Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

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3
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1answer
403 views

Quantization of symplectic vector space and choice of lagrangian subspaces

My question is related to Geometric Quantization. I don't undrestand the philosophy of following assertion If $(V,\omega)$ be a symplectic vector space then the quantizations of $V$ ...
6
votes
1answer
283 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, ...
14
votes
1answer
599 views

The cone on a manifold

I believe that I have run across the statement that if $X$ is a compact smooth manifold and $CX$ is the cone on $X$, i.e. $[0,1] \times X$ modulo $(0,x)\sim(0,y)$ for all $x,y \in X$, then $CX$ admits ...
8
votes
0answers
299 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 $M$. ...
3
votes
0answers
77 views

Analytic stuctures on $\mathbb R^n$ and the nilpotent ideal of supermanifolds

I have two questions which are somewhat related: (a) It is a well known result (of Freedman?) in differential topology that $\mathbb R^4$ has exotic smooth structures. Apparently, it is known that ...
10
votes
3answers
784 views

Is the class of n-dimensional manifolds essentially small?

Question: Consider the proper class of all $n$-dimensional smooth manifolds. If we take the equivalence classes where two manifolds are identified if there exists a diffeomorphism between them, is ...
3
votes
1answer
559 views

Is it possible to define a structure of differentiable manifold on the Hilbert cube $[0,1]^\mathbb{N}$?

Is it possible to define a structure of differentiable (smooth) manifold on the Hilbert cube $[0,1]^\mathbb{N}$ ? Has it been done in the literature? In textbooks, only the Banach case is treated, ...
11
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1answer
244 views

Hyperbolic Manifolds which fiber over the circle

If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by ...
1
vote
1answer
177 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 ...
1
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0answers
89 views

Proving that two given functionally structured spaces are isomorphic

The relevant definitions are listed below. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups; and Section 2, Chapter II of Bredon's Topology and ...
9
votes
1answer
400 views

Do partitions of unity exist if we impose additional conditions on the derivatives?

Let $ ~~\cup_{k=-1}^{\infty} U_k = \mathbb{R} $ be an open covering of $\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to the cover exists, i.e. there exists smooth ...
7
votes
2answers
229 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
2answers
213 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
3answers
437 views

Configuration spaces of the torus

I would like a reference that calculates the rational homology of the unordered configuration spaces of the torus.
2
votes
3answers
270 views

Fatou sets and topological entropy

Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from ...
1
vote
1answer
282 views

The space of generalized complex structures in sense of N.Hitchin is contractible?

Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...
3
votes
1answer
154 views

Lipschitz Approximation to a PW Smooth Map

Suppose I have a triangulated smooth manifold, $\tau : |K| \rightarrow M$ (so that $\tau | _{\sigma}$ is smooth for each $\sigma \in K$), and a piecewise smooth map, $f: M \rightarrow \mathbb{R}^n$. ...
2
votes
2answers
150 views

intersection of Whitney stratifications

Let $X$ be an oriented smooth manifold with dimension $n$. If $U$ and $V$ are two oriented closed submanifolds of $X$ and $U$ is transverse to $V$ in $X$. Then $U\cap V$ (suppose the intersection is ...
0
votes
0answers
117 views

transverse intersection of Whitney stratifications

Let $M$ be a smooth manifold. If $X$ and $Y$ are two Whitney objects, i.e. subsets with a given Whitney stratification, then $X$ and $Y$ are transverse if each stratum of $X$ is transverse to each ...
9
votes
1answer
530 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 ...
3
votes
2answers
287 views

Is it impossible for the dimension of a topological space to increase under a smooth map?

First let me make a definition. Let $M$ be a smooth manifold and $S \subset M $ a topological subspace of $M$. We say that $S$ has "dimenion" at most $k$ if $S$ is a subset of $$ X_1 \cup X_2 \ldots ...
3
votes
1answer
116 views

Hilbert Manifolds and embedding

In the Wikipedia article on Hilber manifolds, it is claimed that every Hilbert manifold can be smoothly embedding onto an open subset of the model Hilbert space. However, no explicit reference is ...
4
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0answers
146 views

formal smooth morphism with a formal smooth source

Let $f:X\rightarrow Y$ a morpism between $k$-schemes ( $k$ a field). We suppose that X is formally smooth and f is formally smooth and surjective. Do we have that $Y$ is formally smooth? Or if it's ...
11
votes
1answer
691 views

Does the gluing procedure in Robert Wald’s book *General Relativity* yield a Hausdorff spacetime?

Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity. Definition 1: A triplet $ (\Sigma,h,k) $ is called an initial data set if $ ...
20
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0answers
593 views

Do there exist exotic 4-tori?

More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds? Related: if such a ...
0
votes
1answer
132 views

A version of implicit function theorem when sections are not everywhere smooth?

Let $V_1, V_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$ a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section ...
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0answers
228 views

Can one always extend a smooth section defined on a non compact submanifold to the whole manifold, provided it extends continuously to the closure?

Let $V \rightarrow M$ be a smooth vector bundle over a smooth compact manifold $M$ (without boundary) and $X \subset M$ a smooth submanifold of $M$, that is not necessarily closed. Suppose $s: X ...
5
votes
0answers
224 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 ...
2
votes
1answer
469 views

Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space?

Suppose that $X$ is a compact, finite dimensional manifold and $Y$ is an infinite dimensional, second countable ($C^\infty$-)Banach manifold. Let $\nu \in \mathbb{N}$. Question: Is the space ...
8
votes
1answer
357 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 ...
24
votes
1answer
1k views

Strong Whitney embedding theorem for non-compact manifolds

$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds. The strong ...
5
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0answers
121 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). ...
4
votes
1answer
641 views

Differentiable manifolds by Serge Lang question

I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ...
2
votes
1answer
162 views

Vector fields on a simplicial manifold.

Is there a known definition of vector fields on a simplicial manifold? For me, it seems natural that the definition should be something along the lines: Let $M_{\bullet}$ be a simplicial manifold ...
8
votes
2answers
593 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
2answers
512 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 ...
14
votes
3answers
779 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 ...
1
vote
0answers
158 views

Uniqueness of the Smooth Structure on a Handle Attachment [closed]

I posted this question on math stack exchange and didn't receive an answer. If it is too elementary for this forum I will be happy to delete it. Let $M^m$ be a smooth manifold with boundary. We may ...
21
votes
4answers
1k 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 $k=1$ ...
9
votes
0answers
241 views

Homology classes represented by $J$-holomorphic curves

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if $$ du \circ j = J \circ ...
1
vote
1answer
164 views

Can elements of Weil algebras be detected by maps into truncated symmetric algebras?

Recall that a Weil algebra is a finite-dimensional real unital algebra that admits exactly one homomorphism to R. Such algebras form the basis of the Weil approach to differential geometry, pioneered ...
2
votes
0answers
76 views

Non-clean fiber products

Usually, the most general condition for fiber product of manifolds (or vector bundles) to exist is that we require the images cleanly intersects. See e.g. When do fibre products of smooth manifolds ...
3
votes
2answers
318 views

if $S \times \Re$ is diffeomorphic to $T \times \Re$ then are S and T diffeomorphic?

Suppose that $S$ and $T$ are two smooth manifolds and '$ \Re$' be the reals with the normal manifold structure. And here I use '$=$' to mean diffeomorphism. Is the statement below true? $ S \times ...
1
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0answers
107 views

Functional Analysis Generalizations: indeterminated inner product and functions over manifolds

There are books or articles that deal with generalizations of functional analysis in the sense that the inner product need not be positive-definite or that works with functions over manifolds?
3
votes
1answer
309 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 ...
11
votes
1answer
469 views

Pullbacks as manifolds versus ones as topological spaces

My question is: Does the forgetful functor F:(Mfd) $\to$ (Top) preserve pullbacks? Detailed explanation is following. A pullback is defined as a manifold/topological space satisfying a universal ...
1
vote
1answer
142 views

finding effective 2-form corresponding to an equation

What is the effective 2-form corresponding to the equation $det Hess v=(v-q_1v_{q_1}-q_2v_{q_2})^4$ you can find the definition of effective forms here
11
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3answers
671 views

Characterizing Hessians among symmetric bilinear tensors

I apologize in advance if this is somewhat elementary, but: Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ are ...
1
vote
2answers
366 views

Is this surface diffeomorphic to a sphere(S^2)? [closed]

let $f:R^3 -> R \ \ \ \ \ \ \ \ f(x,y,z)=x^4 + y^6 +z^8 \\$ $M = f^{-1}(1)$ Is M is diffeomorphic to a sphere $S^2$ ? I tried to solve this problem, but I realized that I have no tools to ...
12
votes
1answer
318 views

What is the status of the PL-pseudoisotopy stability theorem?

Suppose that $M$ is a compact PL-manifold (possibly with boundary) and let $C^{PL}(M)$ denote the (simplicial) group of PL isomorphisms of $M \times I$ relative to $M \times \{0\} \cup \partial M ...