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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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. ...
3
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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 ...
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1answer
41 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 ...
2
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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
27 views

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 ...
3
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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 ...
2
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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
84 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 ...
2
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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 ...
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0answers
28 views

Let $X=\{0,1\}$ be equipped with the indiscrete topology; Why is every $f:Y \to X$ continuous? [duplicate]

By continuity of $f$, I understand that $f^{-1}X$ must be open in $Y$. Well, the statement is general, i.e. for any space $Y$. Don't know what's in it, don't know what topology it has. Regardless, ...
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0answers
20 views

Localization in equivariant cohomology theory for groups other than ($p$-)tori

Recall the following localization theorem, as stated in Hsiang's Cohomology Theory of Compact Transformation Groups: Theorem. Let $G=(S^1)^k$ be a torus, $X$ a paracompact $G$-space with finite ...
2
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2answers
51 views

Triangulation of torus - understanding why

Note: in relation to the answer of the duplicate question, I see that the second picture below refers to the triangulation when we consider simplicial complexes. I do not understand why the triangles ...
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2answers
40 views

How to write the real projective plane as a pushout of a disk and the mobius strip?

I heard in topology class that the real projective plane is obtained by gluing a disk along the boundary of the mobius strip. I was wondering - how can I write this as a pushout? Also, how can I ...
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0answers
30 views

How does the Whitehead quadratic functor act on morphisms?

I am trying to understand the universal quadratic functor $\Gamma$ that appears on the Whitehead exact sequence (J. Whitehead, A certain exact sequence, Annals of Mathematics 52 (1950)) particularly ...
2
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1answer
35 views

Coverings of a three-manifold

He guys, I have two questions regarding the following: Consider the three-manifold $\mathbf{T}^3 = S^1 \times S^1 \times S^1$ and let $S_n$ be the permutation group acting on $n$ letters. Let ...
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0answers
26 views

Lifting paths in a fibration in families

Given a fibration (or a fiber bundle or whatever might make my question true) $f\colon E\rightarrow B$ and basepoints $e$ in $E$ and $b$ in $B$, such that $f$ preserves them. By the homotopy lifting ...
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0answers
29 views

Need a reference book for stokes theorem

I am studying singular homology, I would like a good reference for the proof of stokes theorem for chains in manifolds. Thank you!
1
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1answer
31 views

Number of path components for products; Is my conjecture right?

For two topological spaces, I am wondering how the product of the two would make the number of path components in them... Let $X,Y$ be topological spaces and say there are $n$ and $m$ path ...
0
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1answer
40 views

Does a continuous map between $X,Y$ imply…that they have the same number of path components?

I am having trouble with understanding the "degree" of maps, which involves one of my previous questions. I've decided to sit down for however many hours it might take for me to ram it down my ...
0
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1answer
61 views

Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$?

The first question is Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$? In other words, is it obvious? The question stems from a theoretical physics ...
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0answers
32 views

Minimality of the Spatial C*-Norm

Given C*-Algebras $A,B$ and $x \in A \otimes_* B$ show that: (a) If $(\tau \otimes_* \rho)(x)=0$ for all $\tau \in A_+^*$ and $\rho \in B_+^*$, then $x=0$. PS: $\tau \in A_+^*$ means that $\tau$ is ...
4
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1answer
61 views

Continuous map in $\mathbb{R}^2$ has a (scaled) fixed point

Let $\phi:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a continuous map. How do I prove that there exist $a>0$ and $x\in\mathbb{R}^2$ such that $\phi(x)=ax$? What I know: I thought maybe this can ...
0
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1answer
64 views

Quotient topology on unit sphere

Let $\sim$ be the equivalence relation $$a\sim b\iff a=b\text{ or }a=-b,$$ for $a,b$ on the unit sphere $S^2$. Let $Q$ be the quotient space. How do I show that the quotient map is a covering ...
0
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1answer
22 views

Mind numbing homotopies; why is this a homotopy rel?

Proposition. every map $\alpha: [s_0,s_1] \to \mathbb{R}^n$ is homotopic rel $\{s_0,s_1\}$ to the linear map $$\beta=\frac{(s_1-s_0)\alpha(s_0)+(s-s_0)\alpha(s_1)}{s_1-s_0}:[s_0,s_1] \to ...
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2answers
42 views

Free cocompact action of discrete group gives a covering map

I'd like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah's paper "Elliptic operators, discrete groups, and von Neumann algebras." ...
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2answers
34 views

$f$ is null homotopy if and only if $f_*$ is the trivial induced homomorphism

I want to show that $f:S^1\to S^1$ is homotopic to the constant map if and only if $f_*:\pi_1(S^1,x_0)\to \pi_1(S^1,x)$ , $f_*([\gamma])=0$ This seems like it should be an obvious fact but I am having ...
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1answer
30 views

Computing fundamental group of the mapping torus $\pi_1(T_{z\mapsto z^r}(S^1))$

I am trying to compute the fundamental group of the mapping torus of $f(z) = z^r$ for $r\in \mathbb{R}$ on the domain $S^1$. So, the space is $S^1\times I$ with $(z, 0)$ identified with $(z^r, 1)$. ...
3
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0answers
48 views

How to construct a G-extension of a category C?

Note: I'm a physicist so this will be phrased somewhat in physics language. Suppose we have a unitary modular tensor category $\mathcal{C}$. In physics language, we can think of $\mathcal{C}$ as ...
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0answers
31 views

(Readings on) fundamental group and boundary identifications

I'm trying to solve some problems on topology, and I'd like to know where to find methods to calculate fundamental groups of objects like, e.g., the union of two solid tori where the two boundaries ...
0
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1answer
31 views

Visualizing the quotient of a torus and a circle

We were asked to compute the homology for the double torus, $X$, and a circle around one of the loops, $B$, of the torus (not a circle between the two halves of the torus) and were told that this ...
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0answers
20 views

Help understanding remark on Greenberg and Harper text on cohomology chapter

I am working through Greenberg and Harper text and I don't understand a remark. I think it should be very easy to gasp but I don't see what it means: My problem is with the remark 23.0. Could ...
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0answers
42 views

Why abelian groups instead of modules in Algebraic Topology

I am studying Algebraic Topology, homology and cohomology to be concrete. I am reading\working through Hatcher, Rotman, Harper and sometimes I combine them with other books when none of them give a ...
0
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1answer
42 views

Trivial second homology group

Let $\Omega\subset\mathbb{R}^3$ be a an open bounded set. Let us consider the following statement: every closed surface in $\Omega$ is the boundary of a suitable subdomain $D\subset\Omega$. ...
1
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1answer
33 views

How to Pair Generators in the Presentation of Fundamental Group of a Surface

The fundamental group of a surface with genus $g$ is widely given by the group presentation (for example Hatcher p.51): $$\langle a(1),b(1),a(2),b(2),..,a(g),b(g) \mid ...
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0answers
25 views

Compute the homology group

Let X be the space obtained by removing two out of three coordinate axes from $\mathbb{R}^{3}$. I don't know how to compute its homology group. Actually I even don't know how to find the n-simplices. ...