1
vote
1answer
33 views

Simultaneous coordinate representation of a submanifold and its sub-submanifold

Suppose $Z\subset X\subset Y$ are manifolds and $z \in Z$. Prove that there exist an independent function $g_1,...g_l$ on a neighborhood $W$ of $z$ in $Y$ such that $$Z \cap W =\{y\in ...
0
votes
0answers
17 views

Orientation of the intersection of manifolds

From Guillemin and Pollack Differential Topology: Compute the orientation of $\mathbf{X}\cap\mathbf{Z}$ in the following examples by exhibiting positively oriented bases at every point: a) ...
9
votes
0answers
98 views

$C^{k}$-manifolds: how and why?

First of all, I have a specific question. Suppose $M$ is an $m$-dimensional $C^k$-manifold, for $1 \leq k < \infty$. Is the tangent space to a point defined as the space of $C^k$ derivations on the ...
3
votes
3answers
57 views

$T^2\times S^n$ is parallelizable

This is taken from a UCLA Geometry/Topology qualifying exam. How would one prove that $T^2\times S^n$ is parallelizable for all $n\geq 1$? Is there a way to find $n+2$ linearly independent vector ...
2
votes
1answer
77 views

If $S\subseteq M\times N$ is embedded, and $S$ and $\{p\}\times N$ intersect transversely in one point, then $\pi_M|_S$ is a diffeomorphism?

I'm trying to prove the equivalence of the following statements: Suppose $M^m$ and $N^n$ are smooth manifolds, $S\subseteq M\times N$ immersed, and $\pi_M$ and $\pi_N$ the projection maps. TFAE: ...
2
votes
1answer
33 views

Does Heine-Borel hold for smooth manifolds?

If $M$ is a smooth $n$-manifold, the famous Whitney embedding theorems show that we can view $M$ as an embedded submanifold of some Euclidean space $\mathbb{R}^N$. Does the Heine-Borel theorem still ...
4
votes
0answers
97 views

Surgery presentations and gluing

It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial ...
2
votes
1answer
53 views

If $b$ is a regular value of $f$, $f^{-1}(-\infty,b]$ is a regular domain?

I'm trying to prove the first part of Proposition 5.47 of Lee's Smooth Manifolds, which is left to the reader. It says Suppose $M^m$ is a smooth manifold, and $f\colon M\to\mathbb{R}$ smooth. For ...
4
votes
1answer
73 views

Example of a submanifold $S\subseteq M$ that is an immersed submanifold is more than one way?

Known uniqueness results say that an embedded submanifold has a unique smooth structure making it an embedded submanifold with the subspace topology, and immersed submanifolds have a unique smooth ...
2
votes
1answer
43 views

When gluing maps are isotopic?

Let $M$ and $M'$ be compact orientable connected topological 3-manifolds. (One may need more conditions to answer the question.) Suppose we have two homeomorphisms $f$ and $g$ from the boundary ...
2
votes
1answer
51 views

Boundary connected sum of manifolds

I have two related questions about the boundary connected sum of manifolds with boundaries. Let $T=S^1 \times S^1$ be a torus and let $X=T \times [0, 1]$ be the cylinder over the torus. Let $X'$ be a ...
3
votes
0answers
30 views

Existence of orientation-reversing automorphisms

Let $M$ be a connected orientable manifold (I don't really care about which category of manifolds we should work in). Since I read that there are nice manidolds without orientation-reversing ...
1
vote
0answers
34 views

A question about the boundary points of manifolds.

A topological $n$-manifold is a second countable Hausdorff space such that every point has a neighbourhood which is homeomorphic to an open ball centred at the origin in $\Bbb{R}^n$. The "centred at ...
0
votes
1answer
35 views

Condition for Orientability of Manifold

Let $M^n$, $n>2$ be a manifold and let $f:D\rightarrow M$ be an embedding of the closed $n-$disk in $M$. Prove or Disprove: $M$ orientable iff $M-f(D)$ is orientable. $M$ is orientable iff all ...
2
votes
1answer
43 views

Quotient of Unit Quaternions by Subgoup (Lie Groups)

Let $G \leq Sp(1)\cong S^3$ (unit quaternions) be a discrete subgroup of order 120 (the Binary Icosahedral Group, not the other one), with presentation $G=<s,t| s^2=t^3=(st)^5>$. $\hspace{2mm}G$ ...
2
votes
1answer
48 views

Is $\mathbb{R}P^n$ “two-sided” in $\mathbb{R}P^{n+1}$

i.e. Does $\mathbb{R}P^n$ have a tubular neighborhood $N$ such that $N-\mathbb{R}P^n$ is disconnected. My guess is yes, but don't know how to show it convincingly ( or maybe only for $n$ odd, I'm ...
1
vote
1answer
23 views

Inverse Image of a Regular Value an Orientable Submanifold

Let $f:M^n \rightarrow \mathbb{R}$ be a smooth map, and let $c\in N$ be a regular value. When is $f^{-1}(c)$ an orientable manifold? Note: I know by regular value thm, $f^{-1}(c)$ is a smooth $n-1$ ...
3
votes
1answer
62 views

What's the Differential of this Map $f:S^3\rightarrow \mathbb{R}$

$f:S^3 \rightarrow \mathbb{R}$ is defined as $f(x,y,z,w)=x+zw$, where $S^3= \{(x,y,z,w) | x^2 +y^2 +z^2 +w^2 =1\}$ I tried using a stereographic chart but that got ugly. The function is so simple I ...
4
votes
0answers
51 views

Gluing two solid tori along their boundary resulting in a topological manifold

The following question is from a past qualifying exam. Take two solid tori $D^2 \times S^1$, and construct the space $X$ by identifying their boundaries via the map $f \colon \partial D^2 \times S^1 ...
9
votes
2answers
200 views

Is there a retraction of a non-orientable manifold to its boundary?

It's easy to show using Stokes theorem that a compact orientable manifold with boundary cannot retract to its boundary, by choosing a volume form. But for the non-orientable case I don't know if this ...
1
vote
1answer
29 views

A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
3
votes
0answers
64 views

Submanifold of $2\times 2$ Complex Matrices Using Transversality?

Let $M$ be the set of all $2\times 2$ complex matrices with $a_{21}=\bar a_{12}$. $M$ is a smooth manifold diffeomorphic to $\mathbb{R}^6$. Let $W$ be the subset consisting of all matrices in $M$ with ...
4
votes
2answers
51 views

Geometrically, what is the stereographic projection of a closed $n$-ball?

To show $\overline{B^n}$ is a $n$-manifold with boundary, apparently there is a trick to use stereographic projection after subtracting out the radius connecting $0$ to the north pole. I'm familiar ...
2
votes
0answers
70 views

Role of the Thom space in the Pontryagin-Thom construction

I am trying to understand the Pontryagin-Thom theorem; especially how the Thom space comes into play. Just to bring everyone on the same page: I am specifically talking about the construction of an ...
4
votes
1answer
85 views

Orientation of hypersurface

Some books on mean curvature flow (e.g. Mantegazza, Ecker) state that an embedded hypersurface in $\mathbb{R}^{k+1}$ is orientable (Mantegazza page 3, Ecker page 110). In other words, they assume the ...
1
vote
1answer
18 views

natural projection on a slice

I'm currently studying Warner's book "Foundations of Differentiable Manifolds and Lie Groups". Within the proof of the Frobenius Theorem he is constructing a slice $S$ of a coordinate system ...
0
votes
0answers
20 views

Question on framed bordism classes definition

I was reading recently about cobordism, and in specific about the Thom-Pontraygin theorem which states $\pi_{k}(S^n)$ is isomorphic to the cobordism classes of framed $n$-manifolds in $R^k$. In ...
2
votes
1answer
38 views

Wedge product of basis elements of cohomology

Let $M$ be a compact, connected, oriented 4-manifold without boundary. If $H^2(M)\cong \mathbb{R}^2$ and I have a basis $\{[\omega_1],[\omega_2]\}$ for $H^2(M)$, is it the case that $[\omega_1\wedge ...
0
votes
0answers
42 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
1
vote
2answers
163 views

Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
1
vote
0answers
43 views

A basic question on intuition of second countability for definition of topological manifold

What is the intuition behind the requirement of second countability in definition of topological manifold? It seems that it is relevant to the $\mathbb{R}^n$ has second countable basis. Is there any ...
2
votes
1answer
58 views

Euler characteristic for non-compact manifolds

How can one generalize the Euler characteristic to non-compact manifolds? Furthermore, is there a way to generalize the notion of an intersection number to non-compact manifolds, so that one could ...
1
vote
1answer
136 views

Another differential topology lemma

Another lemma (1) Why can we assume $z=f(z)=0$ and that $U$ is convex? (the coordinate domains of the manifolds can be taken to be balls?) (2) Why is it enough to consider the special case of a ...
0
votes
1answer
167 views

A differential topology lemma

Consider the following lemma (1) How come he talks about degrees here, after all he doesn't assume $X$ to be oriented? (2) Why is $\bar{v}|\partial X$ homotopic to $g$? (NOTE: we consider them as ...
6
votes
1answer
86 views

Is there any embedding theorem for fibre bundles?

I would like to know whether there is an embedding theorem for fibre bundles, like Whitney embedding theorem. When can a given fibre bundle be a subbundle of some higher dimensional bundle?
3
votes
2answers
41 views

smooth surjective map from lower dimensional manifold onto higher dimensional manifold?

I'm thinking about whether there exists a smooth surjective map $f: M^m \twoheadrightarrow N^n$ where both $M$ and $N$ are smooth manifolds, and $\dim M = m < n = \dim N$. (We might assume that $M$ ...
0
votes
0answers
27 views

express skew-commutative product of Whitney sum of vector bundles in tensor products

Let $\xi$ and $\eta$ be vector bundles over a paracompact space $B$ and $\xi\oplus\eta$ be their Whitney sum. Can we write $\Lambda(\xi\oplus\eta)\cong \Lambda(\xi)\otimes \Lambda(\eta)$ as (graded) ...
0
votes
1answer
19 views

generation of sub bundle

Let $M$ differentiable manifold with $\dim M=n$. If $(TM,\pi,M)$ be the fiber bundle tangent. Consider the family $E=\lbrace E_x\rbrace _{x\in M}$ such that $E_x \subset T_xM$ and $\dim E_x=k$ for ...
3
votes
1answer
37 views

Finding zeros of maps between manifolds with different dimensions

One of the questions for which the notion of degree is useful is: does this map have a zero. For example, one can prove the Fundamental Theorem of Algebra using the following fact involving the ...
7
votes
3answers
137 views

let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
3
votes
2answers
112 views

tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
0
votes
0answers
27 views

Orientation of manifolds with boundary

I have an ambiguity about how to orient the boundary of a manifold. In particular : Consider the example $M=B^2 \subset \mathbb{R^2}$ be the manifold with boundary. suppose positive orientation for M ...
1
vote
0answers
25 views

define distance in a manifold over the reals

G is a Hausdorff manifold over the reals with a finite atlas: $\exists m$ $G=\bigcup_{1 \leq i \leq n}U_i$, $g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}^m$. Can I somehow define a metric inside G, ...
1
vote
0answers
49 views

Sequence of critical values has no cluster point (Milnor, Morse Theory)

The following Claim is used in the proof of Theorem 3.5 in John Milnor's "Morse Theory": Claim: Let $f: M \rightarrow \mathbb{R}$ be a differentiable function on a manifold $M$ with no degenerate ...
0
votes
0answers
32 views

Real analytic manifold

I found in some lecture notes such a definition of real analytic manifold: Let $X$ be a complex manifold (ringed space, locally isomorpic to...), $i: X \rightarrow X$ - a conjugation ...
1
vote
1answer
74 views

Definition of smooth maps between manifolds

Here is a page from Guillemin-Pollack's differential topology: My question is: At the bottom he defines $df=d\psi\circ dh\circ d\phi^{-1}$. Why doesn't he just define $df=dh$, like here: ...
1
vote
0answers
41 views

Upper-half space of a manifold with boundary

Suppose that $X$ is a manifold with boundary and suppose that $x$ is a boundary point. Define the upper half space $H_x(X)$ in $T_x(X)$ to be the image of $\mathbb{H^k}$ under $d \phi_0:\mathbb{R^k} ...
3
votes
1answer
54 views

submanifold of Euclidean space is oriented if and only if normal bundle is an oriented vector bundle.

Let $f:M\longrightarrow \mathbb{R}^{n+k}$ be an immersion of $n$-dimensional manifold $M$ into $\mathbb{R}^n$. Let $\nu(M)$ be the normal bundle of $M$. Prove that $M$ is oriented if and only if ...
2
votes
1answer
65 views

measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently ...
1
vote
1answer
56 views

Tangent bundle of a manifold [duplicate]

can anyone help me with this problem: Show that for a manifold $M$, the tangent bundle $TM$ also has the structure of a manifold. If $M$ is an n-manifold, what is the dimension of $TM$? for the 1st ...