The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key ...

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2
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1answer
27 views

Fuchsian group with parabolic element

I'm interested in this problem: let $\Gamma \subset PSL(2, \mathbb{R})$ a Fuchsian group (i.e. it is a discrete subgroup) which contains the trasnformation $\gamma \colon z \mapsto z+1$ then the ...
1
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0answers
27 views

natural map to the homotopy fibre

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 280, paragraph 4, line 2-line 3 and Configuration spaces of positive and negative particles, McDuff, page 105, line ...
4
votes
1answer
53 views

Topological information from metric tensor

Suppose I am working with a Riemannian manifold $(M,g)$, and I have a particular coordinate expression for the metric $g$. What topological information can I infer about the manifold $M$? For ...
2
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0answers
25 views

universal bundle of topological monoids

Let $M$ be a topological monoid. There is a classifying space $BM$ (cf. canonical map of a monoid to its classifying space). When $M$ is a group $G$, there is a principal $G$-bundle $EG\to BG$ such ...
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0answers
22 views

pontrjagin ring of the homology of iterated loop suspension

In The homology of C n+1 spaces, n>=0, F. Cohen, proof of Theorem 3.1 and proof of Theorem 3.2 (p. 228 - 243) I totally do not understand the proofs of these two theorems from page 228 to page 243 ...
5
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1answer
56 views

What are the 8 non-compact Euclidean 3-manifolds?

I have found several sources that mention that there are eight non-compact Euclidean 3-manifolds, with four orientable and four non-orientable. There are two standard references for this, but ...
3
votes
1answer
66 views

Is every open cover of a smooth manifold finer than a cover built from the union of disjoint open sets?

Let $M$ be a finite dimensional smooth manifold and $M=\bigcup_{i\in I}U_i$ an open cover of $M$. Does there exist a finite open cover $M=\bigcup_{k=0}^l V_k$, such that each $V_k$ is the disjoint ...
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0answers
38 views

configuration space model for classifying space of monoid

Let $M$ be a monoid and $BM$ be its classifying space. There is a model for $BM$ based on labelled configuration spaces of the line $[0,1]$. Points of the configurations are labelled by elements of ...
31
votes
6answers
489 views

Is the complement of countably many disjoint disks path connected?

Let $\{D_n\}_{n=1}^\infty$ be a family of pairwise disjoint closed disks in $\mathbb{R}^2$. Is the complement $$ \mathbb{R}^2 -\bigcup_{n=1}^\infty D_n $$ always path connected? Here ...
31
votes
3answers
407 views

Does every bijection of $\mathbb{Z}^2$ extend to a homeomorphism of $\mathbb{R}^2$?

Given a bijection $f\colon \mathbb{Z}^2 \to \mathbb{Z}^2$, does there always exist a homeomorphism $h\colon\mathbb{R}^2\to\mathbb{R}^2$ that agrees with $f$ on $\mathbb{Z}^2$? I don't see any ...
4
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2answers
94 views

Why the lens space L(2,1) is homeomorphic to $\mathbb{R}P^3$?

According to one definition of lens space $L(p,q)$, which is gluing two solid tori with a map $h:T^2_1 \rightarrow T^2_2$. And $h(m_1)=pl_2+qm_2$, $l_i$ means longitude and $m_i$ means meridian of the ...
4
votes
1answer
47 views

Are there compact manifolds homotopy equivalent to a wedge sum of compact manifolds?

One example given by Hatcher as an application for the cohomology ring is to distinguish $\mathbb{CP}^2$ from $S^2 \vee S^4$ up to homotopy equivalence despite their cohomology groups being the same. ...
6
votes
1answer
76 views

knot theory: two definitions of equivalence (ambient isotopy and homeomorphism)

I am looking into knot theory and have found two different definitions stating that two knots $K_1$ and $K_2$ are equivalent, namely the concept of an ambient isotopy: These two knots are ambient ...
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0answers
25 views

Alexander polynomial of sums

A standard argument of knot theory reveals that the Alexander polynomial of the sum of two knots $K= K_1 \# K_2$ is equal to the product of the Alexander polynomial of the two summands $K_1$ and ...
2
votes
1answer
31 views

Definition of a rim torus

We know a torus is $S^1 \times S^1 =T^2$. We know a solid torus is $D^2 \times S^1$ whose boundary is a torus $S^1 \times S^1 =T^2$. What is the definition of a rim torus?
4
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0answers
63 views

Non trivial homomorphism from $SU(2)$ to the diffeomorphism group of the circle

Is there a non trivial homomorphism $f: SU(2) \to \operatorname{Diff}(S^1)$? (From the comments) By a previous question, we know that there is no nontrivial homomorphism $SU(2) \to O(2)$. Since ...
3
votes
1answer
58 views

cup product in cohomology ring of a suspension

Let $X$ be a CW-complex. Let $\Sigma$ be suspension. Let $R$ be a commutative ring. Is the cup product of $$ H^*(\Sigma X;R)$$ trivial? How to prove? Where can I find the result?
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1answer
35 views

Confused with The Transversality Theorem when all manifolds are boundaryless

In Guillemin-Pollack's book Differential Topology, the Transversality theorem states that The transversility Theorem. Suppose that $F:X \times S \to Y$ is a smooth map of manifolds, where only $X$ ...
3
votes
1answer
47 views

Remove one ring of Borromean rings in 3-sphere: linked or unlinked?

We know Borromean rings in a 3-sphere $S^3$ can be unlinked if we remove one of the three rings. Here let us consider a slight different procedure. If we remove the neighbored solid torus $B^2 \times ...
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0answers
33 views

Integral Over the N-Sphere in the framework of chains:

Integration over manifolds is commonly defined with object called chains. What about if I want to integrate the exterior derivative of a $k-form$ over the n-sphere and use Stokes theorem: ...
2
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0answers
78 views

Which spaces admit bump functions?

Let me first fix terminology. Let $X$ be a topological space and $A\subseteq B\subseteq X$ be subsets. Let's say X admits a bump function relative to $A\subseteq B$ if there's a continuous function ...
4
votes
1answer
70 views

If $M$ is a 4-dimensional, compact, simply connected manifold with boundary, what is the topology of $\partial M$?

I know if we have a compact simply connected 3-manifold with boundary, then the boundary will be $S^2$. The argument is as following 1) $H_1(M)\simeq 0$; 2) Poincare duality $H_2(M,\partial ...
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votes
1answer
54 views

What is a simply connected space and a multiply connected space?

I was reading the book Hyperspace by Michio Kaku when I read about simply connected space and multiply connected space. I just tried searching the answer using Wikipedia but I wasn't able to ...
0
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1answer
52 views

About the Chern class of the determinant line bundle

Is it true that the first Chern class of a rank $k$ complex line bundle is equal to the first Chern class of its determinant line bundle?
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0answers
54 views

cohomology ring of a subspace of real projective spaces

I learned $H^*(\mathbb{R}P^n;\mathbb{Z}_2)=\mathbb{Z}_2[a]/(a^{n+1})$, $|a|=1$, in topology class, when studying cell complex and cohomology. Now I want to find the cohomology ring ...
1
vote
1answer
57 views

Is my graph a tree?

Let M be a smooth connected manifold. G is a group act on M cocompactly and suppose there is a harmonic function $h$ on M with minimal energy.$h:\rightarrow [0,1]$ such that h is nonconstant and ...
4
votes
1answer
43 views

Nonorientable manifolds being a boundaries

I will say that a manifold (smooth, compact, without boundary) is itself boundary if there is some smooth compact manifold $W$ with boundary, such that $M=\partial W$. I'm interested in nontrivial ...
3
votes
1answer
46 views

Showing that gluing two knot exteriors together contains subgroups isomorphic with the knot groups.

I'm working through Rolfsen's "Knots and Links" and section 9D exercise 10 has me stumped: Let $K_1$ and $K_2$ be knots in two separate copies of $S^3$ with respective meridians $m_1$ and $m_2$ and ...
5
votes
1answer
77 views

Can a trefoil knot be stretched to look like a triangle with three knots at the vertices?

Can a trefoil knot be stretched to look like a triangle with three knots at the vertices, like in the right side of the image below, or is that transformation impossible to happen? If possible, what ...
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vote
0answers
16 views

About Lefschetz fibration signature

Does there exists a Lefschetz fibration over $S^2$ for any given number admitting it as a signature? I think it is not possible so I need an counter example.
6
votes
1answer
101 views

Homotopic equivalence for $S^5$ without three $S^1$

Consider a standard embedding of $S^5$ in $\mathbb R^6$: $S^5: \; x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 = 1.$ And consider three circles, which are sections of $S^5$ by $x_1 x_2$, $x_3 x_4$, ...
2
votes
0answers
89 views

Surgery to unlink $S^1$ and $S^2$ in $S^4$ [closed]

Let us start with a $S^1$ and a $S^2$ are linked in $S^4$. Can I unlink the $S^1$ and $S^2$ by doing some surgery (with certain constraints described below, and let us say both $S^1$ and $S^2$ ...
3
votes
1answer
80 views

Examples of connections between bounded cohomology and geometric properties of groups

I think that the question is self explanatory. The only example that comes to my mind is the characterization of hyperbolic groups given by Mineyev ("Straightening and bounded cohomology"). There is ...
3
votes
1answer
34 views

Embeddability of connected sum of non-embeddable surfaces

Let $X$ be a surface which can not be embedded into $\Bbb R^n$. Let $X \# X $ denotes the connected sum of two copies of $X$. Then is it true that the connected sum $ X \# X $ is also not embeddable ...
2
votes
1answer
47 views

A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph

The stable manifold theorem tell us: A local stable manifold $W^{s}_{loc}(x^{*})$, which is a differentiable manifold of class $\mathcal{C}^{r}$ and dimension $n_{-}, $ tangent to the ...
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0answers
30 views

Linking of $S^p$ and $S^q$ in the $\mathbb{R}^d$ space

Can we have a nontrivial linking of a $S^p$ sphere and a $S^q$ sphere in the $\mathbb{R}^d$ space (or in the ${S}^d$ space)? I suppose that it can happen only if $p+q<d$. For example, we can have: ...
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0answers
12 views

Homology group and homotopy group of the standard twin

Given a 4-sphere, if we cut out a solid 3-torus $B^2 \times S^1 \times S^1$ from a 4-sphere $S^4$ (with an unknotted torus), the remained exterior is called "the standard twin," say $M$. What are ...
2
votes
0answers
58 views

Embed $S^{p} \times S^q$ in $S^d$?

Can we embed $S^{p} \times S^q$ in $S^d$ with all the nice properties, what are the allowed values of $p$ and $q$ for $d=2,3,4$ where $p+q \leq d$? =For $d=2$= I suppose that we cannot embed $S^{1} ...
1
vote
1answer
34 views

1-surgery on the figure-eight knot: reference request

As far as I know, 1-surgery on the figure-eight knot gives ($\pm$) the Brieskorn sphere $\Sigma(2,3,7)$. However, is there a citeable source for this? Sometimes Thurston's notes are mentioned, but I ...
4
votes
3answers
214 views

Question about closed sets

Let $A$ and $B$ be subsets of $\mathbb{R}^n$ (where $\mathbb{R}^n$ is Euclidean n-space). Define $A + B = \{ x + y : x \in A , y \in B \}.$ Now If $A$ and $B$ are closed sets, is $A+B$ also a closed ...
0
votes
2answers
54 views

Orbits of mapping class group on four-punctured sphere

Let $\mathcal{M}_{0,4}$ be the mapping class group of the four-punctured sphere $S_{0,4}$. Denote the simple closed curve around boundary components $b_1$ and $b_2$ by $x$, the one around $b_2$ and ...
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0answers
44 views

Decomposition of ball in Banach Tarski paradox and covering a soccer ball

Banach Tarski paradox says that it's possible to decompose a ball in $R^3$ into a finite number of disjoint subsets, which can be then reassembled into 2 identical copies of the original ball. ...
8
votes
1answer
64 views

Parallelizing lines

Let $n \geq 1$ be an integer, and $L_1,\ldots,L_n$ be $n$ lines in $\mathbb{R}^3$ which are pairwise disjoint. Is it possible to move all $n$ lines continuously so that they never cross, and so as to ...
2
votes
1answer
43 views

Are knot complements prime 3-manifolds?

Well, that is basically the question. Is the complement of a knot, i.e. an smoothly embedded copy of $S^1$ in $S^3$ a prime $3$-manifold? Here I mean by prime: A connected $3$-manifold $M$ is prime ...
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0answers
41 views

Determining slice knots

Lately I have been thinking about slice knots. Is there any known effective procedure for determining whether a knot is a slice knot?
3
votes
1answer
99 views

Stiefel-Whitney class of complex projective spaces

Let $T\mathbb{C}P^m$ be the tangent bundle of complex projective space. What is the total Stiefel-Whitney class $w(T\mathbb{C}P^m)$? Let $a_m$ be the maximal integer such that the $a_m$-th dual ...
2
votes
0answers
65 views

A homotopy equivalence between two sets

I was trying to prove that the set consisting of the union of the circles $\{\langle x,y\rangle\mid(x-10)^2 +y^2 = 1\}$, $\{\langle x,y\rangle\mid(x+10)^2 +y^2 = 1\}$ and line segment $\{\langle x, ...
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0answers
34 views

Jordan curve interior and curvilinear coordinates

One of the popular ways of evaluating areas of some plane subsets is change of variables. One classic example is the lemniscate of Bernoulli $\gamma: (x^2+y^2)^2=x^2-y^2, x>0$. Using polar ...
14
votes
1answer
159 views

How equivalent are the homeomorphism and diffeomorphism groups of 3-manifolds?

Every topological 3-manifold carries a smooth structure, unique up to diffeomorphism. In addition, "up to homotopy", the maps between them are the same: every continuous map is homotopic to a smooth ...
13
votes
1answer
309 views

Maps from $D^n$ to $D^n$ with a single inverse set are open.

Let $D^n$ denote the closed unit ball in $\Bbb R^n$. In multiple sources proving Brown's generalized Schoenflies theorem (including a version in the original paper), the following consequence of ...