Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

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1
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1answer
46 views

Pull-back of a fibration along a homotopy equivalence

Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\rightarrow B$ and $g:B\rightarrow B'$ be homotopy inverses. Denote by ...
5
votes
1answer
69 views

How general $ [X,[Y,Z]] \cong [X \times Y, Z] $ is?

In some algebraic categories (abelian groups, modules on a ring) and in pointed spaces categories the following holds: $$ [X,[Y,Z]] \cong [X \times Y, Z] $$ In wich generality this lemma holds?
4
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2answers
51 views

Torsion of homology group

If $U$ is an open connected subset of $\mathbb{R}^n$ where $n\ge 2$,is it true that $H_1(U,\mathbb{Z})$ is torsion free?Or in general,$H_i(U)$ is free?I am thinking whether it has deformation retract ...
0
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0answers
31 views

The complement of the image of the zero section is still a $\mathbb{G}_m$-torsor?

This came up while doing some reading Schneps text on Galois Groups and Fundamental groups, but it's glossed over. In any case, suppose that you have a line bundle over a scheme $L\to X$, with zero ...
0
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2answers
53 views

Does the singular homology functor preserve injectivity and surjectivity?

I was wondering if the singular homology functor preserve injectivity and surjectivity? I've been trying to figure out a proof or counterexample for ages now and I just can't. This came up when I was ...
2
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1answer
31 views

degree of a self map on the sphere

could you help me with this one? How do I determine the degree of the continous map $\mathbb{S}^n\to \mathbb{S}^n$ induced by multiplication with an orthogonal matrix $A$? I think it should be ...
2
votes
1answer
73 views

Higher homology groups of infinite cyclic cover

Prove that all homology groups of the infinite cyclic cover of a knot complement are trivial except $H_1$. I've posted an answer below using Mayer-Vietoris. If you know of other arguments, please ...
0
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1answer
38 views

Small doubt about the connecting homomorphism on the long homology sequence

When you consider the long homology sequence (of spaces $A,X$ , with $A$ subspace of $X$) you need to define an homomorphism from $H_q(X,A)$ to $H_{q-1}(A)$ to obtain the long homology sequence from ...
5
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1answer
52 views

Tautological line bundle over $\mathbb{RP}^n$ isomorphic to normal bundle? Also “splitting” of transition functions

Hallo fellow mathematicians. I try to understand why the normal bundle of $\mathbb{PR}^n$ is isomorphic (in the category of vector bundles) to the tautological line Bundle. More aptly, why ...
0
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0answers
52 views

cohomology ring of a subspace of real projective spaces

I learned $H^*(\mathbb{R}P^n;\mathbb{Z}_2)=\mathbb{Z}_2[a]/(a^{n+1})$, $|a|=1$, in topology class, when studying cell complex and cohomology. Now I want to find the cohomology ring ...
5
votes
0answers
47 views

No retract $X \wedge \mathbb{R}P^2 \to X \wedge \mathbb{R}P^1$

Let $X$ be a finite CW complex, and suppose $\Sigma X \cong X \wedge \mathbb{R}P^1$ is not contractible. By considering the fundamental group or otherwise, it is easy to see that there can be no ...
3
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1answer
42 views

almost complex structures on $R^4$

How should I see that the set of almost complex structures on $R^4$ preserving the positive orientation, namely $\{J\in GL^{+}(4,R), J^2=-I\}$ is homotopy equialent to $S^2$. There is a similar ...
1
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0answers
28 views

$\beta_{q}=\dim H_{q}(X,\mathbb{Q})$

Let's define $\beta_{q}$ to be $q^{th}$-Betti number of X, i.e. the rank of of $H_{q}(X,\mathbb{Z})$, the $q^{th}$-homology of $X$. How can I see that $\beta_{q}=\dim H_{q}(X,\mathbb{Q})$, where ...
88
votes
18answers
9k views

How to distinguish walking on a sphere or on a torus?

Imagine that you're a flatlander walking in your world. How could you distinguish if the world is a sphere or a torus ? I can't see the difference from this point of view. If you are interested, this ...
3
votes
0answers
36 views

Cobordism of two manifolds [closed]

Is $\mathbb RP^4 \times \mathbb RP^{12} \times \mathbb RP^{15}$ cobordant to $\mathbb RP^6 \times \mathbb RP^{9} \times \mathbb RP^{9} \times \mathbb RP^{7}$?
-4
votes
1answer
83 views

Homology groups of retracts of algebraic topological spaces

Sup, recently I had an extremely vivid dream about algebraic topology. In it, I computed the homology group of the Klien Bottle $K$ to be $$H_1(K) \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$$ ...
5
votes
1answer
41 views

What is the relation between cohomology of $K(\mathbf{Z}_2, n)$ and $K(\mathbf{Z}_2, n+1)$?

I'm reading a proof to a theorem, "Suppose a, b: $H^*(-, \mathbf{Z}_2)\to H^{*+k}(-, \mathbf{Z}_2)$ are two stable (commuting with suspension isomorphism) cohomology operation of degree k. If ...
4
votes
1answer
51 views

Chern classes mod 2 equal Stiefel-Whitney classes via Milnor/Stasheff language

I'm having truble with Exercise 14B of Milnor/Stesheff Characteristic classes: prove that the total Chern class of a comple bundle is mapped to the Stiefel-Whitney class by the coefficient ...
3
votes
1answer
33 views

Relation between cohomology of Eilenberg- MacLane space and product of projective space

In an article, it says that "Consider the map $\mathbf{RP}^\infty\times\cdots\times\mathbf{RP}^\infty$(n copies) $\to$ $K(\mathbf{Z}_2,n)$", I think this map is the map related to killing homotopy to ...
1
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1answer
48 views

There is no equivariant map $f:S^2 \to S^1$

To fix some notation, let $n \geq 2$ and let $p:S^n \to P^n$ be the canonical double cover. Let $\gamma:I \to S^n$ be a lift of a representative of a nontrivial element in $\pi_1(P_n) \cong ...
2
votes
2answers
49 views

Find all covering spaces of $\mathbb{RP}^n \times \mathbb{RP}^n$, $n>1$

Let $X = \mathbb{RP}^n \times \mathbb{RP}^n$. I know the following: the universal cover of $X$ is $Y = \Bbb S^n \times \Bbb S^n$ the fundamental group of $X$ is $G = \Bbb Z/2 \Bbb Z \times \Bbb Z/2 ...
1
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1answer
41 views

Difficulty in understanding the Dyadic map and its application

The Dyadic map also called as the Bernoulli Shift map is expressed as $$x(k+1) = 2x(k) \bmod 1$$. Consider a discrete map $F : X \rightarrow X$ in the interval. Let this map be the Tent Map. In Link1: ...
0
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0answers
18 views

Theory around the Cellular Sheaf

I have lately stumbled upon cellular (co)sheaves, which look very interesting. To understand them better, I would like references that systematically develop the theory behind them (preferably in ...
1
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1answer
21 views

Prob. 10 (b), Sec. 25 in Munkres' TOPOLOGY, 2nd ed: How to show that components and quasicomponents are the same for locally connected spaces?

Let $X$ be a topological space; let us define $x \sim y$ if there is no separation $X = A \cup B$ of $X$ into disjoint open sets such that $x \in A$ and $y \in B$. This relation is an equivalence ...
2
votes
1answer
38 views

Prove $H_0 (X,A)=\bigoplus H_0 (X_i,X_i\cap A)$ for $X_i$ the path components

Let $X$ be a topological spcae, $X_i$ its path components, and $A\subset X$ a subspace. I'm interested in proving $H_0(X,A)=\bigoplus H_0(X_i,X_i\cap A)$. my work by now (I have completely proved ...
3
votes
0answers
24 views

Multiplicative structure in the cohomological Leray-Serre spectral sequence — please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
1
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1answer
38 views

why is $H_0(A)\overset {i_*}{\to} H_0(X)$ injective?

Let $X$ be a a topological space, $A\subset X$. I've been told that it is "trivial" that if each path component of $X$ contains at most one path component of $A$ then $H_0(A)\overset {i_*}{\to} ...
2
votes
3answers
190 views

Is a path connected covering space of a path connected space always surjective?

If $X$ is a path connected topological space, a covering space of $X$ is a space $\tilde{X}$ and a map $p:\tilde{X} \to X$ such that there exists an open cover $\left\{ U_\alpha \right\}$ of $X$ where ...
1
vote
1answer
44 views

is the homomorphism induced by the inclusion map is the inclusion map?

let $X$ be a topological space, $A\subset X$ subspace. Consider $i:\:A\to X$ the inclusion, and $i_* :\:H_n(A)\to H_n(X)$ the induced homomorphism. is $i_*$ the natural homomorphism $[a]\mapsto ...
0
votes
0answers
38 views

What does it mean if two maps are not homotopic?

If you have two maps, $f,g:X\rightarrow Y$ between cell complexes and the maps are homotopic, they will clearly induce the same map on homotopy groups. But what if they are not homotopic? What can we ...
0
votes
1answer
57 views

Why is the inclusion an isomorphism?

Consider $X$ a path-connected space, $A\subset X$ a non-empty subset. My textbook makes the following claim without any explanation, and I wondered if you could help: it says that the inclusion $H_0 ...
0
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0answers
64 views

Comments on Eilenberg and Steenrod's “Foundations of algebraic topology” and other similar books for recomendation

The biggest obstacle for me to learn geometry and topology is the haziness of textbooks. I took algebraic topology last semester and the textbook we used in class was Rotman's "An introduction to ...
2
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0answers
42 views

what do the branch points of a function tell about the function?

If we know the zeros of an entire function, we have the Weierstrasse construction of the function. How about the branch points? Is there any topology involved?
1
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0answers
73 views

Covering Space of $\mathbb{C}-\{a,b\}$ via Multivalued Function

Consider the multivalued complex function $f(z)= \sqrt{z-a}+\sqrt{z-b}$, where $a\neq b$, defined in the domain $U=\mathbb{C}-\{a,b\}$. The graph $W$ of $f$ defines a regular covering space $W ...
3
votes
0answers
38 views

The relation between homotopy equivalence and contractible mapping cone?

In this MO thread, the OP claimed that it is obvious that homotopy equivalence implies the mapping cone contractible, whereas the converse proposition is wrong. I hate to admit that it's not obvious ...
2
votes
0answers
64 views

Maps of degree $k$ and their behavior on higher homotopy groups

Suppose $f \colon S^n \to S^n$ is a map of degree $k$ and suppose we know what group $\pi_j (S^n)$ for $j > n$ is. Is it possible to know what kind of map is induced on $\pi_j(S^n)$ by $f$? For ...
2
votes
0answers
44 views

Kernel of an homomorphism between two free groups

I am trying to prove that $G=\langle \alpha,\beta,\gamma \mid \alpha\beta\alpha^{-1}\beta^{-1}\gamma \rangle$ is isomorphic to $H=\langle \delta,\varepsilon \mid \hspace{0.5cm}\rangle$. Let N be the ...
1
vote
1answer
29 views

Direct sum of nontrivial vector bundles?

When is it true that the direct sum (or whitney sum) of two nontrivial vector bundles is nontrivial? Also, if you have a direct sum of vector bundles, with $a$ and $b$ global sections respectively, ...
3
votes
2answers
111 views

homotopic maps?

For cell complexes, Whitehead's theorem says that a weak homotopy equivalence is an actual homotopy equivalence. More generally, if I have two maps between cell complexes which agree on homotopy ...
3
votes
1answer
43 views

Showing that gluing two knot exteriors together contains subgroups isomorphic with the knot groups.

I'm working through Rolfsen's "Knots and Links" and section 9D exercise 10 has me stumped: Let $K_1$ and $K_2$ be knots in two separate copies of $S^3$ with respective meridians $m_1$ and $m_2$ and ...
0
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1answer
534 views

Locally star-shaped space and piecewise linear path

I'm trying to do exercise 4 on page 38 in Hatcher. Can you tell me if this is right? claim: $X$ locally star-shaped, $\gamma$ a path in $X$ then there is a path consisting of a finite number of line ...
7
votes
0answers
164 views

Fundamental group of the complement homogeneous variety in $\mathbb{C}P^{2}$

Let $f,g:\mathbb{C}^3\to \mathbb{C}$ are two irreducible homogeneous polynomials. If there is a homeomorphism $h:\mathbb{C}^3\to \mathbb{C}^3$ such that $h(X)=Y$ and $h(0)=0$, where $X=f^{-1}(0)$ and ...
0
votes
1answer
35 views

Universal covers of the rotation group

Is it correct to say that the universal cover of the rotation group SO(n) is always a double cover? Does this hold when we generalise to SO(p,q)?
1
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1answer
66 views

Covering $S_2$ with $S_3$(or $S_n$)

How can I construct a covering map $p : S_3 \to S_2$? I can construct coverings for $S^1\vee S^1$, but same ideas don't work for $S_2$ ($S_g$ is sphere with $g$ handles).
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0answers
23 views

Automorphisms of simple covers of Riemann surfaces

Can anybody give me a simple proof that simple covers of a Riemann surface have no covering automorphisms?
2
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1answer
47 views

Getting fiber bundles from short exact sequences

Are there conditions that guarantee that a split short exact sequence of groups $$ 1 \rightarrow K \rightarrow G \rightarrow Q \rightarrow 1 $$ gives rise to a fiber bundle $$ F \rightarrow E ...
0
votes
2answers
55 views

Regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $.

I tried to draw the regular and non-regular covering spaces of $ \Bbb{S}^{1} \vee \Bbb{S}^{1} \vee \Bbb{S}^{1} $. I think the regular covering space is: Is it true? How do you draw the non-regular ...
2
votes
1answer
35 views

Direct proof of decomposition of real vector bundle of odd degree into the direct sum of a trivial bundle and another of even degree

The real splitting principle tells us that when taking a real, oriented vector bundle of odd dimension $\zeta$ over a manifold $M$ you can always write $\zeta$ as $\tilde{\zeta} \oplus \varepsilon^1$, ...
0
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0answers
58 views

Gromov's defenition of Content of Ball

Let $B(p, R)$ denote the metric ball of radius $R$ centered at $p$ in a manifold. Then Gromov defined the content of the ball by $$Cont(B(p,R))=rank(H_*(B(p, R/5))\to H_*(B(p,R))) $$ and he remark ...
1
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1answer
20 views

Intersection of an open neigbourhood of a mobius strip and and a disk

I have seen that the real projective plane of dimension 2 can be expressed as a union of a disk and a mobius strip but what I don't understand is that why their intersection is a cylinder? Can ...