# All Questions

Tagged with differential-topology manifolds

37 questions

**22**

votes

**1**answer

512 views

### Closed manifold with non-vanishing homotopy groups and vanishing homology groups

Is there a closed connected $n$-dimensional topological manifold $M$ ($n\geq 2$) such that $\pi_i(M)\neq 0$ for all $i>0$ and $H_i(M, \mathbb{Z})=0$ for $i\neq 0$, $n$? The manifold $S^1\times S^2$ ...

**4**

votes

**1**answer

242 views

### Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?

Let $M$ be a compact connected manifold-with-boundary such that $\circ M \neq \emptyset$, where $\circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $\circ ...

**9**

votes

**0**answers

265 views

### History of the definition of smooth manifold with boundary

I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it ...

**8**

votes

**0**answers

146 views

### $k$ times differentiable but not $C^k$ manifold

I asked the following question on Math Stack Exchange 3 months ago but got no answer. So maybe Math Overflow is a more suitable place for such a question:
I cannot find the notion of $k$ times ...

**7**

votes

**1**answer

202 views

### Action of diffeomorphism group on non-vanishing vector fields

Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0)...

**5**

votes

**0**answers

90 views

### Induced new structures on Poincare dual manifolds

"R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows
Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$...

**5**

votes

**1**answer

292 views

### Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure

The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) ...

**4**

votes

**0**answers

179 views

### Non-spin 5-manifold and $2^2$-Bockstein homomorphism

The $2^2$-Bockstein is $\beta_4$ is associated to
$$0\to\mathbb{Z}/2\to\mathbb{Z}/{8}\to\mathbb{Z}/{4}\to 0,$$
(The $2^n$-Bockstein homomorphism
$$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\...

**8**

votes

**1**answer

261 views

### Non-triangulable 4-manifold as a boundary of some 5 manifold

We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold.
Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What ...

**6**

votes

**2**answers

474 views

### Any 3-manifold can be realized as the boundary of a 4-manifold

We know
"Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...

**11**

votes

**1**answer

383 views

### Is there a PL, or topological, bordism hypothesis?

The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with ...

**6**

votes

**0**answers

182 views

### Differential topology on arbitrary fields

What do the differential topology theories on arbitrary fields have in common?
Different differential topology theories
There is "ordinary" differential topology on real manifolds, with its rich ...

**5**

votes

**1**answer

278 views

### Is the connected sum of a triangulable manifold with a non-triangulable manifold a non-triangulable manifold?

Let $M,N$ be topological manifolds such that $M$ does not admit a $PL$ structure and $N$ does. Is $M\#N$ still a triangulable manifold?

**7**

votes

**0**answers

207 views

### Atiyah duality with coefficients and boundary

Looking at Atiyah's paper "Thom complexes", I find two statements of the Atiyah duality theorem:
Proposition 3.2 Let $X$ be a compact smooth manifold with boundary $Y$ and tangent bundle $\tau$. Then ...

**19**

votes

**2**answers

1k views

### A manifold is a homotopy type and _what_ extra structure?

Motivation: Surfaces
Closed oriented 2-manifolds (surfaces) are "classified by their homotopy type". By this we mean that two closed oriented surfaces are diffeomorphic iff they're homotopy ...