The tag has no wiki summary.

learn more… | top users | synonyms (1)

6
votes
1answer
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 $M$, ...
1
vote
0answers
81 views

Is there a reference for some notions relative to distributions of corank 1?

In a expository text on differential geometry I am reading about the geometry of distributions of a corank one. Here the first properties are reported without proof, and no reference is given. I ...
16
votes
1answer
585 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 ...
4
votes
2answers
1k views

On the smooth structure of the spaces of $k$-jets

I was asking myself, if the following list of conditions is sufficient to determine the usual smooth structure on the spaces of $k$-jets. the map $j^k f:M\ni x\to j_x^k f\in J^k(M,N)$ is smooth, ...
31
votes
4answers
2k views

Can cotangent bundles see exotic smooth structures?

I have two questions that are inspired by a couple of questions here on MO (referenced below), as well as by a conversation with some other grad students at a summer school. Caveat: I'm not a ...
1
vote
3answers
441 views

symplectic form with partition on unity

Assume $M$ is a $2n-$dimensional differentiable manifold. Let $(U_{i})$ be a open covering of $M$. With respect to this covering let $\rho_{i}$ be a partition of unity. Assume that on each $U_{i}$ we ...
3
votes
2answers
922 views

The double of a smooth manifold with boundary?

$\def\mc#1{\mathcal#1}\def\seq#1{\langle#1\rangle}\def\bbR{\mathbb R}\def\gt{>}\def\dom{{\rm dom\ }}$In some instances, I have seen an appeal to the concept of "the double" of a smooth manifold ...
0
votes
1answer
406 views

immersion: submanifold of complex manifold

Let $\alpha : \mathbb{C} \rightarrow M$ be an immersion and $M$ a $n$ dimensional complex manifold with complex structure $I$. Does then follow that $\alpha (\mathbb{C})$ is a one dimensional ...
0
votes
0answers
330 views

covariant derivative complex manifold

Assume we have $X$ a complex manifold and $Y = Y^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ and $Z = Z^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ two vector fields on $X$. Let $\nabla$ be the ...
2
votes
1answer
205 views

biholomorphism complex manifold induced structure

Let $X$ be a $n$ dimensional complex manifold with complex structure $I$ and assume one has a diffeomorphism $f : \mathbb{C} \rightarrow X$ of some open set $U$ in $\mathbb{C}$ into its image $f(U)$. ...
9
votes
1answer
474 views

Is every closed embedded codimension-n submanifold cut out by a section of a rank-n vector bundle?

Let $M$ be a smooth manifold (over $\mathbb R$) and $N \hookrightarrow M$ a closed embedding. Locally near any point in $N$, I can find coordinates $x^1,\dots,x^{\dim M}$ on $M$ so that $N$ is the ...
6
votes
1answer
469 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 ...
1
vote
1answer
622 views

About the geometry of completely integrable systems

During a conversation I heard an assertion that I found at least dubious for the lack of adeguate hypothesis, but I am not able to imagine a counterexample, even if it is probably obvious to some of ...
22
votes
2answers
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 ...
6
votes
0answers
254 views

Universal property for complex blowup in smooth category

If $M$ is a smooth complex manifold and $N$ is a smooth complex surface, we may ask when a holomorphic map $f:M\rightarrow N$ lifts to a map $f:M\rightarrow [N:p]$, where $[N:p]$ denotes the blowup of ...
3
votes
1answer
407 views

Fixed points of the action of an algebraic group

Hello! If a compact Lie group $K$ acts smoothly on a smooth manifold $M$, then the set $M^K$ of fixed points under this action is a smooth submanifold of $M$. This is proved for example in ...
3
votes
2answers
352 views

uniqueness of regular/tubular neighborhood with equivariant boundary

Let $N$ and $N'$ be regular neighborhoods of a subpolyhedron $P$ in a closed PL manifold $M$, and suppose that $t$ is a free PL involution on $M$ such that each of $\partial N$, $\partial N'$ is ...
11
votes
2answers
551 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 ...
23
votes
1answer
1k views

Can you flip the end of a large exotic $\mathbb{R}^4$

Can you flip the end of a large exotic $\mathbb{R}^4$ Background Definition (Exotic $\mathbb{R}^4$): An exotic $\mathbb{R}^4$ is a smooth manifold $R$ homeomorphic but not diffeomorphic to ...
14
votes
13answers
3k views

What should be taught in a 1st course on smooth manifolds?

I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic ...
9
votes
1answer
801 views

Looking for a simple proof that R^2 has only one smooth structure

So not so long ago, I asked for a simple proof that $\mathbf{R}$ has only one smooth structure. A proof that was communicated to me by Ryan Budney (link text) was the following: So let me recall his ...
3
votes
1answer
425 views

Do the focal points of a submanifold $M$ in $\mathbb{R}^k$ form a closed subset?

Let $M$ be a submanifold in an euclidean space $\mathbb{R}^k$, and $\nu(M)$ the normal bundle to $M$, let us denote $\phi$ the restriction to $\nu(M)$ of the exponential map for $\mathbb{R}^k$. A ...
5
votes
0answers
171 views

How do metrics behave under joining along a manifold embedded in the boundary?

How do metrics behave under joining along a manifold embedded in the boundary? This is, more-or-less, part of Problem 4.66 in Kirby's List: Problem 4.66 How do metrics (e.g. Riemannian, Lorentz, ...
10
votes
0answers
373 views

Exotic smoothness and Parallelizability

Regarding the parallelizability of the Milnor's seven dimensional exotic spheres: Parallelizability of the Milnor's exotic spheres in dimension 7 The following question naturally arises: Suppose ...
54
votes
4answers
2k views

Parallelizability of the Milnor's exotic spheres in dimension 7

Are the Milnor's seven dimensional exotic spheres parallelizable?
26
votes
3answers
1k views

Is it possible to improve the Whitney embedding theorem?

Edited to fix the example, as per Zack's suggestion. Edit 2: So it turns out that when I think 'manifold' I tend to assume the nicest possible object. As I believe is standard, I would like to ...
8
votes
1answer
749 views

Finite-dimensionality for de Rham cohomology

I was browsing through the litterature, hoping to find sufficient and necessary conditions for a smooth manifold to have finite-dimensional de Rham cohomology, but I can't find any satisfactory ...
18
votes
2answers
2k views

Universal property of the tangent bundle

If $X$ is a scheme (over some base scheme, but which I will ignore) its tangent bundle $T(X)$ is defined as the relative spectrum of the symmetric algebra of its sheaf of differentials. Combining the ...
13
votes
2answers
775 views

Nice application of generalized smooth spaces

I am a fan of category theory in general, and I appreciate that various brands of generalized smooth spaces (Diffeological spaces, Chen spaces, Frolicher spaces ...) form much nicer categories of ...
6
votes
3answers
516 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
2answers
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. ...
5
votes
2answers
801 views

Is volume--preserving an intrinsic property?

Let $M$ be a compact smooth manifold without boundary. A Riemannian metric $g$ on $M$ induces a volume measure (or Lebesgue measure) $m_g$ on $M$. A diffeomorphism $f:M\to M$ is said to be ...
19
votes
6answers
3k views

Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$. In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$, in Spivak, we find ...
17
votes
1answer
810 views

What's the Kirby Diagram of a universal $\mathbb{R}^4$?

What's the Kirby diagram of a universal $\mathbb{R}^4$? Background Define $\mathcal{R}$ as the set of smoothings of $\mathbb{R}^4$. For two oriented elements $R_1$, $R_2$ in $\mathcal{R}$ we can ...
0
votes
0answers
469 views

A doubt on a problem in Manifolds, Tensor Analysis and Applications

Having tried to solve exercise 4.4-7 to have another proof of Frobenius Theorem, I would ask you a question. This is what I have understood: In Step 2 there is to prove, for any tangent subbundle ...
1
vote
1answer
234 views

Commutativity of pullbacks and the exterior derivative as an unbounded operator on $L^2$

Let $d_c, \delta_c$ be operators with domains $D(d_c) = D(\delta_c) = C_{c}^\infty(\wedge T^\ast M)$. We let $d_c$ be the usual exterior derivative on compactly supported smooth forms, ie., $d_c\omega ...
1
vote
1answer
267 views

why rotation group is a smooth 3-manifold [closed]

Do you need the implicit function theorem? Can you give an explicit charts?
12
votes
1answer
499 views

smoothly varying smooth structures

Can one vary smooth structures on $\mathbb R^4$ smoothly/continuously? This question popped out of Ben's answer here.
7
votes
2answers
642 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 ...
3
votes
0answers
218 views

Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold?

Let $(M,g)$ be a geodesically complete Riemannian manifold that is not necessarily compact. Futhermore, assume that $M$ has at most exponential volume growth (ie., locally doubling property). Let ...
2
votes
3answers
665 views

analytic structure on lie groups

I need a reference for a result I have heard only very vaguely "A lie group (smooth) has a compatible analytic manifold structure". (Would even appreciate a concise way to refer to the result..) I ...
11
votes
2answers
970 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 ...
4
votes
2answers
2k views

A metric for Grassmannians

Hello everybody! I'm reading an article by Ricardo Mane, Hausdorff dimension is dipheomorphism. I'm having a technical problem. Sorry for my ignorance but Would you like a references where I can find ...
12
votes
3answers
714 views

orientations for zero-dimensional manifolds

I am teaching a course on manifolds, and soon I will have to prove the Stokes' theorem which, of course, involves defining oriented manifolds. There are many ways to define an oriented manifold. My ...
5
votes
2answers
496 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 ...
2
votes
2answers
1k views

When are diffeomorphisms from a manifold to itself homotopic to the identity?

I'm sure people in the field know this, but I'm not in the field. Under what conditions (be they on the manifold or the map) is a diffeomorphism from a differentiable manifold $M$ to itself homotopic ...
9
votes
5answers
1k views

Ricci Curvature in Infinite Dimensions?

Is there a good notion of "Ricci curvature" in infinite dimensions? My intuitive understanding of Ricci curvature is that it is some kind of an "average" of the curvature tensor over "different ...
1
vote
1answer
381 views

Understanding manifold GL+(3,R)/SO(3) ?

I'm trying to better understand the manifold GL+(3,R)/S0(3) which is diffeomorphic to positive definite symmetric matrices. My motivation is to understand U in F = RU where F in GL+(3,R) = deformation ...
21
votes
4answers
1k views

Can all n-manifolds be obtained by gluing finitely many blocks?

Fix a dimension $n\geqslant 2$. Let $S= \{M_1,\ldots, M_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is generated by $S$ if it may ...
7
votes
2answers
1k views

Slice knots and exotic $\mathbb R^4$

In the http://arxiv.org/abs/math/0606464v1 I read "If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in possession of a knot which is ...