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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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 ...
-1
votes
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." ...
0
votes
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 ...
1
vote
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 ...
0
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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
votes
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 ...
1
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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 ...
1
vote
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
votes
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
vote
1answer
34 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 ...
1
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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. ...
1
vote
0answers
41 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 ...
0
votes
0answers
32 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
vote
1answer
35 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 ...
1
vote
0answers
19 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 ...
2
votes
0answers
37 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 ...
1
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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 ...
1
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0answers
23 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
votes
1answer
22 views

What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold for $k = n$ and $k = n-1$?

What can you say about the $k$-th cohomology group of a closed orientable $n$-manifold for: (1) $k = n$, and (2) $k = n - 1$. Poincaré Duality tells us that for $M$ a closed ...
0
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0answers
18 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
votes
1answer
35 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?
0
votes
1answer
49 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
votes
1answer
27 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 ...
-2
votes
1answer
30 views

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

I was thinking of splitting up the cases of orientable and non-orientable surfaces.
0
votes
1answer
19 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
49 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
votes
2answers
87 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
votes
0answers
37 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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0answers
28 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
vote
1answer
38 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
votes
1answer
33 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
votes
0answers
41 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, ...
1
vote
1answer
50 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
37 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
votes
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
votes
0answers
17 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
votes
1answer
72 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
votes
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
votes
0answers
11 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
votes
0answers
39 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 ...
0
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0answers
47 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
votes
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
votes
0answers
38 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
votes
1answer
72 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 ...
0
votes
1answer
27 views

Group action on coset space is continuous

I found this exercise in various places, but I could not find the answer anywhere. As I am quite new to topology, I would appreciate any help. Let $G$ a topological group and $H$ a subgroup. Let the ...
6
votes
1answer
79 views

Is it possible for $R \oplus M$ and $R \oplus N$ to be isomorphic to each other if $M$ and $N$ are not isomorphic?

Suppose $M$ and $N$ are non-isomorphic $R$-modules (where $R$ is a commutative ring with a unit element).Can we conclude that $R \oplus M \not\simeq R \oplus N$ ? If not in this most general ...
0
votes
1answer
71 views

Unit sphere without a point is contractible

Let $a$ be a point on the unit sphere $S=\{(x,y,z)|x^2+y^2+z^2=1\}$. How do I show that $S\backslash\{a\}$ is contractible? How do I show that a non-surjective loop $\phi\in P(S,s)$ with ...
0
votes
1answer
48 views

Unique homomorphisms between fundamental groups of topological spaces

Let $u:A\rightarrow B$ be a continuous map of topological spaces, $a\in A$, $b=u(a)$. How do I prove that there exists a unique group homomorphism $$u':\pi_1(A,a)\rightarrow\pi_1(B,b)$$ ...
0
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0answers
27 views

Cellular homology for 3-sphere and $L^3$ lens space

When trying to follow the professor's computation of the cellular homology for the lens space $L^3(4)=S^3/\mathbb{Z}_4$ I became aware I had some trouble understanding the definition of the (cellular) ...