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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1answer
41 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 ...
1
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0answers
9 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
37 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
28 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
16 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 ...
0
votes
2answers
30 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." ...
0
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2answers
29 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 ...
1
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1answer
27 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
41 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
23 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
26 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 ...
1
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0answers
18 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 ...
1
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0answers
31 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
40 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
24 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
20 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. ...
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0answers
33 views

Computing homology of a torus

I'm trying to calculate homology groups of a a torus using Meyer-Vietoris sequence. Let $A,B$ be a half of a torus homeomorphic to $S^{1} \times I $. Let's enlarge them so that they intersect and $A ...
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0answers
27 views

Fundamental Group of Orientable Surface

On p.51 Hatcher gives a general formula for the fundamental group of a surface of genus g. I have one specific question, but would also like to check my general understanding of what's going on here. ...
1
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1answer
28 views

Why is this mapping not contractible?

We define the relative homotopy for a pair $(X,A)$ to be the homotopy classes of continuous maps $$(D^n, S^{n-1},s_0) \to (X,A,x_0)$$ This is technically a continuous map from $D^n \to X$ with the ...
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0answers
16 views

Showing that the Mobius strip is non-orientable using obstruction theory

In the notes available here, the first example (p. 2) says that the Mobius band is non-orientable because there is an obstruction to a map sending a disk to the band. I don't really understand what's ...
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0answers
27 views

Homology, addition of homology classes in construction of Poicare Sphere

I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one exercise. I have spoken with a professor and he encouraged me to ask here or look for ...
3
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0answers
17 views

Computing $E^*(\mathbb{C}P^n)$, for $E$ an oriented spectrum via AHSS

I'm trying to understand the proof of Prop. $4.3.2$ in Kochman's Bordism, Stable Homotopy and Adams Spectral Sequences, in particular the claim b): $$ E^*(\mathbb{C}P^n)\cong ...
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0answers
16 views

circle actions on spheres

I'm considering the following action of $S^1$ on $S^3$: $$ e^{i\theta}.(z_1,z_2)=(e^{i\theta}z_1,e^{iq\theta}z_2) $$ It is clear that when $q=1$ the quotient space is $S^2$. Is there any description ...
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0answers
21 views

a algebraic concept in reimannian geometry?

let $(M,g)$ is a $C^{\infty}$ manifold with product form of $M=R \times \Theta $ the for each element $\varepsilon (r,\theta )\in M$, the tangent space $T_{\varepsilon}M$ is natural isomorphic to ...
0
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0answers
13 views

Non-Inductive formula for subdivision operator

This problem is from hatcher 2.1.25. Find an explicit, noninductive formula for the barycentric subdivision operator. I have no idea how to get that formula. The only way I see it geometrically is ...
2
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1answer
33 views

From Dividing open bounded sets in $\mathbb{R}^2$ into equal areas, what goes wrong if $U,V$ not connected?

Dividing open domains in $\mathbb R^2$ in parts of equal area From this question, what can go wrong if $U,V$ are not connected?
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1answer
45 views

Showing $\mathbb{R}^3$ minus $n$ parallel lines is homotopic to $\mathbb{R}^2\setminus\{p_1,\dots,p_n\}$

I want to show that if I remove $n$ parallel lines from $\mathbb{R}^3$ then I get $\mathbb{R}^2\setminus \{p_1,\dots,p_n\}.$ There is also some underlying structure I wish to also understand. That ...
0
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1answer
22 views

Properties that remain invariant under deformation retraction map

I am studying Algebraic topology where I came across the kind of maps called retraction, specifically deformation retraction. What kind of properties are conserved under such maps? From what I could ...
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1answer
28 views

How many closed surfaces (up to homeomorphism) are there with Euler characteristic -2? [on hold]

I was thinking of splitting up the cases of orientable and non-orientable surfaces.
0
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1answer
17 views

Computing the degree of a one-variable map

Edit: This question originally contained a typo where the function $f$ specified below was equal to $x$, not $x^2$ as currently written, outside an interval $[-T,T]$, and the accepted answer was ...
3
votes
1answer
46 views

Torus with a point deleted is not a retract of the torus.

Show that the 2-torus with a deleted point $T\setminus \{ x_0\}$ is not a retract of $T$. I know that we can prove the torus with a point removed deformation retracts to the wedge of two circles. ...
6
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2answers
83 views

Examples of Manifolds such that $\chi (X)=-3$

I am trying to come up with an example of a closed oriented manifold with euler characteristic equal to $-3$. I have tried to use $\chi (\underbrace{T^2\mathbin{\#}\cdots \mathbin{\#} T^2}_{\text{$g$ ...
2
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0answers
35 views

Abelianization and analysis of Fundamental Groups

I am working through Hatcher on my own, and currently doing problem $9$ on $p53$. This problem brings up the strategy of abelianization of groups to solve problems of fundamental groups and ...
0
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1answer
31 views

Is $S^n\times S^n\setminus (S^n\times \{pt\})$ homotopic or homeomorphic to $S^n\times S^n\setminus\{(x,-x)|x\in S^n\}$ [on hold]

when n=1, it is homotopic and homeomorphic. when n=2, the first one is trivial $\mathbb{R}^2-$ bundle. The second should be nontrivial $\mathbb{R}^2-$ bundle. But I cannot prove it.
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0answers
27 views

Is there genera for algebraic cobordism?

For real and complex manifolds, we can form the (oriented) cobordism ring $\Omega$, and a genus is defined to be a ring homomorphism $$\varphi:\Omega\otimes\mathbb{Q}\to R$$ where $R$ is any ...
1
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1answer
37 views

$\operatorname{S}X \approx I\times X/ \{0\}\times X\cup I\times \{x_0\}\cup \{1\}\times X$

The reduced suspension of the pointed space $(X,x_0)$ is the smash product $(\mathbb S^1\wedge I, *)$ of $(X,x_0)$ with the $(\mathbb S^1,s_0)$ and is denoted by $\operatorname{S}X$. My problem is to ...
0
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1answer
28 views

How to lift an action along a covering

Took from Lawson-Michelsohn "Spin Geometry", pages 80-81. Let $E\to X$ be an oriented $n$-dimensional vector bundle over a manifold $X$, and let $P_{SO}E$ be the associated $SO(n)$-bundle. Assume ...
4
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0answers
39 views

a topological space with finite integer first cohomology group?

One more problem preparing for a PhD exam! It states "describe a space such that $H^1(X,Z)=Z_5$." I thought this was impossible by the universal coefficient theorem since $H_0(X;Z)$ is always free, ...
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0answers
37 views

Parallelization of a Sphere gives Division Algebra

Is there an elementary proof of the fact, that a parallelization of $S^n$ can turn $\mathbb{R}^{n+1}$ into a division algebra? My guess was something like this: Let $v_1(x),\dots, v_{n}(x)$ denote ...
3
votes
1answer
34 views

Fundamental Group and DeRahm Cohomology from Group of Covering Transformations

Old qual problem here, test tomorrow in topology and we barely got to DeRahm Cohomology so I'm not sure how to do this. Let $G$ be the group of transformations of $\mathbb{R}^3$ generated by ...
0
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1answer
15 views

How does a function influence an induced homomorphism?

Let T be continuous and surjective from X to Y. Is the induced homomorphism $T_*$ surjective? Does injectivity of T imply injectivity of $T_*$? I have a feeling that this is trivial to answer and ...
0
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0answers
16 views

intersection between elements in first homology of 2-genus torus

Could you please explain for me when two non-trivial elements in homology of the 2-genus torus intersect? I know that in case of the torus, two non-trivial circles in the torus intersect if and only ...
3
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1answer
71 views

A question about covering space

Let $p: T \to X$ be a covering and let $f:Y\to X $ be a continuous function we define $f^*T$ as $$ f^*T=\{(y,\tilde{x})\in Y\times T|f(y)=p(\tilde{x})\} $$ let $p':f^*T\to Y$ be the map given by ...
0
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0answers
25 views

If $X$ is a topological space then $X \times X$ is not homeomorphic to $S^1$. [duplicate]

If $X$ is a topological space then $X \times X$ is not homeomorphic to $S^1$. How can I show this? I know that $\pi_1(S^1)=\mathbb{Z}$ and $\pi_1(X \times X)=\pi_1(X)\times \pi_1(X)$, but I got stuck ...
2
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0answers
9 views

When does the cohomological Atiyah–Hirzebruch–Leray–Serre spectral sequence converge?

Given a Serre fibration $F \to E \to B$ of spaces homotopy equivalent to CW complexes, with $B$ simply-connected, and a generalized homology theory $h_*$ with respect to which the fibration is ...
0
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0answers
38 views

identity for quaternions' group Sp(n)=Sp(2n,C)∩U(2n)

Could you help me for solving this: Let $Sp(n)$ be the group of linear transformations of $H^n$ such that preserve hermitian form $$\sum_{i=1}^n \overline{q_i}r_i,$$ that $H$ is the quaternions _ the ...
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0answers
42 views

Computing relative de Rham cohomology?

Let $f: N \to M$ be a smooth map. Then in Bott and Tu they define a relative de Rham complex $\Omega^{\bullet}(f)$ where $\Omega^k(f) = \Omega^k(M) \oplus \Omega^{k-1}(N)$ with coboundary map given in ...
3
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1answer
66 views

Trivialised sub-bundle of a trivialised bundle and its orthogonal bundle

I'm reading Kirby's The Topology of $4$ Manifolds, and I encountered the following claim: Recall that a trivialised sub $k$-plane bundle of a trivialised $(n+k)$-plane bundle determines a ...
2
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0answers
37 views

Homeomorphisms on a finite connected graph $X$ with $H_1(X; \Bbb Z)$ free abelian

For context, this is exercise 2.2.42 in Hatcher's Algebraic Topology. Let $X$ be a finite connected graph having no vertex that is the endpoint of just one edge, and suppose that $H_1(X; \Bbb Z)$ ...
4
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1answer
70 views

True or False: there is a space $X$ such that $S^1$ is homeomorphic to $X\times X$

I had an exam this morning, one of the questions asked about the truth of the statement There is a space $X$ such that $S^1$ is homeomorphic to $X\times X$. I said that this was false and this ...