the boundary is isotopic to the identity?

Consider the set of homeomorphisms of the topological n-ball to itself with the compact open topology. Sitting inside this space of homeomorphisms are particular subspaces. The first subspace is those ...

**33**

votes

**0**answers

730 views

### Homotopy type of TOP(4)/PL(4)

It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that ...

**0**

votes

**0**answers

193 views

### Killing homotopy groups by removing subsets

Let $X$ be a locally finite CW-complex and let $U$ be an open subset of $X$. Given a non-zero homotopy class $x\in\pi_i(U)$ say, is it possible to find a closed subset $Z\subset U$ whose removal from ...

**6**

votes

**3**answers

630 views

### Homotopy class of a homeomorphism

I was reading a book and find this statement:
''It is well kown that each homeomorphism $f:S\rightarrow R$ between compact surfaces is homotopic to a diffeomorphism''
I would know some references ...

**2**

votes

**2**answers

480 views

### Unreduced Suspension Isomorphism

Tending to a lecture on homotopy theory, the following question occured to me (is that a correct sentence?):
Given a pointed space $(X,x)$, is the UNREDUCED suspension map $S:\pi_k(X,x) \rightarrow ...

**3**

votes

**0**answers

238 views

### Smooth manifold with non-trivial inertia group? (wrt homotopy spheres)

Let $\Theta_n$ be the set of orientation-preserving diffeomorphism classes of homotopy spheres, with abelian group structure given by #. Then for any smooth manifold $M^n$ one defines the "inertia ...

**6**

votes

**1**answer

417 views

### Embedded (framed) cobordisms

[The title initially was "Actions of gauge groups on framed cobordisms. This has been changed.]
This question is a follow-up to my answer to When is a submanifold of $\mathbf R^n$ given by global ...

**23**

votes

**3**answers

2k views

### In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?

In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra.
He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie ...

**10**

votes

**3**answers

911 views

### Is there a good definition of (topological) K-Theory over arbitrary spaces?

Hi
(this is my very first question here, so please don't hurt me...)
for some time now i've been looking for a sufficiently aesthetical definition of (topological) K-theory of arbitrary spaces, yet ...

**4**

votes

**2**answers

328 views

### Does a homeomorphism of $S^1 \times S^2$ which is homotopy to the identity has to isotope to it?

I guess the question can be asked for all manifolds. But I am particularly interested in $S^1 \times S^2$ right now. Concrete example preferrd.

**6**

votes

**0**answers

256 views

### homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equivalence

Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with ...

**7**

votes

**1**answer

298 views

### Aspherical homotopy orbit space of configurations on the 2-sphere

The group SO(3) acts naturally on $S^2$ and thus on $Conf(S^2, q)$, the configuration space of $q$ distinct points on the 2-sphere, via the diagonal action on $S^2 \times...\times S^2$. This is a ...

**5**

votes

**1**answer

669 views

### Exotic spheres detected in higher homotopy

Thinking about exotic 7-spheres, one can look at the maps $\cdots \rightarrow \Omega^2Diff(D^4, rel \space \partial) \rightarrow \Omega Diff(D^5, rel \space \partial) \rightarrow Diff(D^6, rel \space ...

**4**

votes

**3**answers

479 views

### Natural examples of finite dimensional spaces with interesting 2-type

Riemann surfaces provide interesting examples of 1-types - interesting as they have roles in diverse areas. However, apart from 2-dimensional lens spaces, I can't readily bring to mind natural ...

**1**

vote

**2**answers

413 views

### Interesting representations/cohomology of surface groups?

For purposes of my own, I'm interested in constructing connected spaces, without recourse to geometric realisation or the like, that have non-trivial homotopy groups in dimension 1 and 2 and are not ...

**7**

votes

**3**answers

253 views

### An obstruction theory for promoting homotopy equivalences that are equivariant maps to equivariant homotopy equivalences?

Say I have a map of $G$-spaces $f : X \to Y$ and I know it is a homotopy-equivalence in the plain sense that there exists a map (maybe not equivariant) $g : Y \to X$ such that the two composites are ...

**19**

votes

**2**answers

2k views

### Does this approach for the Poincare conjecture work?

Several months ago a paper was posted at
http://arxiv.org/abs/1001.4164
called "Another way of answering Henri Poincare's fundamental question." The author gave a talk on it today at my institution. ...

**18**

votes

**6**answers

2k views

### Failure of smoothing theory for topological 4-manifolds

Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is ...

**9**

votes

**1**answer

450 views

### Smooth structures on PL 4-manifolds

Is it known whether $O(4) \to PL(4)$, the map from the orthogonal group to the group of piecewise linear homeomorphisms of $\mathbb{R}^4$, is a homotopy equivalence? By smoothing theory for PL ...