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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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
64 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 ...
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1answer
59 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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1answer
46 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 ...
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1answer
30 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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52 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 ...
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1answer
52 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 ...
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1answer
41 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 ...
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1answer
39 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 ...
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47 views

$\mathbb{T}^2 \not \cong \mathbb{R}P^2$ without homologies or fundamental group

I want to prove that $\mathbb{T}^2 \not \cong \mathbb{R}P^2$(not homeomorphic) using only elementary methods(and maybe fact that $K_5$ not planar).
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1answer
413 views

Can one cancel $\mathbb R$ in a bi-Lipschitz embedding?

Let $X$ be a metric space. Suppose that the product $X\times\mathbb R$ admits a bi-Lipschitz embedding into $\mathbb R^{n}$. Does it follow that $X$ admits a bi-Lipschitz embedding into $\mathbb ...
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1answer
74 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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83 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 $S^2$ ...
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13 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.
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1answer
100 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 $x_1 x_2$, $x_3 x_4$, ...
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1answer
70 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 ...
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1answer
139 views

4-manifold: $0$-handle $\cup$ $2$-handles along a framed link in $S^3$ (intersection form = linking matrix = presentation matrix of $H_1(\partial M)$)

Let $L$ be a framed link in $S^3$, consisting of framed knots $L_1,\ldots,L_m$. Let $A=[a_{ij}]\in\mathbb{Z}^{m\times m}$ be its linking matrix, with $a_{ii}=$ framing of $L_i$ and $a_{ij}=$ linking ...
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1answer
32 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 ...
3
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0answers
63 views

John Hempel's proof of residual finiteness of surface groups

John Hempel proved that fundamental groups of $2$-manifolds are residually finite. I want to understand this proof so I have some questions: Why if we have $S(f)=\emptyset$ then $f$ represents ...
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1answer
44 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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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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11 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 ...
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56 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} ...
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1answer
306 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 ...
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1answer
33 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 ...
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2answers
47 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 ...
4
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3answers
210 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 ...
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1answer
130 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 ...
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39 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. ...
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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 ...
3
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1answer
164 views

Covering spaces!

If one wanted to find all connected covering spaces of a product of two spaces, say $S^1\times RP(3)$, how would you go about it? I'm thinking finding the fundamental group of $S^1\times RP(3)$, and ...
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1answer
41 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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2answers
111 views

A complex line bundle is trivial if and only if the first Chern class is zero

Let $\xi$ be a complex line bundle over a CW-complex $B$. I want to prove that $\xi$ is trivial if and only if $c_1(\xi)=0$. My attempt: Suppose $c_1(\xi)=0$. Then the Euler class $e(\xi)=0$. Since ...
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35 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?
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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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1answer
96 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 ...
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31 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 ...
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0answers
41 views

Side identification on a hexagon

Apparently giving a hexagon side identification aabbcc results in a sphere. I'm struggling to see this, can someone explain? perhaps with a diagram? It seems to be all the vertices are identified, but ...
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0answers
28 views

Euler Characteristic of Side identification

Let $S$ be a surface obtained by identifying the sides of a regular hexagon in pairs. I want to show $\chi(S) > -1 $. I can see how we can obtain surfaces with $\chi(S) = 0,1,2$ but I think I'm ...
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1answer
39 views

Is the closure of $[\sigma_1^2,\sigma_2^2]$ in $B_3$ equal to the Borromean rings?

Is the closure of $[\sigma_1^2,\sigma_2^2]\in B_3$ (the braid group with $3$ strings) equal to the Borromean rings? If yes, is there any simple proof?
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1answer
35 views

Description of real projective space $P^3$

I know that the real projective plane $P^2$ can be thought of as a union of a mobius band and a disk, where the union occurs among the common boundary of the two (circle). My question is about $P^3$. ...
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1answer
107 views

Torsion Subgroup of Mapping class group.

What is the cardinality of finite order elements in Mapping class group of a surface $S_{g,n}$ of genus g and n boundary components. 1) If it is infinite then how can I generate a collection of ...
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4answers
617 views

Embedding compact (boundaryless?) n-manifolds in n-dimensional real space

I know the embedding theorems that allow you to embed $n$-manifolds into $\mathbb{R}^k$, provided $k$ is chosen large enough. Here I'm interested in the possibility of taking $k=n$ in the case of ...
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2answers
65 views

A reference for a result by A. Casson

I was reading this article about the disproof of Triangulation conjecture: it says that A. Casson disproved this conjecture in dimension 4 in the '80s In 1982, Michael Freedman, then at the ...
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1answer
51 views

Several questions concerning Alexander's Theorem

I'm reading Hatcher's proof of Alexander's theorem in his 3 manifolds notes. The statement is the following: Let $\Sigma \subset \mathbb{R}^3$ be an embedded $2$-sphere; then $\Sigma$ bounds a ...
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31 views

Zeros of vector field extended from field lines

Suppose I have a finite number of 1-dimensional curves embedded in a 3-dimensional space. What can I say topologically about vector fields chosen so that these closed curves are field lines? For ...
3
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1answer
27 views

Framing of embedding induces an isotopy of embeddings

Let $M$ be a smooth manifold of dimension $m$ and $\phi : S \to M$ a smooth embedding (dim S = k < m) such that the normal bundle $T_S M$ is trivializable. Let $f: T_S M \to S \times ...
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1answer
113 views

About generalized Schoenflies problem in the smooth category

Reading some books and comparing with Wikipedia I found some different statements about how the smooth Schoenflies problem is solved in high dimension, and I wanted to know which one is the correct ...
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70 views

Homotopically equivalent manifolds and product with $\mathbb{R}$

I know that some manifolds which are homotopically equivalent become homeomorphic after taking the product with $\mathbb{R}$, e.g. $\mathbb{T}^{2}$ minus a point and $\mathbb{S}^{2}$ minus three ...
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1answer
28 views

punctured Mobius band in high dimension

Let $M^{n+1}$ be the non-trivial line bundle over sphere $S^n$. When $n=1$, $M^2$ is Mobius band and the punctured space $M^2\setminus *\simeq S^1$. How about $M^{n+1}\setminus *$ in general? Does ...