Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.
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88 views
suggestion for video lectures on algebraic topology
can anyone suggest me any good video lecture series forr algebraic topology other than N.J.wildberger video.If it is equivalent to munkres topology(algebraic topology section)
it should be great.
...
2
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54 views
The fundamental group of an open set in $\mathbb{R} ^n$ does not have nilpotent elements.
I am studying a little of basic algebraic topology and I thought that this statement could be true. If you have an open connected set $U \subset \mathbb{R}^m$ and a loop $\gamma$ that is not ...
2
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38 views
fundamental group of complement in the thickened torus
What is the fundamental group of $(T \times I ) \backslash J$ where $J$ is a closed loop going around the vertical $\mathbb{S}^{1}$ of the thickened torus twice.
2
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54 views
Is this map the Gauss map?
Let $M\subset \mathbb{R}^n$ be a closed compact smooth oriantable manifold. Let $E\to M$ be the normal bundle of the embedding and $N\subset \mathbb{R}^n$ a (closed) tubular neighbourhood. $N$ is an ...
2
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0answers
41 views
Global sections of covering spaces
Let $p:C\to X$ be a covering space having a global section $s:X\to C$. I can show that this implies that $s(X)$ is disconnected from the rest of $C$.
Is there any reference where this is explicitly ...
2
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68 views
Question On Cech Cohomology
In section 16 of the topology book of Bott and Tu, there is a path fibration $\Omega S^2 \to PS^2 \to S^2$. The $E_2$ page of the spectral sequence of this fibration is $$E_2^{p,q}=H^p(S^2,H^q(\Omega ...
2
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92 views
Extending a continuous function on the unit sphere to the unit ball
I'm trying to solve a problem in Lee's topology book. In this book, a closed n-cell $D$ is any space homeomorphic to the closed unit ball. Then $\mathrm{Int} D$ , resp. $\partial D$ are the images of ...
2
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81 views
Torsion-free fundamental group.
Is there a name for spaces whose fundamental group has no torsion? And what, if any, are some nice properties of these spaces?
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36 views
spectral sequence computing invariants
let $G$ be an abelian group (can also assume $G=\mathbb{Z}^n$ for some positive integer $n$). Let $X\stackrel{g}{\rightarrow} Y$ be a $G$-covering where $X,Y$ are schemes, or topological spaces or ...
2
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51 views
To what extent is the global angular form well-defined?
I am reading Differential Forms in Algebraic Topology by Bott & Tu. The authors constructed the global angular form $\psi$ for an oriented $k$-sphere bundle $E$ over a smooth manifold $M$. It has ...
2
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57 views
4D TQFT construction from a modular tensor category
I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category.
I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
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46 views
How to prove that a lie group is simply connected
I need to prove that $O(3,19)/SO(2)\times O(1,19)$ is simply connected. In particular $O(n_{+},n_{-})$ denotes the orthogonal group of $\mathbb{R}^{n_{+}+n_{-}}$ endowed with the diagonal quadratic ...
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55 views
Relationship between the Hopf Fibration and Spinors on $S^2$
The unique spin structre for $TS^2$ is given by the Hopf fibration. We can trivialize the Hopf fibration over open sets $U_1 = S^2 \setminus \{N\}, U_2 = S^2 \setminus \{S\}$ where $N$ and $S$ are the ...
2
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87 views
Properly discontinuous action on a non-locally compact space
Let me begin with some definitions in order to avoid confusion.
An action of a group $G$ on a space $X$ is proper if the map $G \times X \to X \times X$ given by $(g, x) \mapsto (x, gx)$ is proper, ...
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56 views
Products of CW-complexes
I am currently reading through May's "Algebraic Topology" and in the chapter on CW-complexes he shows that a product of CW-complexes is again a CW-complex, because one can define product cells using ...
2
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27 views
How can be computed $H_n(T_i \times T_j)$?
Here $T_g$ denotes the g-fold torus, and $T_i\times T_j$ denotes the cartesian product. My major trouble is that I can't use Kunneth's Theorem. Thanks for helping!
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67 views
Simply connected covering
Question: Construct a simply connected covering which a subspace of $\mathbb R^3$ of union of a sphere and a circle intersecting in two points.
My idea: First of all note that union of a sphere and a ...
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43 views
extending trivializations of plane bundles via obstruction theory
I came across the following statement: A trivialized $k$-plane subbundle of a trivialized $(n+k)$-plane bundle determines a trivialization of the orthogonal $n$-plane bundle over the $(n-1)$-skeleton ...
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45 views
About Thom theorem (representation submanifold for $H_{n-2}(M)$)
Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold.
And in the Harper and ...
2
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71 views
Killing successive homotopy groups via fibrations
Let $X$ be some sort of sufficiently nice space, e.g. a (connected) cell complex. Then $X$ has a universal cover $\tilde{X}$. This is simply connected by definition and it is easy to show that ...
2
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123 views
Top deRham cohomology group of a compact orientable manifold is 1-dimensional
Let $M$ be a compact smooth orientable manifold of dimension $n$. I am looking for a simple proof that $H_{dR}^n(M) \cong \mathbb R$. Equivalently, an $n$-form which integrates to 0 is exact. I can ...
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125 views
Fundamental Group!
Say you have two surfaces of genus 2, say $X$ and $Y$ and you want to attach them via homotopy attaching maps $f$ along their waist curves. Then what will the fundamental group of the adjunction space ...
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67 views
Multiplicativity of the Euler characteristic
One can find all over the internet that it is well-known (and obvious) that given a fiber bundle $F \to E \to B$, the equality $\chi(E) = \chi(F)\chi(B)$ holds ($\chi$ is the Euler characteristic). ...
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60 views
Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius
I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
2
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42 views
Unramified functions between Riemann surfaces
Let $F:X\rightarrow Y$ a unramified holomorphic function between two compact Riemann surfaces. I don't understand why $F$ is a covering map. By a well-known theorem $F$ is surjective; then since the ...
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93 views
Identifying the numbers of degree $n$ covering spaces of $X$
Let $X$ be a path-connected, locally path-connected and semilocally simply-connected space. Can we find a correspondence between degree $n$ covering spaces of $X$ and group homomorphism ...
2
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57 views
Fundamental Groups of Complements of Knots Algorithm
Is there any clear algorithm to compute the fundamental group of complements of knots?
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55 views
Chain map variant
A map $\phi_*:C_*\rightarrow D_*$ of chain complexes is a chain map if it intertwines the boundary operator, i.e. $\phi\partial=\partial \phi$. It is well known that such a map descents to homology, ...
2
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62 views
Massey products in the Adams Spectral Sequence
I've never quite 'got' Massey products - this question, I guess, is to work out a small example that might shed some light for me.
So following Wikipedia, let $\Gamma$ be a differential graded ...
2
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120 views
Importance of well-pointedness (in particular for the pointed mapping cylinder construction)
In a recent question, I asked wether well-pointedness was indeed necessary to make the canonical inclusion
$$X\hookrightarrow M_f^{\bullet}$$
a pointed cofibration. Here $X,Y$ are pointed topological ...
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140 views
Definition of a topological module
A topological universal algebra of type $\Omega$ is a universal algebra $A$ of type $\Omega$ that is also a topological space, such that for any $n\!\in\!\mathbb{N}$ and any operation ...
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75 views
Excision and induced fibration
I've been stuck in the following small detail which is part of the calculation of the $E^2$ term of the Serre Spectral Sequence.
Let $p: E\to B$ be a fibration where $B$ is a CW-complex. Denote by ...
2
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265 views
Homology and cohomology: why does Poincaré duality fail for domains with boundary?
Poincaré duality says that for a compact, orientable manifold without boundary the $k$th and $(n-k)$th homology groups are isomorphic.
For domains with boundary, it's easy to construct examples where ...
2
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67 views
Monodromy groups and the choice of a base point
For a path connected topological space $X$, the fundamental groups at two different base points are isomorphic.
Does the same hold for the monodromy group of a covering map $Y \rightarrow X$, when ...
2
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108 views
Monodromy Theorem and Homotopy Lifting Theorem
I've just come across this proof of the following theorem that I can't convince myself is true. Any ideas whether it's correct?
Suppose $\gamma$ and $\lambda$ are homotopic paths starting at $x$ in a ...
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135 views
How to prove the commutative diagram
Let $p_2:\mathbb S^1 \to \mathbb S^1$ be the two-sheeted covering map $p(z)=z^2$.If $f$ is odd($f(-z)=-f(z)$),show that there exists a continuous map $g:\mathbb S^1 \to \mathbb S^1$ such that $\deg ...
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82 views
Spectral Sequence and Stiefel Manifold
Let $Spin(3)$ be embedded in $Spin(5)$ by the spin embedding then we have a fibration:
$$Spin(3) \rightarrow Spin(5) \rightarrow Spin(5)/Spin(3)$$
Where $Spin(5)/Spin(3)$ is $V_{5,2}$, the Stiefel ...
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92 views
Question about covering spaces
Suppose $\pi:M_1 \to M_2$ is a $C^\infty$ map of one connected differentiable manifold to another.And suppose for each $p\in M_1$,the differential $\pi_*:T_p M_1 \to T_{\pi(p)}M_2$ is a vector space ...
2
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139 views
$X,Y$ are locally path connected and path connected, with universal covers $\tilde{X}, \tilde{Y}$. If $X \simeq Y$ then $\tilde{X} \simeq \tilde{Y}$
Suppose $X,Y$ are locally path connected and path connected, with universal covers $\tilde{X}, \tilde{Y}$. I'd like to prove that if $X \simeq Y$ then $\tilde{X} \simeq \tilde{Y}$.
I've had the ...
2
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0answers
107 views
when is the kth homology group of a space isomorphic to its kth homotopy group?
I'm just thinking about the relationship between homology and homotopy groups of a space. I know that homology is basically an abelianization of the fundamental group (please correct me if I'm wrong). ...
2
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466 views
Torus as double cover of the Klein bottle
Reading through some lecture notes and it says
The torus $T^2$ is the orientation double cover of the Klein bottle $K$, via the covering projection
$p:T^2\to K; [x,y]\mapsto [x,2y]$
Could someone ...
2
votes
0answers
73 views
Image of Thom Class under Sequence of Maps?
So I've been trying to do problems in Milnor & Stasheff's Characteristic Classes as a quick review, not having done anything with them in a while. However, I'm stuck on some parts in attempting ...
2
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0answers
70 views
Cohomology of Grassman manifolds
I am considering the restriction homomorphism
$H^p(G_n(\mathbb{R}^\infty)) \leftarrow H^p(G_n(\mathbb{R}^{n+k}))$, where the $G_n(-)$ are the relevant Grassman manifolds. Does anyone know of ...
2
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181 views
understanding the torus described as a product of two circles
I would like to ask two (related) things about the torus:
The torus can be described as the cartesian product $S^1 \times S^1$ of two circles in $\mathbb{R}^3$. Then one can talk about meridional ...
2
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0answers
163 views
An Exercise in Allen Hatcher's book on Spectral Sequences
anyone knows how to solve Exercise 3 of Chapter 1 of Allen Hatcher's book on Spectral Sequences? The question is as follows:
For a fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$ associated to ...
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0answers
217 views
Topology of wedge products
I have a question about the quotient topology induced on the wedge sum $S^{\,2} \vee S^1$, (where $S^n$ denotes the unit sphere in $\mathbb{R}^n$). In this topological space, the subsets $S^1$ and ...
2
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79 views
A Heegaard splitting of $S^2\times S^1 \# S^2\times S^1$.
For a Heegaard splitting of $S^2 \times S^1$, we can take two copies of genus 1 handlebodies and glue boundaries with the identity map.
I want to generalize this a little bit.
In the case of ...
2
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0answers
76 views
left inverse to trivial fibration is trivial cofibration
It is claimed that in the model category of simplicial sets (with usual model structure), a trivial fibration $X \to Y$ has a section, which is a trivial cofibration.
Now, I see that there is a ...
2
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0answers
123 views
On a canonical homomorphism in cohomology
When $K$ is a simplicial complex, the dual complex $C^*(K)$ to the chain complex $C_*(K)$ has a concrete interpretation: an element in $C^n(K)$ is given by assigning an integer to every oriented ...
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55 views
Is there are a name for a simplicial complex that is homotopic to the clique complex of its 1-skeleton?
A hollow octahedron is a nice triangulation of the sphere, because once you know the edges, you know everything. The vertices are obviously the ends of the edges, and the faces are any collection of ...
