# Tagged Questions

**4**

votes

**2**answers

210 views

### A question about Dehn surgery and Brieskorn homology 3-spheres

I have been learning about Brieskorn homology 3-spheres $\Sigma(a_1,...,a_n)$ and Seifert manifolds. My reference is the first few pages of Saveliev's "Invariants of Homology 3-spheres."
If I ...

**5**

votes

**2**answers

184 views

### What 3-manifolds can be obtained by gluing $ S^1 \times P $ and two copies of $S^1 \times D^2$

Let P denote the pair of pants e.g. a sphere minus three small discs $D_1$,$D_2$,$D_3$ about marked points $x_1,x_2,x_3$. I then consider $P \times S^1$. We have boundary components $T_1$,$T_2$,$T_3$. ...

**7**

votes

**1**answer

264 views

### Homotopy spheres with vanishing and non-vanishing $\alpha$-invariant

I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize.
All manifolds are closed, smooth and have dimensions $n\ge 5$.
The Atiyah-Shapiro-Bott-Orientation gives ...

**3**

votes

**1**answer

243 views

### Compactly supported cohomology of homotopy equivalent manifolds

Are there examples of homotopy equivalent smooth, orientable manifolds $M$ and $N$ of the same dimension with non-isomorphic compactly supported cohomology rings?

**-3**

votes

**1**answer

265 views

### Lie algebraic Grassmannian

Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given.
We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$.
For ...

**6**

votes

**1**answer

278 views

### Can the monoidal structure on manifolds be strictified?

I'm asking this question purely out of curiosity.
Let $\{M_\alpha\}_{\alpha\in A}$ be a collection of closed smooth manifolds, with exactly one in every diffeomorphism class of closed smooth ...

**7**

votes

**2**answers

282 views

### Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex

Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a ...

**-1**

votes

**1**answer

153 views

### A closed manifold with a subset with the same ring cohomology

Is there an example of a closed manifold $M$ with a proper subset $A\subset M$ such the inclusion $i:A \to M$ gives a ring isomorphism $i^{*}$ between $\mathbb{Z}$-cohomologies?
In this question ...

**13**

votes

**1**answer

621 views

### Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map
$f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that ...

**8**

votes

**1**answer

474 views

### Pontryagin numbers on manifolds with an $S^1$-action

Let $M$ is a smooth compact manifold with an $S^1$-action with isolated fixed points. Suppose the representation of $S^1$ at tangent spaces at all fixed points is known. Can one then find all ...

**6**

votes

**1**answer

252 views

### how to obtain a generalized Morse function out of a fiber bundle?

I already asked this question in MSE but did not get any answer/comment yet.
Let $M\to E\to B$ be a smooth fiber bundle. In "Parametrized Morse Theory and Its Applications,(Proceedings of the ICM, ...

**6**

votes

**0**answers

241 views

### Space of embeddings of an $n$-ball into an $n$-manifold

Let $M$ be a smooth $n$-manifold without boundary, and let $B$ be the open unit ball in $\mathbb{R}^n$. I am trying to understand the space $\text{Emb}(B,M)$ of smooth embeddings of $B$ into $M$. ...

**1**

vote

**1**answer

151 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 ...

**7**

votes

**2**answers

220 views

### Pseudofree $T^2$ actions on spheres

Is it possible to construct a smooth action of $S^1\times S^1$ on $S^{2n+1}$ ($n\ge 2$) such that no point on $S^{2n+1}$ has an infinite stabilizer?
Note that if such an action exists, it can not be ...

**6**

votes

**2**answers

211 views

### Fixed component of an $S^1$ action on $S^n$

Suppose $S^1$ is acting smoothly on $S^n$ and $M$ is a connected component of the set of fixed points of the action. What can be said about $M$?
Is it true that $\pi_1(M)=0$? (sorry this first bit ...

**8**

votes

**3**answers

363 views

### Configuration spaces of the torus

I would like a reference that calculates the rational homology of the unordered configuration spaces of the torus.

**9**

votes

**1**answer

466 views

### Does a smooth homeomorphism of closed manifolds preserve cobordism fundamental class?

Let $f:M\to N$ be a smooth map of closed oriented smooth manifolds which is also a homeomorphism. Let $[M]\in H_\bullet(M;\mathbb Z)$ denote the fundamental class (and similarly for $N$). It is ...

**5**

votes

**0**answers

212 views

### Embedding tower in low codimension

If $F$ is a suitably nice functor from manifolds to spaces, it has a degree $k$ "polynomial" approximation $T_k F$ in the sense of embedding calculus. We set $T_\infty F := \mathrm{holim} T_k F$.
The ...

**8**

votes

**1**answer

339 views

### Construction of exotic spheres that do not bound parallelizable manifolds

There are at least two ways to construct homotopy spheres that bound parallelizable manifolds, namely Milnor's plumbing construction and Brieskorn's method of singularities, and each of these methods ...

**5**

votes

**0**answers

108 views

### In cell-decomposed manifolds, how easy is it to arrange for the tubular neighborhood of a diagonal to contract onto the diagonal?

Suppose that you have decomposed a manifold $M$ into cells (I care most, if it matters, about compact oriented smooth manifolds; but if my question can be solved in the PL category, all the better). ...

**8**

votes

**2**answers

458 views

### Waldhausen $K$-theory for $G$-spaces

I would guess that the following is true, and that somebody has worked it out, but I don't recall ever seeing it. Can anyone point me to any literature on it?
Let $G$ be a finite group. We know that ...

**10**

votes

**2**answers

468 views

### When does an even-dimensional manifold fiber over an odd-dimensional manifold?

Are there simple necessary and sufficient conditions for an (oriented) even-dimensional compact smooth manifold to fiber over an (oriented) odd-dimensional manifold (with oriented fibers)?
For ...

**13**

votes

**3**answers

599 views

### Does a *topological* manifold have an exhaustion by compact submanifolds with boundary?

If $M$ is a connected smooth manifold, then it is easy to show that there is a sequence of connected compact smooth submanifolds with boundary $M_1\subseteq M_2\subseteq\cdots$ such that ...

**20**

votes

**4**answers

876 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 $k=1$ ...

**3**

votes

**1**answer

299 views

### Boundaries of smooth manifolds

If one has a smooth simply connected manifold $M^n$ which we know to bound an $n+1$ dimensional manifold $N$, what can be said about a handle decomposition for one in terms of a handle decomposition ...

**0**

votes

**1**answer

355 views

### fundamental group of a compact manifold

why fundamental group of of compact manifold is finitely presented

**21**

votes

**2**answers

692 views

### Cobordism of orbifolds?

Is it possible to setup classical cobordism theory in the context of orbifolds? For example, let's consider the free abelian group generated by oriented smooth orbifolds and quotient by those which ...

**16**

votes

**1**answer

474 views

### Is there a notion of a chain complex with corners?

Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes.
On the topological side, although in the PL ...

**5**

votes

**1**answer

194 views

### Equivariant handle decompositions

Suppose I have some smooth closed high-dimensional manifold $M$ acted on smoothly by a finite group $G$. By a metric averaging procedure, we can equip $M$ with a smooth Riemannian metric so that $G$ ...

**0**

votes

**1**answer

440 views

### sign of the First chern class fundamental group of Kahler Manifolds

We know by some facts from Kobayashi, if the Kahler manifold $M$ has positive first Chern class, i.e., $c_1 (M)>0$ then $M$ is simply connected. So if $c_1 (M)<0$ under which assumption on $M$ ...

**20**

votes

**4**answers

995 views

### Algorithmically unsolvable problems in topology

This question is inspired by a paper by B. Poonen that appeared on the arxiv some time ago: http://arxiv.org/abs/1204.0299. The paper gives a sample of algorithmically unsolvable problems from various ...

**2**

votes

**1**answer

684 views

### Manifolds are paracompact

By Definition, smooth manifolds are assumed to be Hausdorff and to satisfy the second countability axiom.
I have heard (but never seen written) that these assumptions imply paracompactness (and thus ...

**5**

votes

**1**answer

364 views

### Actions of finite groups on exotic smooth manifolds of dimension >4

Let $M_1^n$ and $M_2^n$, $n>4$ be two smooth compact manifolds that are homeomorphic but not diffeomorphic. Suppose that a finite group is $G$ acting faithfully on $M_1^n$ by diffeomorphisms. Is it ...

**12**

votes

**6**answers

825 views

### Does every vector bundle allow a finite trivialization cover?

Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$.
(a) Is ...

**-1**

votes

**1**answer

427 views

### Is this manifold orientable? [closed]

Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy
1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $.
2) $ a\bar{b}+c\bar{d}=0 $
There is a ...

**10**

votes

**1**answer

492 views

### Smooth four-manifolds with contractible universal cover

Let $X$ be a smooth compact four-manifold with definite non-trivial intersection form. Can the universal cover of $X$ be contractible?
It semms to me that the answer is negative when $X$ is simply ...

**28**

votes

**3**answers

2k views

### When is a submanifold of $\mathbf R^n$ given by global equations?

Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...

**6**

votes

**2**answers

293 views

### Cohomology of fixed point subspaces

Suppose $M$ is a smooth manifold and $\phi : M \to M$ is a homeomorphism whose fixed point set is a smooth submanifold $M_{\phi}$. Is there any relation between the cohomology ring of $M_{\phi}$ and ...

**13**

votes

**3**answers

821 views

### How to disjoint two cycles with zero intersection?

Suppose that $M^n$ is an orientable connected (thanks to Greg) manifold and $Z^k$ with $Z^{n-k}$ are two real cycles in $M^n$ with zero index of intersection $Z^k\cdot Z^{n-k}=0$ (for me these cylces ...

**6**

votes

**1**answer

267 views

### Does there exist a model of chains on oriented manifolds with both a strict intersection pairing and strict functoriality for closed embeddings?

Let $M$ be a smooth oriented $n$-dimensional manifold. My favorite model of $\operatorname{Chains}_\bullet(M) \otimes \mathbb R$ is the space of smooth compactly-supported de Rham forms on $M$, ...

**16**

votes

**1**answer

582 views

### How are these algebraic and geometric notions of homotopy of maps between manifolds related?

Let $M$ and $N$ be smooth manifolds, and $f,g: M \to N$ smooth maps. Denote by $(\Omega^\bullet M,\mathrm d_M)$ and $(\Omega^\bullet N, \mathrm d_N)$ the cdgas of de Rham forms in each manifold, and ...

**6**

votes

**1**answer

465 views

### fundamental domain of universal covering

Let $M$ be a connected compact manifold without boundary, $\pi:\widetilde{M}\to M$ be the universal covering map. A fundamental domain of $(\pi,\widetilde{M}, M)$ is a compact subset $D\subset ...

**22**

votes

**2**answers

1k views

### How can we detect the existence of almost-complex structures?

Any smooth $k$-manifold $M$ comes with a well-defined map $f:M\rightarrow BGL_{k}(\mathbb{R})$ (up to homotopy) classifying its tangent bundle. Since $GL_{k}(\mathbb{R})$ deformation-retracts onto ...

**11**

votes

**2**answers

545 views

### Geometric interpretation of the Pontryagin square

The Pontryagin square (at the prime 2) is a certain cohomology operation
$$
\mathfrak P_2: H^q(X;\Bbb Z_2) \to H^{2q}(X;\Bbb Z_4)
$$
which has the property that its reduction mod 2 coincides with ...

**6**

votes

**3**answers

512 views

### Can homologous submanifolds be connected by an immersed manifold with boundary?

Supposed I have an n-dimensional manifold M with a k-dimensional submanifold that is homologous to zero (or, equivalently, two homologous submanifolds). Can I always construct a k+1-dimensional ...

**2**

votes

**2**answers

2k views

### On a proof of the existence of tubular neighborhoods.

Studying analysis on manifolds, I have found, in the proof of the existence of tubular neighborhoods, a reference to theorem 3.1.2 in "Topologie algebrique et theorie des faisceaux" of Godement.
...

**7**

votes

**2**answers

639 views

### Fibrewise homotopy-equivalence of unit sphere bundles vs isomorphism of tangent bundles

Let $M$ be a smooth $m$-dimensional manifold, $TM$ its tangent bundle and $SM$ its unit sphere bundle.
Are there some simple examples where $SM$ is fibrewise homotopy-equivalent to the trivial ...

**11**

votes

**2**answers

964 views

### Smooth structures on the connected sum of a manifold with an Exotic sphere

What can we say about the connected sum of a manifold $M^n$ with an Exotic sphere? Is is possible some of them are still diffemorphic to $M^n$. Is it possible to classifying all the exotic smooth ...

**5**

votes

**2**answers

494 views

### Definition of the Kervaire invariant for normal maps (as in Browder's book)

Browder's book "Surgery on simply-connected manifolds" defines the Kervaire invariant in a very general setting. My question is: how does one get the more usual definition of the invariant for a ...

**5**

votes

**1**answer

676 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 ...