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

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31 views

Homology groups of the complex projective plane of dimension 2 - an affine line and a point not in the line

This is a question from a problem sheet we had in class and the solution says the following : We first note that the complex projective plane of dimension 2 minus an affine line is isomorphic to ...
1
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1answer
37 views

The annulus with with antipodal points on the outer circle identified gives a mobius strip

I ve been told that the real projective plane of dimension two can be expresses as the union of a disk and a mobius strip. The only way that this makes sense to me is that if an annulus with with ...
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0answers
24 views

Pontryagin classes of a tensor product of bundles

This question is related to " How to calculate characteristic classes of tensor products? " but interested in the Pontryagin classes instead of $c_1$. Specifically, given two real vector bundles $E$, ...
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2answers
114 views

Concatenating countably many homotopies

On page 15 of Hatcher's Algebraic Topology, he discusses constructing a homotopy $X \times I \to X \times \{0\} \cup A \times I$, where $(X,A)$ is a CW pair. He does so by concatenating homotopies ...
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1answer
46 views

Let Y and Z subspaces of X such that Y deformation retarcts to Z are their relative homology groups isomorphic?

I was wondering the following let $Y$ a subspace of a space $X$ and suppose there exists another subspace $Z$ of $X$ such that $Y$ deformation retracts to $Z$ does it then follow that ...
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2answers
15 views

Proving that the number of elements in inversed sets are equal

Define a cover mapping $f:Y\to X$ so that for all $x\in X$ the set $f^{-1}(x)$ is finite. Define a function $g:X\to \mathbb{Z}$ with $g(x) = \# (f^{-1}(x))$, as in: the number of elements in the set ...
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3answers
39 views

Let Y and Z be homotopy equivalent subspaces of X, are their relative homology groups isomorphic?

I was wondering the following let $Y$ a subspace of a space $X$ and suppose there exists another subspace $Z$ of $X$ such that $Y$ and $Z$ are homotopy equivalent does it then follow that $H_n(X,Y) ...
2
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0answers
43 views

References for equivariant cohomology

I am studying the paper An introduction to equivariant cohomology and homology, follwing Goresky, Kottwitz, and Macpherson - Julianna S. Tymoczko but there are too many gaps. I can't link most of ...
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1answer
43 views

Coherent Topology and Open Covers

Let $X$ be a topological space, and let $\mathcal{A}$ be an open cover for $X$. To say that $Open(X)$ is coherent with $\mathcal{A}$ means that $$B\in Open(X) \Leftrightarrow B\cap A\in ...
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0answers
89 views

Is a line with all points 'doubled" a differentiable manifold?

The line with two origins is $ X=\mathbb{R}∖\{0\}∪\{0',0''\}$, that is X is the union of the reals minus 0, and two points. Let, $$U_a=(−a,0)∪{0'}∪(0,a)$$ $$V_a=(−a,0)∪{0''}∪(0,a)$$ where $a>0$. ...
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1answer
20 views

Sphere bundle of the tangent bundle of 2-dim sphere

Let sph($\tau S^2$) be the sphere bundle of the tangent bundle of 2-dim$^l$ sphere. Could someone tell me why sph($\tau S^2$)=$\mathbb{R}$P$^2$$\cup$$e^3$ holds? Where $e^3$ is a 3-dim$^l$ cell.
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1answer
36 views

About the Chern class of the determinant line bundle

Is it true that the first Chern class of a rank $k$ complex line bundle is equal to the first Chern class of its determinant line bundle?
2
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1answer
31 views

Covering Map of Torus

how can I show that the following map is a covering map of $T:=$ $S^1$ x $S^1$? $\pi: T\rightarrow T$ with $(x,y)$ $\mapsto$ $(x^ay^b, x^cy^d)$, where $a,b,c,d \in \mathbb{Z}$ and $ad-bc=m\neq 0$. ...
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1answer
20 views

Definition of linear chains

I am reading Hatcher books in algebraic topology and he defined by $LC_n(Y)$ as the subgroup of $C_n(Y)$ generated by linear maps $\Delta^n$ $\to Y$ where $C_n(Y)$ is the abelian group of the $n$ ...
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1answer
88 views

Prove the long line is not contractible.

Given the following definition of the long line: Let $\omega_1$ be the first uncountable ordinal and consider $[0,1)$ as an ordinary set. Define the long ray to be the ordered set $\omega_1 \times ...
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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?
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30 views

Injectivity of precomposition with the suspension of the Hopf map

Let $h_2$ denote the usual hopf map and $h_n$ the $(n-2)$-th suspension of the hopf map i.e $h_n \colon S^{n+1} \to S^n $. Does anyone of you know a proof or a source of a proof of the fact that for ...
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 ...
3
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1answer
57 views

Induced map on homology by $f\colon S^4 \to S^2 \times S^2$

Show that $$f_* \colon H_4(S^4) \to H_4(S^2 \times S^2)$$ is the zero map for any $f\colon S^4 \to S^2 \times S^2$. We are working with integral coefficients. I tried applying the naturality of ...
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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 ...
2
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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 ...
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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 ...
2
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1answer
32 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 ...
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2answers
55 views

What is the homeomorphism type of the surface given by the polygonal presentation $aaa$?

More precisely, I am interested in the mapping cone of the map $S^1 \to S^1,$ $z \mapsto z^3.$ It seems like it should yield a "surface" with the following polygonal presentation: What is this ...
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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 ...
5
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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 ...
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 ...
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 ...
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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 ...
3
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0answers
36 views

Cobordism of two manifolds [closed]

Is $\mathbb RP^4 \times \mathbb RP^{12} \times \mathbb RP^{15}$ cobordant to $\mathbb RP^6 \times \mathbb RP^{9} \times \mathbb RP^{9} \times \mathbb RP^{7}$?
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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 $$H_1(K) \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$$ ...
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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
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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 ...
3
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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 ...
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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 ...
4
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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 ...
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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 ...
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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
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2answers
50 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 ...
3
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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: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
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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} ...
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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 ...
1
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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 ...
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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 ...
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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 ...
3
votes
2answers
101 views

intuition on the fundamental group of $S^1$

I am familiar with the proof that the fundamental group of the unit circle $S^1$ is $\mathbb Z$, yet I couldn't develop intuition for why it is true. For example, why would I fail if I try to find ...
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?
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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 ...
3
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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
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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 ...