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

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2
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0answers
24 views

Brouwer fixed-point theorem on non-convex sets [duplicate]

I read about the Brouwer fixed-point theorem in Wikipedia, and got confused about whether it holds when the domain of the function is non convex. On one hand, convexity is explicitly mentioned as a ...
1
vote
0answers
38 views

Is every regular curve homeomorphic to an interval I $\subset \mathbb{R}$ or to $\mathbb{S}^1$ or are there other posibilities?

Is every regular curve always homeomorphic to an interval or to $\mathbb{S}^1$? If so I would like to know why.
2
votes
3answers
151 views

Winding maps of spheres?

I know that the second homotopy group of $S^2$ is $\mathbb{Z}$. I also know that a simple representative of the class of maps that generates this group is the identity map on the sphere, ...
0
votes
1answer
37 views

How many connected components does the punctured cone of isotropic vectors have?

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
4
votes
1answer
89 views

If the action is free, is it necessarily a covering space action?

Suppose a group $G$ acts simplicially on a $\Delta$-complex $X$, where "simplicially" means that each element of $G$ takes each simplex of $X$ onto another simplex by a linear homeomorphism. If the ...
6
votes
1answer
54 views

When does covering preserve rational cohomology?

Let $X$ and $Y$ be compact manifolds. $p:X \rightarrow Y$ is a covering. Generally it is not true that $$ H^* (X , \mathbb{Q} ) = H^* (Y , \mathbb{Q} )$$ For instance, if $Y$ is a sphere with $y$ ...
5
votes
2answers
56 views

Prob. 5 (a) in Supplementary Exercises in Munkres' TOPOLOGY, 2nd ed: How to show that this map is a homeomorphism?

Let $G$ be a topplogical group, and let $H$ be a subgroup of $G$. Let $G / H $ denote the collection of all left cosets of $H$ in $G$, and let $a$ be a fixed element of $G$. Let the map $f \colon G / ...
3
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2answers
107 views

First book on algebraic topology

Is there a good book on introduction to Algebraic Topology which is self-contained and does not assume any background in topology, only standard calculus and linear algebra courses?
3
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0answers
35 views

The map $\lambda: H^*(\tilde{G}_n)\to H^*(\tilde{G}_{n-1})$ maps Pontryagin classes to Pontryagin classes; why?

In Milnor/Stasheff Characteristic classes on page 180 there is a statement (inside a proof) that I don't fully understand. Let $\tilde{G}_k$ be the oriented real grassmanian, $e$ its Euler class: ...
0
votes
0answers
24 views

An introduction to monodromy in topology and algebra

I'm looking for some basic references on monodromy and monodromy groups, in particular I'd like something which describes the interplay between the topological definition (in the theory of covering ...
6
votes
1answer
73 views

What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
0
votes
0answers
35 views

Cone of projective space?

Is the cone of $\mathbb{C}\mathbb{P}^2$ a familiar topological space? What about $\mathbb{C}\mathbb{P}^3$? I'm having a lot of trouble visualizing it. I just learned the notion of the cone of a ...
1
vote
0answers
38 views

Projection of a covering map over product set.

Let $p,q$ be a covering maps, $p:\tilde X \rightarrow X$ and $q:\tilde Y \rightarrow X$ and let $Z=\lbrace(\tilde x, \tilde y)| p(\tilde x)=q(\tilde y) \rbrace$, I want to proff that $f:\tilde ...
5
votes
1answer
46 views

Spectral sequence for computing the homotopy fixed points in unstable equivariant homotopy theory

I'm reading Carlsson's "A survey of equivariant homotopy theory" and I have a question. Let $G$ be a topological group and $X$ be a $G$-space (for a nice notion of "space"). He defines ...
1
vote
1answer
73 views

Homotopy lifting property of $\mathbb{R} \to S^1$ in Hatcher

I am reading Hatcher's proof of the homotopy lifting property of the covering map $p: \mathbb{R}\to S^1$. Starting with a homotopy $F: Y \times I \to S^1$ and a map $\tilde{F}:Y \times \{0\} \to ...
5
votes
1answer
45 views

If the top Stiefel-Whitney class of a compact manifold is nonzer0, must there be another non-vanishing Stiefel-Whitney class?

I was trying to collect some examples of Stiefel-Whitney class computations, just to make myself more familiar with them. It seems from my (relatively short list) that if the top Stiefel-Whitney ...
2
votes
1answer
36 views

$H_n(S^n,A)$ is not trivial

Let $(H_n)_{n\in \Bbb{Z}}, (\partial_n)_{n\in \Bbb{Z}}$ be an ordinary homology theory with values in the category of $R$-modules. Let $A\subset S^n$ be a proper subset. Then $H_n(S^n, A)$ is not ...
1
vote
0answers
59 views

Understanding cofibration sequence

Recently I'm studing some basic homotopy theory. An important brick of the exact sequence of cofibration is the following sequence: $$X \stackrel{f}{\longrightarrow} Y \stackrel{i}{\longrightarrow} C ...
1
vote
2answers
57 views

Splitting of Singular Homologies

In Singular homology, let $C_n(X)$ be the free abelian group generated by all the $n$-siimplices of the topological space $X$. Let $U$ be a subspace of $X$, then we have a spliting sequence ...
0
votes
2answers
66 views

Why does $\overline{\alpha * \beta}=\bar{\beta} * \bar{\alpha}$

I'm working on this question from Munkres' topology: Let $\alpha$ be a path in $X$ from $x_0$ to $x_1$; let $\beta$ be a path in $X$ from $x_1$ to $x_2$. Show that if $\gamma = \alpha * \beta$ , then ...
2
votes
1answer
57 views

Showing two spaces are homotopy equivalent

So I understand the basics about homotopy, I know a punctured disk or $\mathbb{R}- \{ 0 \}$ are homotopy equivalent to $\mathbb{S}^1$. This can be shown using the deformation retract ...
3
votes
1answer
32 views

Inclusions of CW-complexes are cofibrations.

Has the inclusion from the $ (n - 1) $-sphere in the $ n $-disc the left lifting property for all acyclic Serre fibrations? I am looking for a reference for this proposition, or alternatively, for an ...
0
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0answers
35 views

Proving two spaces are homotopy equivalent

We are given a topologic space X, defined as: $$X= \mathbb{S}^2 \cup \mathbb{D}_2 \cup \mathbb{I} \subset \mathbb{R}^3$$ Where $$\mathbb{S}^1=\{ (x,y,z) \in \mathbb{R}^3 | x^2+y^2+z^2=1 \} $$ ...
2
votes
1answer
46 views

Brouwer's fixed-point theorem and the intermediate value theorem?

In one dimension, Brouwer's fixed-point theorem (BPFT) can be proved easily based on the Intermediate Value Theorem (IVT). Is the inverse also true? I.e, is it possible to prove the IVT directly from ...
5
votes
2answers
47 views

Even number of points in the preimage of a regular value of a map $f:M^n \to \mathbb{R}P^n$

Let $M^n$ be a closed manifold (I think we can also assume it is connected, though it isn't explicitly stated). Let $f:M^n \to \mathbb{R}P^n$ be a smooth map, and let be $a \in \mathbb{R}P^n$ be a ...
0
votes
1answer
67 views

homology groups of a torus

How can I find the homology group of a torus without using cellular homology and the CW complex ? in other words , how can i calculate the homology groups of a torus using only relative homology ? I ...
-1
votes
0answers
15 views

degree of orientation-preserving map

Let's consider $f:X\rightarrow Y$ to be a $m$-covering space, with $X$ and $Y$ compact connected and oriented $n$-topological manifolds. Let $\alpha_{x}$ be a generator of $H_{n}(X,X\setminus\lbrace ...
4
votes
1answer
58 views

If $M$ is a 4-dimensional, compact, simply connected manifold with boundary, what is the topology of $\partial M$?

I know if we have a compact simply connected 3-manifold with boundary, then the boundary will be $S^2$. The argument is as following 1) $H_1(M)\simeq 0$; 2) Poincare duality $H_2(M,\partial ...
2
votes
1answer
48 views

Show that the comb space has fixed-point property

By "comb space", I mean the space $X=([0,1] \times \{0\}) \cup (K \times [0,1])$, where $K=\{ \frac{1}{n} : n \in \mathbb{N}^+ \}$, without the leftmost vertical line segment. How to prove that this ...
1
vote
1answer
46 views

Sufficient condition for $\mathbb{Z}$-orientability

Let $X$ be a topological $n$-manifold. Let's define a R-orientation on $X$ as a choice of generators $\alpha_{x}\in H_{n}(X,X\setminus\lbrace x\rbrace;R)$ that is consistent. Suppose that $X$ is ...
2
votes
0answers
54 views

Assignment - Euler characteristic constant under barycentric subdivision

I am working on an assignment and I am stuck, mostly I have no clue how to quite attack it. I don't want the answer or anything just advice on angels at which I can go about this. For an n-dimensional ...
6
votes
3answers
128 views

$S^m * S^n \approx S^{m+n+1}$

I'm interested in showing that $S^m * S^n \approx S^{m+n+1} $, as discussed in exercise 0.18 of Hatcher's Algebraic Topology. One way to show it would be to show that $X * Y \approx \Sigma(X \wedge ...
0
votes
0answers
35 views

Algorithm for finding zero of an odd function from n-sphere -> R^n

There is a well-known Borsuk-Ulam theorem stating that each continuous mapping $f : S^n \rightarrow \mathbb{R}^n$ that is odd in sence of $f(v) = -f(-v)$ for each $v \in S^n$ (where $-v$ denotes the ...
5
votes
1answer
54 views

Does every even-dimensional sphere admit an almost complex structure?

We know that there is an almost complex structure on $S^6$ which is not integrable. Is it always possible to find almost complex structures on $S^{2n}$? In particular does $S^4$ admit one?
0
votes
0answers
26 views

Where can I find Kan's paper “On c.s.s. categories” from 1957.

Does anyone know how I can find the following article by Daniel Kan: Kan, D. M. On c.s.s. categories Bull. Soc. Math. Mexicana (1957), 82-94. Quillen lists it as a reference in his paper Rational ...
1
vote
1answer
35 views

Understanding the generator of $H_1 (S^1 \times I)$

I'm trying to work with the space $X=S^1\times I$. It is obvious that $X\simeq S^1$ and therefore $H_1(X)=H_1(S^1)=\mathbb Z$, but I want the properties of $X$ itself. I would assume that a ...
0
votes
0answers
29 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
vote
1answer
35 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 ...
0
votes
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 $E$, ...
5
votes
2answers
92 views
+50

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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
3answers
38 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) ...
0
votes
2answers
115 views

Choosing a topic for a short talk in algebraic topology [closed]

I'm attending my second course in Algebraic Topology. Exam consists in preparing (and present) a talk. Course's arguments are classical results of Homotopy Theory. I had selected "Homotopy is not ...
2
votes
0answers
41 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 ...
0
votes
1answer
40 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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votes
0answers
80 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 $a>0$. ...
2
votes
1answer
19 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.
0
votes
1answer
29 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
votes
1answer
30 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 $ad-bc=m\neq 0$. ...