# Tagged Questions

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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 $H_1(X; \Bbb Z)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$u':\pi_1(A,a)\rightarrow\pi_1(B,b)$$ ...
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### 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) ...
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### the definition of relative Thom spaces

I find the definition of Thom space on the book Characteristic classes, J.W. Milnor, J.D. Stasheff, page 205: For a vector bundle $\xi$ with a Riemannian metric over a $CW$-complex $B$, let $A$ ...
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### Show that if $\widetilde{H}^n(X;\mathbb{Q})$ and $\widetilde{H}^n(X;\mathbb{Z_p})$ are zero, then $\widetilde{H}_n(X;\mathbb{Z})$ is zero.

Show that if $\widetilde{H}^n(X;\mathbb{Q})$ and $\widetilde{H}^n(X;\mathbb{Z_p})$ are zero for all $n$ and all primes $p$, then $\widetilde{H}_n(X;\mathbb{Z})$ is zero for all $n$. My Try: ...
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### A circle with a line bisecting it is homotopic to the wedge sum of two circles

Let $C$ be the circle centered at $0$ with radius $2$. $L$ is the segment connecting $(0,2)$ and $(0,-2)$. Prove that $F = C \cup L$ is homotopic to the wedge sum of two circles. Intuitively, I ...
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### Homology Poincare Homology Sphere by Mayer-Vietoris

I am working through some pages of Dale Rolfsen, Knots and Links, AMS Chelsea Publishing, 2003, in order to understand Dehn approach to the original Poincaré conjecture. To be concrete with what I ...
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### Fundamental group of a wedge sum, in general (e.g. when van Kampen does not apply)

What is the fundamental group of a wedge sum in general? e.g. including the times when van Kampen cannot help us. The Wikipedia article on wedge sums mentions that Van Kampen's theorem gives ...
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### limit of covering spaces

Say we have $X$ a manifold with a compact exhaustion of embedded submanifolds $X=\cup K_n$ with $K_n\subset K_{n+1}$. Let $H\subset \pi_1(X)$ a infinite index subgroup that is finitely generated, ...
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### $\mathbb{CP}^2$ realizable as boundary of compact smooth $5$-manifold? [duplicate]

As the title says, can $\mathbb{CP}^2$ be realized as the boundary of a compact smooth $5$-manifold?
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### Euler characteristic is the alternating sum of dimensions of the homology

Give a direct proof that Euler characteristic is the alternating sum of dimensions of the homology, using the rank of the boundary maps. This is my instructor's question of today's topology ...
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### Show that $\Bbb R^{2n+1}$ is not a division algebra over $\Bbb R$ for $n>0$

This is an exercise from Hatcher's Algebraic Topology (exercise 2.B.8). Here is the problem statement: Show that, for $n>0$, $\Bbb R^{2n+1}$ is not a division algebra over $\Bbb R$ by showing ...
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### Proposition of CW complexes

I am trying to prove the following result: Suppose $X$ is a connected finite CW complex. Fix a vertex $x\in X^0$. Prove that the inclusion $X^2\subset X$ induces an isomorphism ...
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### Show that Riemann Surface is connected?

I was reading Artin's Alegbra when this question came into my mind. Consider $f(t,x)=x^{2}-t$ , The locus X of zeros in $\mathbb C^{2}$ of a polynomial is called Riemann surface of f. I understood ...
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### Quick question: Line bundle on union of two lines

Let $l_1$, $l_2$ be two lines in $\mathbb{P}^n$. What is the meaning of $\mathcal{O}_{l_1}(a_1)\cup\mathcal{O}_{l_2}(a_2)$ as a sheaf on the union $C=l_1+l_2$ of two distinct lines and why do we ...
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### CW-complex via composition of pushouts and there characteristic maps

Strom defines CW-complexes in his book Modern Classical Homotopy Theory (p. 47, ch. 3.2.1) via composition of pushouts, i.e. given a discrete topological space $X_0$, he constructs $X_{n+1}$ from ...
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### Loops homotopic relative to A

Let $(X,A)$ be a pair of path-connected spaces, where $A\subset X$. How do we see/prove that this statement is true? Two loops $\gamma_0,\gamma_1\in\pi_1(X,x_0)$ are homotopic relative to $A$ if ...
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### Fundamental groups of path connected subspaces

Does every path connected subspace of $\mathbb{R}^2$ have a fundamental group the trivial or an infinity group? For example, for convex subspaces we know that, but if we take only path connected ...
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### Contradiction in spectral sequence calculation of $H_*(BO(2))$

$\newcommand{\Z}{\mathbb{Z}}$ For this post I am going to assume the answer namely $H_*(BO(2))=\Z_2[w_1,w_2]$. Consider the fibration $S^1 \hookrightarrow BO(1) \to BO(2)$. The $E^2$ page has ...
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### Questions on CW complex structure

REMARK: I had already posted these questions, about one hour ago, but one of the questions was not what I meant. I am in the beginning of my studies in Algebraic Topology and am studying CW complexes ...
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### Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.

I have some questions about this proof that "Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.": By the example (12), we just have to consider the ball $B(0,1)$, we ...
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### Local coefficients involved in the obstruction class for a lift of a map

I'm interested in understanding the importance of the local coefficients in the definition of the obstruction cocycle for a lift of a map $f\colon X \to B$ along a fibration $p \colon E \to B$. I'm ...
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### Finding a problem book in algebraic topology

I simply need book with problems solved with greatest explanation possible. I know about Hatcher and have a great lecturer, so I do not need theory. I need problems solved in detail.
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### Deck group of a connected n-fold cover must have at most n elements

Let $p:Y\to X$ be an $n$-fold covering map, with $Y$ connected. Show that $Deck(p)$ has at most $n$ elements. My thinking was to prove this by contradiction, i.e. suppose we have distinct ...
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### Deck transformation of the $n$-sheeted covering $\Bbb S^1 \to \Bbb S^1$

From Hatcher: For the $n$-sheeted covering space $S^1\to S^1$, $z\mapsto z^n$, the deck tranformations are the rotations of $S^1$ through angles that are multiples of $2\pi/n$. Why is this so? I ...
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### retraction of two surfaces (Hatcher 3.3.13)

This is problem no 13 in page no 258 of Hatcher's algebraic topology: Let $M'_h$ be a compact subsurface of genus $h$ with a boundary circle ,so $M'_h$ is homeomorphic to $M_h$ with one open disc ...
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### $\mathbb Q$ is totally disconnected. What is the open set in subspace of $\mathbb Q$?

I am trying to understand the proof that $\mathbb Q$ is totally disconnected. If $Y$ is a subspace of $\mathbb Q$ containing two points, $p$ and $q$, we can choose irrational a lying between $p$ and ...
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### Hatcher's notation and definition for the boundary of a singular chain complex

I have a hard time following Hatcher in general, but especially his notation in this case with the bar (page 108 in my pdf copy): For the boundary of a singular $n$-simplex, $\sigma$,he writes: ...
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### Every loop space $(\Omega Y,w_0)$ has the structure of an $H$-group.

The most important example of an $H$-group is the loop space $(\Omega Y,w_0)$ of any pointed space $(Y,y_0)$. Let $\mu:\Omega Y\times \Omega Y\to \Omega Y; \;\; \mu(\alpha,\beta)=\alpha \star\beta$, ...
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### Homotopy connectedness

I am trying to prove the following statement: Let $X,Y$ be two homotopy equivalent topological spaces. If $X$ is connected then $Y$ is connected. So far, this is my attempt: If $X,Y$ are homotopy ...
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### Fundamental group of a covering space

I understand the correspondence between the subgroups of a fundamental group $\pi_1(X)$ and the covering spaces of $X$. However, I do not understand what is implied about the fundamental groups of ...
I am reading a paper that states the following theorem without proof: Poincaré duality in middle dimension: Let $M$ be a connected oriented manifold of even dimension $2d$. Then the cup product ...
Is there an orientable, irreducible, non-compact, 3-manifold $M$ with $\partial M\cong \Sigma_2$ , genus 2 orientable surface, with $\pi_1(M)\cong \pi_1(\Sigma_2)$ and $M$ not \$\Sigma_2\times ...