Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

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3
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46 views

Serre spectral sequence and locally constant coefficients

I have a brief question - In the Serre Spectral sequence for a fibration $$F \rightarrow E \rightarrow B$$ one can require, to avoid using local system of coefficients, that the action $\pi_1(B)$ on ...
3
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183 views

Galois correspondence of covering spaces of spaces not necessarily semilocally simply-connected

I've been trying to solve the following exercise (1.3.24) from Hatcher's Algebraic Topology: Given a covering space action of a group $G$ on a path-connected, locally path-connected space $X$, ...
3
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212 views

Want to show two maps are homotopic

I am trying to solve the following problem but so far I cannot do it. Let $X$ be a connected CW-space such that its homotopy group is 0 except for the fundamental group. Let $M$ be a closed manifold ...
3
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28 views

Relative Hopf degree theorem

If $f,g$ are two maps from $(D^n,S^{n-1})$ to $(D^n,S^{n-1})$ such that they have the same degree, that is $f_*[\mu]=g_*[\mu]$ where $[\mu]$ is a generator of $H_n(D^n,S^{n-1})$, then can we find a ...
3
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55 views

Understanding J homomorphism

I was trying to understand J homomorphism $J:\pi_r(SO(q)\rightarrow \pi_{r+q}(S^q)$from the Wikipedia page http://en.wikipedia.org/wiki/J-homomorphism. It's clear that an element of $\pi_r(SO(q)$ ...
3
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76 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
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80 views

Euler characteristic of affine space

Sorry for the trivial question.. but what is the (topological) Euler characteristic of $\mathbb{A}^n$? Also, is there a genus-degree formula for affine curves similar to $g={d-1\choose 2}$ for smooth ...
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63 views

What is a $c_1$-map for Riemann-Roch theorems?

Atiyah and Hirzebruch define a $c_1$-map to spell out Riemann-Roch theorem for (compact and connected) smooth manifolds. The definition is following: a map $f:Y \to X$ is called a $c_1$-map if we are ...
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86 views

What big families of theories/structures are there?

To gain a big picture of (pre-categorical) mathematics, is it correct to divide mathematical theories resp. structures in two big families? universal algebra: classes of objects with arbitrary ...
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97 views

Homotopy of a CW complex

I have a CW complex constructed as follows: (The circle and the rectangles are 2-cells, different 1-cells are denoted by different colors, and there is one 0-cell). We can see it as gluing two Klein ...
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66 views

Proof that continuous $f: S^1 \rightarrow S^1$ satisfying $f(x) = -f(-x)$ represents an injection on homology

I'm looking to prove that continuous functions $f: S^1 \rightarrow S^1$ satisfying $f(x) = -f(-x)$ for all $x \in S^1$ represent injections on homology. I'm trying to prove this fact on the way to ...
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111 views

Simplicial homology for n-simplex

I've just started to study homology theory. And I'm trying to calculate all $H_n(\Delta_N)$ for some $N$. I know that the number of $m$-simplex in $N$-simlex is $b_{N,m}={N+1 \choose ...
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63 views

Does the boundary of a handle decomposition obtain a handle decomposition?

Let $M$ be the $4$-manifold $D^4\cup2\text{-handle}\cup\ldots\cup2\text{-handle}$, where the attachment of the handles is specfied by an oriented framed link $L=L_1\cup\ldots\cup L_n\subseteq S^3$. By ...
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80 views

Subset of $\mathbb{R}^3$ with an element of finite order in its fundamental group

Is there a subset of $\mathbb{R}^3$ with an element of finite order (not the identity!) in its fundamental group? I think the real projective plane is such a subset as its fundamental group is ...
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69 views

Understanding the proof of the Seifert-van Kampen theorem

Here is the bit of the proof I didn't really understand: Let $X=X_1\cup X_2$, where $X_1$, $X_2$, $X_1\cap X_2$ and $X$ are path-connected and $X_1, X_2$ are open subsets of $X$. Moreover, assume ...
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87 views

Almost complex structures on a manifold and its exotic copy

Consider a compact simply-connected smooth $4$-manifold $M$ and its exotic copy $M'$. Identify the underlying topological manifolds. Dold-Whitney theorem guaranties ($\mathbb R$-linear) isomorphism of ...
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47 views

Topology of a 3D wired Mandala?

There is a so called 3D-wired Mandala, based upon $2$ large circles each flowered symmetrically on its circumference by two sets of each $8$ half-circles. The circles are interconnected together by ...
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34 views

How to give the coproduct of differential graded algebras explicitly?

Let $X$ and $Y$ be based spaces such that their respective loop spaces $\Omega X$ and $\Omega Y$ are connected. In the first paragraph of this article by Dula and Katz, it is given that ...
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94 views

The monodromy and cut planes.

I am trying to understand the following example from the book Riemann surface by Donaldson (p.48). This was given after introducing the notion of monodromy of the covering. Consider the Riemann ...
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104 views

simple proposition about homotopy group

Let $A$ be a topological subspace of $X$. Then we have exact sequences of homotopy groups for pair spaces: $$ \ldots \rightarrow \pi_n(A,x_0) \xrightarrow{i_{*}} \pi_n(X,x_0) \xrightarrow{j_{*}} ...
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63 views

Homotopy operator for the Gysin sequnce

Let's consider some sphere bundle $π:E ↦ M$ with fiber $S^{r}$. What is homotopy operators in case of the Gysin sequence? $$ \ldots \rightarrow H_{dR}^p(B) \xrightarrow{\wedge e} H_{dR}^{p+r+1}(B) ...
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81 views

deck transformations and covering spaces

Let $p:\tilde X\rightarrow X$ be a universal covering space, and let $H\leq G$ where $G$ is the group of covering transformations. Let $q:\tilde X \rightarrow \tilde X/G$ be the quotient map which is ...
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124 views

Leray-Hirsch theorem

I'am studying the book "Bott, Tu Differential forms in algebraic topology." I don't understand the proof of Leray-Hirsch theorem via Cech-de Rham complex. Lets consider some bundle $\pi: E \mapsto ...
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45 views

eilenberg-steenrod for pairs in any model category?

The Eilenberg Steenrod axioms are functors on the homotopy category of pairs of "spaces" $(X, A)$. Typically they are introduced when $X$ and $A$ are some sort of topological spaces. My question is ...
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76 views

About Linking Number

I'm looking for some references about Linking Number. I already know these http://mathworld.wolfram.com/LinkingNumber.html http://www.matapp.unimib.it/~ricca/publications/2011/JKTR11.pdf Anything ...
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79 views

Applications of a theorem of Cartier and Gabriel

In a representation theory course I took we stated and proved the following Theorem due to Cartier and Gabriel: Theorem: Suppose $H$ is a cocommutative Hopf algebra over a field $k$ such that $ ...
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75 views

Is a complex polynomial a regular covering? What is its group of deck tranformations?

We know that a complex polynomial $P$ of degree $n$ is an $n$-sheeted covering from $$\{\mathbb{C} - P^{-1}\{\text{critical values of }P\}\} \to \{\mathbb{C} - \{\text{critical values of }P\}\}. $$ ...
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72 views

Help calculating the second homology group of $\mathbb{R}P^2 \times S^1$

I need help calculating the second homology group of $\mathbb{R}P^2 \times S^1$. I found all the other homology groups using the Mayer-Vietoris sequence. Any suggestions? I can't use Kunneth.
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50 views

PBW theorem for restricted Lie algebras

I'm looking at the proof of the PBW theorem for restricted Lie algebras to be found in Ponto and May's "More Concise Algebraic Topology", page 361 (367 in linked file). I either see an error in their ...
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95 views

Linking number of curves in SO(3)

Suppose you have two closed curves in $\mathbb{R}^3$, and allow them to continuously deform and possibly pass through themselves, but not each other. The linking number is an invariant of such ...
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73 views

How should I imagine the cup product on a topological space/manifold/variety

Let $X$ be a "space" with a vector bundle $E$. Let $n$ be the dimension of $X$. Then there is a map $$H^1(X,E)\otimes \ldots\otimes H^1(X,E)\to H^n(X,\Lambda^n E).$$ This map is given by taken the ...
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52 views

Removing the star without changing homology

I know that if the link of a simplex $\sigma$ in a finite simplicial complex $K$ is contractible then the two complexes $K$ and $K\setminus \text{Star}(\sigma)$ share the same homotopy type. ...
3
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58 views

What is the geometric meaning of powers of the first Stiefel-Whitney class?

If $M$ is an $n$-dimensional manifold, I know $\\w_1^n(M)$ is a Stiefel-Whitney number of the manifold, but what does its vanishing geometrically mean? More generally, does the Stiefel-Whitney height ...
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174 views

Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
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110 views

Number of composition cycles on honeycomb graph embedded on torus

I am reading a lecture about dimers by R.Kenyon (http://arxiv.org/pdf/0910.3129v1.pdf). I have a question concerned to the honeycomb graph $H_n$ embedded on a torus (see the picture at page 24 of the ...
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53 views

Reference for couple facts in algebraic topology (tautological line bundle, principle G bundle)

I have seen the following couple of (basic and possible obvious) facts written when I look up sources on the internet, but I'm not so sure why they are true or where to find them. 1) We mentioned ...
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98 views

Principal connection & curvature

Let $(P, \pi, B)$ be a principal $G$-bundle over $B$ and $\omega$ a principal connection. Then the curvature is defined as $$ \Omega_\omega = d \omega + \frac{1}{2} \omega \wedge \omega$$ With the d ...
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40 views

some help on the group of unknotted

Show that the group of the unknotted $K=\{(z_z,z_2)\in \mathbb{S^3} : |z_1|=1 \}$ is infinite cyclic. where $\mathbb{S^3}$ is to be considering as the unit vectors in $\mathbb{C^2}\cong \mathbb{R^4}$. ...
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56 views

Basic computation of a double graded spectral sequence: $^I E^0_{pq}$

Let $C=\oplus_{p\geq0, q\geq0}C_{pq}$ be a double graded group with two differentials: $d^I_{pq}:C_{pq}\rightarrow C_{p-1,q}$ and $d^{II}_{pq}:C_{pq}\rightarrow C_{p,q-1}$, with the usual assumption ...
3
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88 views

Relation between complex and real sphere

I want to understand relation between complex and real spheres. How to show? $S^1(\mathbb{C}) \approx \mathbb{R} \times S^1$ $S^3(\mathbb{C}) \approx \mathbb{R} \times S^3$ $\approx$ means homotopy ...
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100 views

Topology Qualifying Exam Problem 42

I was going through some old qualifying exam problems and I have been struggling with this one. Any help would be great, thanks. Consider the 2-dimensional torus $\mathbb{T}^2$ and the topological ...
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84 views

cellular chain complex of sphere

The cellular chain complex $C_{\ast}(X)$ of an $n$-sphere $X=S^{n}$ (with any CW-complex structure), gives rise to an exact sequence $$ 0 \rightarrow \mathbb{Z} \rightarrow C_{n}(X) \rightarrow ...
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159 views

Mackey functor structure on equivariant homotopy groups

I have read that the equivariant stable homotopy groups $\pi_n^{-}(X)=\pi_n(X^{-}) $ of a $G$-space or $G$-spectrum $X$ have a Mackey functor structure. Can somebody please explain how the covariant ...
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97 views

Groups not arising from certain centralizers

There's a lot of fuss in certain subfields of algebraic topology about giving fancy interpretations to the rings coming from the cohomology of groups, where "cohomology" is allowed to be taken to be ...
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58 views

If $S_1$ is orientable and $S_2$ it isn't,

Let $S_1$ and $S_2$ be two closed surfaces. Demonstrate that the following conditions are necessary for there to be a $k$-sheeted covering $p: S_1 \rightarrow S_2 $. a) $\chi(S_1)=k \chi (S_2)$. b) ...
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132 views

fundamental theorem of algebra proof

I have seen some proofs of the fundamental theorem of algebra using algebraic topology. But I have seen an exercise to prove the theorem by defining this map $f_t: S^1\longrightarrow S^1$ by $f_t(z) ...
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95 views

Extension of Brouwer's degree to continuous functions.

I am studying the first chapter of this book: Topological Degree Theory and Applications At page 13 of this document, Definition 1.2.5, it is essentially said that to define the degree of a ...
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84 views

k3 surface as ramified double cover of $\mathbb{P}^2$

I read that one example of k3 surface is a double cover of $\mathbb{P}^2\mathbb{C}$ ramified over a sextic. My question is why a sextic? i believe that the sextic is isomorphic to the ramification ...
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140 views

Cech Cohomology Of Pullback Linebundle

my question is as follows. Let $\chi$ a compact Calabi-Yau 3-fold and $A,B \subset \chi$ two 2-complex dimensional manifolds such that their intersection $C := A \cap B$ is a 1-complex dimensional ...
3
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43 views

isomorphisms of $\pi_{1}(T^2, x_{0})$ with itself.

I1m studying fundamental group and its relation with covering maps, I was thinking about an exercise: every isomorphism of $\pi_{1}(T^2, x_{0})$ with itself is induced by a homomorphism $f:T^2 ...