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
19 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
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0answers
14 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 ...
1
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2answers
19 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 ...
63
votes
1answer
2k views

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. ...
10
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2answers
259 views

Are there any simply connected parallelizable 4-manifolds?

On pp. 166 of Scorpan's "The Wild World of 4--manifolds", he gives an example of a parallelizable 4-manifold ($S^1\times S^3$) and then asserts: "there are no simply connected examples". It's ...
2
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1answer
20 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
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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, ...
2
votes
2answers
37 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
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0answers
10 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 ...
0
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0answers
16 views

How does the universal 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 ...
0
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0answers
10 views

Coverings of $\mathbf{T}^3$

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 ...
0
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0answers
14 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
votes
1answer
30 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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0answers
27 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
171 views

How do I prove that the union of two simply connected open sets whose intersection is path connected is simply connected?

I'm trying to understand Ronnie Brown's answer here: union of two simply connected open , with open and non empty intersection in $R^2$ Let $X$ be a topological space and $U,V$ be simply connected ...
0
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1answer
20 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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1answer
57 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 ...
0
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1answer
38 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 ...
7
votes
3answers
619 views

deformation retract of $GL_n^{+}(\mathbb{R})$

Well, I need a deformation retract from $GL_n^{+}(\mathbb{R})$ to $SO(n)$ Here is what I tried, let $A\in GL_n^{+}(\mathbb{R})$ $A=(A_1,\dots,A_n)$ where $A_i$'s are collumn vectors, Recall that the ...
4
votes
1answer
152 views

Fundamantal group of a regular covering space

Let $B$ be the space of figure $\infty$ (with $x$(the red circle) and $y$(the black one) as generators) and $E$ its covering space (in the picture below). let $P_{*}: \Pi(E,a) \to \Pi(B,b) $ be the ...
1
vote
2answers
79 views

A detail about reconstructing covering space from the action $\pi_1(X,x_0)\to S_{p^{-1}(x_0)}$ in Hatcher's book

I'm struggling understanding a small sentece from Hatcher's Algebraic topology book (available online for free). In page 70 Hatcher wants to reconstruct the covering $p:\tilde X\to X$ from the ...
1
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1answer
26 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
31 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 ...
0
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2answers
34 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
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 ...
0
votes
1answer
34 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 ...
4
votes
1answer
41 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 ...
1
vote
1answer
139 views

Direct limit of $CW$ complex and infinite Stiefel manifold

Let $V_{n}(\mathbb{R}^k)$ be the Stiefel manifold of ortogonal $n$-frames in $\mathbb{R}^k$ and $G$ a compact Lie group. A classifyng space for group $G$ is a connected topological space $BG$, ...
1
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1answer
28 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 $(z^r, 1)$. ...
0
votes
1answer
29 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 ...
0
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2answers
33 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 ...
0
votes
1answer
47 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 ...
3
votes
0answers
47 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
24 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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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 ...
1
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0answers
19 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 ...
0
votes
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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0answers
37 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
31 views

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

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.
1
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0answers
36 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 ...
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 ...
1
vote
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 ...
1
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0answers
24 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. ...
0
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0answers
28 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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0answers
17 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
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0answers
33 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
22 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 ...
2
votes
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?
0
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0answers
15 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 ...