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

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Cohomology exact sequence induced from exact sequence of coefficient

This might be a trivial question, but I was wondering if the following was true or not. Suppose that we have an exact sequence of groups $$0\rightarrow G\rightarrow H\rightarrow K\rightarrow 0$$ Does ...
4
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1answer
62 views

What would an infinite dimensional projective space look like as a scheme?

In topology, we can construct $\mathbb{CP}^\infty$ as the direct limit of $\cdots\rightarrow \mathbb{CP}^n \rightarrow \mathbb{CP}^{n+1}\rightarrow \cdots$ with the embedding given by $[x_0: x_1: x_2: ...
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0answers
35 views

Questions about complexes and homology

I just learn about the simplicial and delta complexes and computing homology group. But I have a few questions: Is there any topological space which cannot be given a delta compplex structure? Is ...
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2answers
79 views

de Rham cohomology on finitely smooth manifolds

In all of the places I've looked, de Rham cohomology is defined on $\mathcal{C}^\infty$ manifolds with $\mathcal{C}^\infty$ differential forms. What about de Rham cohomology on $\mathcal{C}^r$ ...
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1answer
27 views

Two different kinds of actions on fibers of a universal cover

This concerns Hatcher's exercise 1.3.26. It says almost this: Given a universal cover there are two actions of $\pi_1$ on the fibers, one given by lifting loops and one given by restricting deck ...
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1answer
20 views

Covering Torus by torus

This question concerns example 1.41 in Hatcher's Algebraic topology. There he constructs a covering for the genus 3 surface by the genus 11 surface by shaping it like a star with 5 arms with two holes ...
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1answer
47 views

Homotopic type of $GL^+(n)$, $SL(n)$ and $SO(n)$

Question: Consider $GL^+(n) \supset SL(n) \supset SO(n)$ the groups of matrices $n \times n$ with positive determinant, determinant $1$ and orthogonal with positive determinant, respectively. Show ...
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31 views

Constructing normal covering spaces

This is an exercise from Hatcher 1.3.12. It says given a,b the generators of $\pi_1(S^1\vee S^1)$, draw a picture of the covering space that corresponds to the normal subgroup generated by ...
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150 views

Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$. Let $M$ and $M'$ manifolds of dimensions $m>0$. It is true that if ...
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1answer
31 views

Connected Linear Graph not Path-Connected

Given a set of vertices $\{x_\alpha\}$ whose cardinality exceeds $\aleph_1$, (assume the axiom of choice) connect each vertex with its successor by an edge, forming a linear graph. Choose two vertices ...
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17 views

Every vector bundle can be induced from a principal bundle? its frame bundle?

If it is a theorem could somebody tell me the name? If it is wrong could somebody give a counterexample to illustrate what the obstruction is? I am wondering this because in Clifford Taubes' book on ...
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32 views

Universal Cover of a non-Hausdorff space

It is widely known that if $X$ is a path-connected Hausdorff space, and the universal cover $\widetilde{X}$ is compact, then $\pi_1(X)$ is finite. What happens if we don't assume that $X$ is ...
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35 views

Gluing Balls to get $CP^k$?

As we all know, the $2$-sphere can be obtained by gluing together two discs along their boundary. One way to generalize $S^2 \simeq CP^1$ is a $CP^k$. Does there exists a generalised construction of ...
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2answers
64 views

Homology of the $n$-torus using the Künneth Formula

I'm trying to apply the Künneth Formula $$H_{n}(X \times Y) \simeq \displaystyle \bigoplus_{r+s=n} H_{r}(X) \otimes H_{s}(Y)$$ to compute the homology groups of the $n$-torus. For the double torus, ...
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0answers
23 views

Topology: every continuous function on $\mathbb{R}^2$ scales a point

The question is simple: Suppose $f : \mathbb{R}^ 2 \to \mathbb{R}^ 2$ is continuous. Show that there exist $\lambda > 0$ and $x \in \mathbb{R}^2$ such that $f(x) = \lambda x$. So basically, we ...
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2answers
48 views

Existence of a (n-1)-connected map beween CW-spaces

I have two finite CW-spaces $K$ and $L$ (K is n-dimensional and L is (n-1)-dimensional), a topological space $X$ and two maps $\phi:K\to X$ and $\psi: L\to X$, while $\phi$ is n-connected and $\psi$ ...
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0answers
27 views

Set of equilibrium points generically finite and odd

Let $\Sigma$ denote a finite product of unit simplices and $E:\mathbb{R}^k \rightrightarrows \Sigma$ an upper-hemicontinuous and compact valued correspondence with graph $\Gamma$. By $\pi: \Gamma \to ...
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2answers
34 views

injective open map between two euclidean spaces

Does there exists an injective function from $R^2\ to\ $R such that image of an open set is open ? Where $R^2$ and $R$ are usual euclidean spaces. Please help.Thank you.
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1answer
21 views

Fundamental group of cylinder -triangulation method

Is this correct? Can we conclude that the fundamental group is trivial since there are no remaining generators on 1-simplices?
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24 views

Covariant derivative induced by a connection

I am studying Ralph Cohen's lecture notes on the topology of fiber bundles and encountered the following definition of covariant derivative induced by a connection on principal bundles: the covariant ...
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1answer
16 views

Fundamental group Klein Bottle triangulation

I have been trying to find the FG of the Klein bottle, and I was wondering if someone could verify that this process is correct. After triangulating it, I then found a maximal tree (shown in yellow) ...
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1answer
36 views

Fundamental group of the sphere via triangulation

I know that the fundamental group of the sphere is zero, i.e. $\pi(S^2)=0$ I want to show this by triangulation, i.e: Triangulate the sphere Draw maximal tree Draw maximal contractable subspace ...
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53 views

$U \subset \mathbb{R}^{2}$ be a bounded open set such that $\mathbb{R}^{2} -U$ is not connected. Then $U$ is not contractible .

Let $U \subset \mathbb{R}^{2}$ be a bounded open set such that $\mathbb{R}^{2} -U$ is not connected. Then I want to conclude that $U$ is not contractible (to a point). My attempt We proceed by ...
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1answer
40 views

Fundamental group of a tree?

Find the fundamental group of the space $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$. $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$ where $T$ is a graph $T$ is the graph made of $3$ edges with a ...
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1answer
66 views

Homology group of non orientable manifold

I think that if $M$ is a non-orientable, connected, compact, n-manifold, then $H_n(M,\mathbb{Z}/k)=0$ if $k\neq 2$. My proof is the following: $H_n(M,\mathbb{Z}/k)\neq 0$ then $H_n(M,\mathbb{Z}/k)$ ...
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1answer
48 views

Short Exact Sequence of Vector Bundles

Just wish to clarify, is it true that in order to show some vector bundles (over the same space) fit into a short exact sequence we just need to check that their fibers fit into a short exact sequence ...
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1answer
64 views

Why does this have to be $f(0)=g(0)$?

For the problem, I am not given any solution so no idea Prove that any two continuous maps $f,g; I \to X$ such that $$f(0)=g(0) \in X$$ are homotopic where $I=[0,1]$ is the unit line. ...
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1answer
48 views

What do paths have anything to do with homotopy equivalence?

I don't understand how to solve this problem, it seems disconnected from the definition of homootpy equivalence Let $X,Y$ be spaces with the underyling set $\{a,b\}$ for both but ...
0
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1answer
38 views

Homology group of $CP^n$ and $\mathbb{R}P^n$

I am trying to compute the homology groups for the real and complex projective spaces but without use the cw-complex structure. My idea would be to use the transfer sequence, because we already know ...
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1answer
34 views

Every fiber bundle with Cantor set fiber is the suspension of a homeomorphism of the Cantor set.

I've heard that every fiber bundle (over $\mathbb S^1$?) with Cantor set fiber is the suspension of a homeomorphism of the Cantor set. Can someone explain the intuition behind the fact? Is there a ...
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1answer
19 views

$f:M\to N\times N$ is continuous and $\Delta = \{(y,y):y\in N\}\subset N\times N$ then $f^{-1}(N\times N-\Delta)$ is an union of open balls in $M$

I need to show the following: $f:M\to N\times N$ is continuous and $\Delta = \{(y,y):y\in N\}\subset N\times N$ then $f^{-1}(N\times N-\Delta)$ is an union of open balls in $M$ But I have no idea of ...
3
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2answers
53 views

connected sum of surfaces is well defined proof attempt

Suppose $S_1$ and $S_2$ are compact surfaces (connected 2-dimensional manifolds). If we cut out of them two closed disks, and glue the surfaces along disk boundaries we get new surface, their ...
2
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1answer
24 views

Contradictory; Homotopy equivalence and deformation retract problem

The question Show that $S^1$ is a deformation retract og $D^2\setminus\{(0,0)\}$ the unit punctured disc. The solution the inclusion map $i:S^2 \to D^2\setminus\{(0,0)\}$ and ...
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1answer
20 views

Winding map doesn't make sense to me

I am looking at fundamental groups and about $S^1$, I was given the following Regard $S^1=\{z \in \mathbb{C};|z|=1\}$. For all $N \in \mathbb{Z}$ let $\omega_N:S^1 \to S^1; z \to z^N$ be the ma ...
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1answer
66 views

Is $S^3\times S^2$ orientable?

The question comes from another question, I am asked to calculate the dimension and check orientability of the manifold $$ V_2(\mathbb{R}^4) = \{(v_1,v_2) \in \mathbb{R}^4\times \mathbb{R^4} \mid ...
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0answers
33 views

I don't understand what a Fundamental group is

I have been staring at the definition for days, drew diagrams but I don't understand as to what its elements are The fundamental group $\pi_1(X,x)$ at a base point $x$ is a set of rel $\{0,1\}$ ...
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1answer
19 views

Explicit homotopy that takes antipodal map to a map with fixed point

homotopic maps from the sphere to the sphere The link above gives a very intuitive way to show that the result in question holds but could someone give me please the explicit homotopy he is using? ...
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2answers
40 views

Embeddings of surfaces into a 3-manifold

Say we are given a disconnected closed orientable surface $S=S_1\coprod S_2$ with $f=f_1\coprod f_2:S\rightarrow M$ such that the $f_i$ are embeddings and the images are incompressible. Suppose that ...
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0answers
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Computing the “limit” of a SSeq with $E_{\infty}^{*,*}$ a free graded $\Gamma$-module.

I'm trying to prove the following proposition from Kochman's book. For completion I will write it here the relevant part: Let $E$ be an oriented spectrum with orientation class $x\in ...
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0answers
23 views

Computing the group of deck transformations w.r.t. a polynomial

Let $p: \mathbb{C}\backslash Y' \to \mathbb{C}\backslash X'$ be a polynomial where $Y'$ is the set of branch points and $X'$ is the image of $Y'$ under $p$. If $\deg p = n$ then $p$ is an unbranched ...
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2answers
55 views

Fundamental group of $\mathbb{R}^3$ minus trefoil knot

Let $ \ T \subset \mathbb{R}^3 \ $ be the trefoil knot. A picture is given below. I need a hint on how to calculate the fundamental group of $ \ X = \mathbb{R}^3 \setminus T \ $ using Seifert-van ...
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1answer
27 views

Null-homotopic map from $SL_2(\mathbb{R})$

I need to prove that a smooth map $f\colon SL_2(\mathbb{R})\rightarrow S^4$ is homotopic to the constant map. I think that computing the corresponding homotopy groups may help, but I don't see how to ...
5
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1answer
82 views

Property similar to connectedness

Recall that $X$ is connected if $X$ cannot be written as the union of nonempty open sets with empty intersection. Consider the following similar property: $X$ is good if $X$ cannot be written as ...
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1answer
33 views

Compact Poincaré dual of $S^{n-1}$ in $\mathbb{R}^n \backslash \{0\}$

I have been asked to find a sub manifold $S \subset M := \mathbb{R}^n \backslash \{0\} $ s.t. its compact Poincaré dual is a basis of the first cohomology group $H^1_c(M) = \mathbb{R}$. Now, $S$ must ...
5
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1answer
58 views

How to show $\mathbb{R}^2/\mathbb{Z}^2$ is homeomorphic to $\mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z}$

I know that $\mathbb{R}^2/\mathbb{Z}^2$ is homeomorphic to $S_1 \times S_1$ and $\mathbb{R}/\mathbb{Z}$ is homeomorphic to $S_1$ thus the product is homeomorphic to $S_1 \times S_1$. But I wonder if ...
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1answer
18 views

Simply connected covering space is a covering of other covering

Prove that a simply connected covering space of X is also a covering space for any other covering space of X. Actually I don't have an idea how to start with. But if X has a universal cover, then the ...
2
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1answer
34 views

Duality of diagrams for fibration and cofibration

According to May's A Concise Course in Algebraic Topology, the diagrams in the following represent cofibration and fibration, respectively if there exists an arrow diagonally (to the upper right ...
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1answer
30 views

What does path-connectedness of $I$ have to do with this at all?

I am utterly confused. Q. Show that $X=\{0,1\}$ with the discrete toplogy is not contractible. Well i need to show that $X$ isn't homotopy equivalent to $\{0\}$. My argument is this We ...
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1answer
35 views

“Reduction to finite case” arguments in algebraic topology

Hello I was studying the corollary to the excision property in Homotopy theory (Hatcher 4K.2) and the thing I can't understand is why the injectivity argument works when moving from an infinite ...
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1answer
31 views

How to present $S_g$ as a $(4g+2)$–gon?

I know we can present $S_g$ (compact surface of genus $g$) as a $4g$–gon with opposite sides identified, but how to present $S_g$ as a $(4g+2)$–gon with opposite sides identified? There is an ...