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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$X = S^1 \vee S^1$ so $\pi_1(X) = F\{a,b\}$. Given homo. $\varphi: \pi_1(X) \rightarrow \mathbb{Z}/3$, draw associated cover of $\ker\varphi$.

Let $X=S^1 \vee S^1$ and so that $\pi_1(X)=F\{a,b\}$, the free group on two generators. Let $\varphi:\pi_1(X) \rightarrow \mathbb{Z}/3$ be the homomorphism induced by $\varphi(a)=1$ and $\varphi(b)=0$....
6
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1answer
485 views

Homology groups of orientable surfaces.

Edit: I have a proof here but when I spoke last with my professor, she told me something was close, but not quite. Can someone help me patch this proof? I've been trying to get this down for quite a ...
2
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1answer
37 views

Existence criterion of $\operatorname{Spin}_{\mathbb{C}}$ structure via determinant line bundle

In Dan Freed's notes Exercise 9.30 he outlines the proof of the existence criterion which is that there exists $\tilde{c} \in H^2(M;\mathbb{Z})$ such that $2\tilde{c} = c_1(E)$. His approach is to ...
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1answer
15 views

Universal Cover of wedges $S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2}$ and $\mathbb{R}P^{2} \vee \mathbb{R}P^{2}$.

We are asked to find the universal cover of the wedges $S^{2} \vee S^{2}, \mathbb{R}P^{2} \vee S^{2}$ and $\mathbb{R}P^{2} \vee \mathbb{R}P^{2}$. I am second guessing myself on this problem because I ...
0
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1answer
50 views

Does $T^3$ double cover $\mathbb{R}P^3$?

So $T^2$ double covers $S^2$, and $S^2$ double covers $\mathbb{R}P^2$. Therefore, $T^2$ quadruple covers $\mathbb{R}P^2$. I am looking at $\mathbb{R}P^3$ because I am interested in rotations. The ...
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4answers
387 views

The circle is not contractible

I know that the circle is not contractible because I know that $\pi_1(S^1)\cong \mathbb Z$. But something is going wrong in my head. Choose a basepoint $*$ on the circle and chose an orientation (...
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1answer
28 views

Existence criterion of $\operatorname{Spin}_{\mathbb{C}}$ structure

In deriving the existence criterion of $\operatorname{Spin}_{\mathbb{C}}$ structure Ralph Cohen in his notes on the topology of fiber bundle (pp.169) uses the following diagram $\require{AMScd}$ \...
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0answers
38 views

Modular curve, $G$-Galois branched cover

I was reading this answer by Pete L. Clark: "For "most" finite simple groups $G$ it is indeed the case that $G=⟨x,y⟩$ where $x$ has order $2$ and $y$ has order $3$. Equivalently, $G$ is a ...
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0answers
39 views

(Non-trivial) local coefficient system which is not a bundle of groups

I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients. For my question the following definition of a local coefficient system is needed ([2, p. 257], [3, p. 35]): Let $X$ ...
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1answer
19 views

Counterexample: Homology with Local Coefficients versus Homology with Module Coefficients

Definitions Let $X$ be a nice space with universal covering $\widetilde{X}$. Homology $H_*(X,R)$ with Ring Coefficients $R$ is the homology of \begin{align*} C_n(X;R) :=&\; \text{ free left $R$-...
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0answers
12 views

Does the inclusion of $n$-connected $\infty$-groupoids into $(n-1)$-connected $\infty$-groupoids admit adjoints?

Denote the $(\infty ,1)$-category of $n$-connected $\infty$-groupoids by $\operatorname{Grpd}^{\infty}_{n}$. There is clearly a suitable inclusion $\iota:\operatorname{Grpd}^{\infty}_{n}\...
3
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1answer
72 views

Jordan curve theorem for a polygon

Let P be a Jordan polygon on the complex plane. I would like to prove Jordan curve theorem for P using an elementary method. I think that we can prove it using the winding number with respect to P. Am ...
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0answers
13 views

The fiber of a covering map in a point is homeomorphic to a quotient of the fundamental group of the base space.

It's well known that if a compact topological group $G$ acts transitively on a Hausdorff space $X$, then $G/G_x$, the quotient of $G$ by the stabilizer of $x$, is homeomorphic to $X$, for every $x \in ...
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2answers
31 views

Homotopy: equivalence relation (continuity of homotopy in symmetry)

Let $f_0, f_1: X \rightarrow Y$ be continous on topological spaces $X,Y$. Let $F$ be a homotopy between $f_0, f_1$. Which argument shows that $G(x,t):=F(x,1-t)$ is continous? Seems to be hard to ...
0
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1answer
35 views

Retractions and Isomorphisms of Fundamental Groups

Suppose there is a retraction from $$S^1 \times D^2 \to S^1 \times S^1.$$ Does that then induce an isomorphism $$\pi_1(S^1) \times \pi_1(D^2) \cong \pi_1(S^1) \times \pi_1(S^1)?$$ Which is obviously ...
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0answers
21 views

Covering space of wedge of two circles generated by $\{a^{3},b^{2},aba^{-1}b^{-1}\}$

I've been drawing graphs for too long trying to figure it out. I know how to find the generating set using maximal trees, but is there a trick for the reverse problem of given a generating set, come ...
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0answers
32 views

Covering Spaces of Wedge of two Circles

I am working on some qual questions, and I don't really comprehend this. Let $X = S^{1} V S^{1}$. The question is asking about covering spaces 7,9, 11, and 12. a. Determine whether or not the ...
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0answers
31 views

Any use to thinking of the pullback of a fiber bundle as an integral?

We can think of the pullback of fiber bundle an integral as follows. Suppose $p:E \to B$ is a fiber bundle and $f:A \to B$ is a map. Then we have the pullback bundle $A \times_f E \to A$. The bundle ...
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0answers
18 views

Normal covering spaces of wedge of two circles

On page 58 of Hatchers textbook, PDF here: https://www.math.cornell.edu/~hatcher/AT/AT.pdf He says the graphs with "maximal symmetry" are normal covering spaces. This is graphs 1,2,5, 6,7,8 and 11 ...
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1answer
87 views

Ham Sandwich Theorem - intuitive proof

Ham Sandwich Theorem. Given 3 measurable "objects" in $\mathbb{R}^3$, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single 2-dimensional plane. Can ...
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1answer
54 views

Why are spherical objects so named?

Let $S$ be an object in an abelian category. Then we say S is spherical if $Ext^p(S,S)$ is 0 unless $p = 3$. I know that the cohomology of the three sphere bears some formal resemblence, but it doesn'...
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2answers
443 views

Is it possible to define orientability using orientation preserving loops?

Wikipedia says that the orientable double cover corresponds to the subgroup of orientation preserving loops in $\pi_1$ (which is of index 1 or 2 apparently). My questions are: What is an orientation ...
2
votes
1answer
412 views

Deck transformation acting properly discontinuously assumed covering space is path-connected

Let $p:E\rightarrow X$ be a covering space, $x\in X$ fixed point, $E$ path-connected and $\Delta(p)$ – Deck transformation group of $p$, that is $\Delta(p) = \{f\in \text{Homeo}(E):pf=p\}$. Let $\pi_1(...
5
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3answers
76 views

Fixed-point free map of the 2-sphere which has order 4

The antipodal involution of $\mathbb{S}^2$ clearly has no fixed points. However I cannot think up an example of homeomorphism of order 4 which has no fixed points. Could you help me?
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0answers
20 views

About homomorphism which appears in universal coefficient theorem (simple question)

$h: H^n(C, R) \rightarrow Hom(H_n(C), R)$ which appears in universal coefficient theorem: Certainly, $h$ is group homomorphism and my question is if this also can be considered as $R$-module ...
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0answers
37 views

Universal Abelian Covering Space of genus two surface

Let M be a surface M, i am concerned with Abelian covers. These are the covering spaces for which the deck group is Abelian. The largest such cover corresponds to the commutator subgroup of the ...
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0answers
17 views

How to give to a tetrahedron an ordering as simplicial complex?

In order to calculate homology groups of the tetraedron by hand, first of all I need to order its simplexes. But I start ordering the 1-simplexes, then the 2-simplexes and the 3-simplexes. In the 3-...
3
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1answer
56 views

Geometric construction of $J$-homomorphism

In D. Freed's notes eqn (5.32), he defines the $J$-homomorphism geometrically by considering the equatorial $n$-sphere as an $n$-submanifold of $S^m$, and giving it a framing that makes it null-...
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2answers
26 views

What are some examples of applications of integral quadratic forms in $n$ variables in algebraic topology?

I'm reading the wiki page of qudratic forms. It simply seems curious to me what are some concrete examples of applications of integral quadratic forms in algebraic topology. I've searched a bit but a ...
2
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0answers
52 views

rho invariant of manifold

If $G$ is a finite group, then the rational oriented cobordism group $\Omega_{2k-1}^{Stop}(BG)\otimes{\mathbb Q}=0$, so if $N^{2k-1}$ is an orientable odd-dimensional Top manifold with fundamental ...
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0answers
23 views

If $\alpha$ and $\beta$ are homotopic curves with same initial and final points on $X$ a topological space, them lifts are homotopic

Let $X$ be a topological space and $\alpha, \beta$ curves on $X$ that have fixed the initial and final points. Show that them lifts are homotopic as well. I can't find a good reference to prove this. ...
1
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1answer
41 views

Proof that classifying spaces for discrete groups are the Eilenberg-MacLane spaces

A classifying space $BG$ of a topological group $G$ is the quotient of a weakly contractible space $EG$ by a free action of $G$. The claim is that if $G$ is a discrete group then $EG/G$ is an ...
2
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0answers
43 views

Applications of Topological Complexity of configuration space

I'm starting to work on Topological Complexity of configuration spaces . Articles say that it has applications in robotic and control theory . One of important article was belong to Michael Farber ...
0
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1answer
35 views

$f: S^n \to S^n$ is the restriction of a continuous mapping $F: \overline{B^n_1} \to S^n$ iff $deg(f) = 0$.

this is a theorem from wikipedia (https://en.wikipedia.org/wiki/Degree_of_a_continuous_mapping#Properties): A self-map $f: S^n \to S^n$ of the $n$-sphere is extendable to a (continuous) map $F: ...
2
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1answer
381 views

Fundamental group obtained by attaching a n-cell with n ≥ 2

I am having trouble with Hatcher's Algebraic Topology P39, Problem 18: Show that if a space $X$ is obtained from a path-connected subspace $A$ by attaching a cell $e^n$ with $n ≥ 2$, then the ...
0
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1answer
47 views

Why is there no retraction between $D^2 \times S^1$ and $S^1\times S^1$?

Why is there no retraction between $D^1 \times S^1$ and $S^1\times S^1$? I have no idea how to prove. I just know that if there is then $\mathbb{Z}\times \mathbb{Z}$ contains a copy of $\mathbb{Z}$. ...
1
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1answer
38 views

Definition of Coboundary

I was reading these lecture notes from Duke University and found a typo I think. If it's not a typo then I'm really confused. Anyway, on page $95$ shouldn't $$B^p=\operatorname{im}\delta^{p+1}:C^{...
6
votes
1answer
102 views

example of toric varieties with nontrivial first cohomology group

If I remember correctly, when a toric variety is smooth or simplical (the moment polytope is simplicial rational), then there is no odd dimensional cohomology group and the dimension of the even ...
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0answers
34 views

Why is topological K-theory equivalent to nonabelian cohomology with respect to the stable unitary group?

I was reading on the $n$Lab page for topological K-theory that taking cohomology of a smooth space with respect to the smooth $\infty$-stack $\mathbf{Vect}$ is equivalent to taking its cohomology with ...
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0answers
16 views

Isomorphisms between images of the Honda formal group law on Morava $K$-theory

Let $L = MU_*$ be the Lazard ring, which represents formal group laws, and let $W = MU_* MU$ be the ring that represents strict isomorphisms between formal group laws. Let $K(n)_* = \mathbb{F}_p[v_n^\...
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1answer
101 views

Is there a general way to tell whether two topological spaces are homeomorphic?

We know that if two topological spaces $X$ and $Y$ are homeomorphic, then they have the same fundamental groups, and the same homology. In other words, we have functors $$\pi_1 : \mathsf{Top} \to \...
3
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1answer
87 views

Why are geodesically convex sets diffeomorphic to $\Bbb R^n$?

In the construction of a good cover for a manifold $M^n$, Bott & Tu use the fact that each point in $M$ is contained in a geodesically convex set (after picking a Riemannian metric). They then ...
3
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2answers
30 views

Is there a contractible space with a free circle action?

Question in title. Seems no to me (some vague intuition here about contracting orbits to a fixed point), but I can't prove it. I'd prefer to be wrong. (I'm curious because I am thinking about group ...
0
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0answers
44 views

Can we prove invariance of dimension directly from the Jordan-Brouwer separation theorem?

Is the following proof correct? Consider spaces $\mathbb{R}^n$ and $\mathbb{R}^m$, where $n<m$, and sphere $S^{n-1}\subset \mathbb{R}^n$. Suppose that we have a homeomorphism $f:\mathbb{R}^m \...
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0answers
41 views

What can be said about a space with solvable fundamental group?

Let $X$ be a topological space. Suppose $\pi_1(X)$ is solvable, can we say something about $X$ ? This question is probably broad, however I am interested in knowing if there is anything at all that ...
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0answers
32 views

Alternate Proof of Unique Lifting Property of Covering Spaces

I proved one of Hatcher's propositions on my own and my proof is quite a bit different than his. The Unique Lifting Property says: Given a covering space $p:\tilde{X} \rightarrow X$ and a map $f: ...
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0answers
108 views

On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
0
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2answers
48 views

Strong Topology is the strongest topology?

In his article Construction of universal bundles. II (1956), John Milnor defines the strong topology in a join of spaces, but his definition is By a strong topology in $A_1\circ A_2\circ \dots \...
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1answer
35 views

Topological Join of Unit Balls

I have seen that apparently one has for spheres that $S^n*S^m=S^{n+m+1}$. Is there a similar result for unit balls? Thank you.
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2answers
139 views

Lifting homeomorphisms covering

Hello I had a question regarding a lemma from the paper: http://www.math.columbia.edu/~jb/bir-hilden-annals.pdf I don't understand the proof of Lemma 5.1. Notation: $T_{0,0}$ is the 2-sphere, $T_{...