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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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 ...
1
vote
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 ...
0
votes
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
votes
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 ...
1
vote
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, ...
1
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0answers
21 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
votes
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 ...
1
vote
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 ...
1
vote
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
votes
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 ...
1
vote
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 ...
0
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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
vote
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
votes
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
votes
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 ...
1
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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
votes
1answer
62 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
votes
1answer
66 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
votes
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
43 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
votes
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
votes
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
35 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
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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
vote
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
vote
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
votes
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
votes
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 ...