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

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0
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65 views

How much algebra one needs to study algebraic topology and homotopy theory?

I wonder how much algebra(group theory, abstract algebra, linear algebra, ring theory, field theory) is assumed as a prerequisite in most of the modern algebraic topology text. For example, these ...
6
votes
1answer
88 views

Can one prove that $\Bbb{CP}^\infty$ is a $K(\Bbb Z, 2)$ without invoking the long exact sequence of a fibration?

In trying to remind myself why $\Bbb{CP}^\infty$ is a $K(\Bbb Z, 2)$, the natural argument that comes to mind is to take the long exact sequence associated to the fibration $S^1 \rightarrow S^\infty ...
0
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2answers
60 views

Is $S^{\infty}$ contractible?

Recently I was reading this post: Unit sphere in $\mathbb{R}^\infty$ is contractible? Then a doubt came across to me: why I can't consider the linear homotopy $H:I\times S^{\infty}\to S^{\infty}$ ...
0
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1answer
59 views

What is the general structure of the complex curve $xy=y^2$?

How can you determine how a complex curve looks like in four dimensions, especially near singularities? In my example, the curve $xy=y^2$ consists of the lines $y=x$ and $y=0$ ($x,y$ complex). I think ...
3
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2answers
38 views

Trivial loop on the $1$-Skeleton

Following Hatcher's proof of Hurewicz Theorem (version of 1999) we arrive at the point that we must show that the loop in the picture, created following the path $0,1,2,3,1,0,1,3,0,3,2,0,2,1,0$, is ...
3
votes
1answer
58 views

Is the cohomology ring (coefficients in a field) functor right adjoint to something? Or, why does it commute with products?

Take coefficients in a field, so as to not have the correction from Tor. I am thinking about the functor sending a topological space $X$ to its cohomology ring $H^*(X)$. So specifically, I am ...
0
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0answers
21 views

Calculating fundamental groups using retracts

I want to know if my calculations are correct. I am trying to find the fundamental group of the figure $X_n$ in $\mathbb{R}^2$ obtained by drawing $n$ lines between two points (the lines only meet at ...
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1answer
50 views

Exact sequence splitting naturally

So I encountered a term that I don't quite recognize from lecture. The professor stated that a certain short exact sequence splits naturally, but I don't understand what the naturally condition is in ...
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0answers
25 views

Properties of loops under lifting

Let $p \colon (\tilde{X},\tilde{x}_0) \to (X,x_0) $ be a covering space. Is it always true that if the image of a path $\tilde{\gamma}$ under $p_*$ is a loop $\gamma$ based at $x_0$, then ...
0
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1answer
18 views

Matching faces in Simplicial Set theory

Let $X$ be a simplicial set. The elements $$x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$$ are said to be matching faces with respect to $i$ if $$d_jx_k=d_kx_{j+1}$$ for $j\geq k$ and $k,j+1\neq i$. ...
2
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0answers
27 views

Fibrant (Kan complex) geometric meaning

A simplicial set $X$ is said to be fibrant (or Kan complex) if it satisfies the following homotopy extension condition for each $i$: Let $x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$ be any ...
2
votes
1answer
35 views

Intersection of the sphere with the first octant is homeomorphic to the ball

I'm trying to show that $B$, which is the intersection of the n-sphere $S^n$ with the nonnegative octant of $\mathbb{R}^{n+1}$ is homeomorphic to the ball $B^n$. I see how to do this when $n=2$. ...
0
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0answers
25 views

Semi-infinite forms?

I am reading Vafa's paper 'Topological Mirros and Quantum Strings'(arXiv:hep-th/9111017). In this paper, the author says the Hilbert Space of a fermionic string theory corresponds to the space of ...
1
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0answers
13 views

Munkres positive linear map definition of path product (page 328)

I'm confused by Munkres' definition of the path product using the positive linear map. He defines the positive linear map $p: [a,b] \rightarrow [c,d]$ to be the unique map of the form $x \mapsto mx + ...
4
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1answer
42 views

Can we prove $H^1(X,\mathbb{Z})\cong Hom(\pi_1(X),\mathbb{Z})$ using torsors?

Let $X$ be a topological space, its first cohomology group $H^1(X,\mathbb{Z})$ classifies $\mathbb{Z}$-torsors over $X$. I think they are special kind of infinite sheet covering space of $X$. How can ...
0
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2answers
49 views

Compute explicitly a fundamental group

I want to compute the $\pi_1(X)$ where $$X=\mathbb{R}^2-(([-1,1]\times \{0\})\cup (\{0\}\times [-1,1]))$$ my only tools at the moment are the basic definitions and the fundamental group of a circle, I ...
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0answers
21 views

Prove exactness on the stalk

There are a number of results that were showed all in the same way : you have to prove that a certain sequence of sheaves is exact, for example : $0\longrightarrow F_{U}\longrightarrow ...
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0answers
29 views

Any solutions to problems in Harris book moduli of curves

Is there any place where I can find solutions to problems of the book Moduli of Curves. I am learning the subject, and want to do some of the problems. Also if there is any good source for problems ...
0
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0answers
24 views

Finding a map from $X = (0,\infty) \times (0,\infty)$ to a cone

Determine the quotientspace $X / \Gamma$, where $\Gamma = <\phi>$, $\phi(x,y) = (x/2,2y)$ and $X = (0,\infty) \times (0,\infty)$. I think the quotient space has to be a cone, but I can't figure ...
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0answers
24 views

why some locally constant sheaves aren't constant?

Many times you have to show that a certain sheaf, maybe obtained as gluing of other sheaves, is not constant ; there are methods or tricks immediate or generally to do this? What is special about a ...
2
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1answer
29 views

Making a Klein bottle from 2 Möbius bands

I thimk this can be done by idemtifying points on the boundary but I am not sure how to show this Any ideas? E.g. By drawing nets..
18
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4answers
1k views

Why Cohomology Groups?

Why do we need cohomology groups? Homology groups are easier to compute and given two topological spaces, there is an isomorphism in homology groups if and only if there is an isomorphism in ...
3
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0answers
34 views

$\mathbb{R}^n$ bundle over $B$ being compact and having finite covering dimension implies has finite type? [closed]

Let $\xi$ be a $\mathbb{R}^n$-bundle over $B$. If $B$ is paracompact and has finite covering dimension, does it follow that every $\xi$ over $B$ has finite type?
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0answers
37 views

Three complex and Euler characteristic zero

So this is an excerpt from Thurston's three manifolds text. He goes onto state that by constructing a complex by gluing faces of polyhedra we have the following condition. Such a complex is a manifold ...
0
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0answers
38 views

Properly discontinuous action of a group

Let $\Gamma=\{\varphi^n\mid n\in\mathbb{N}\}$ where $\varphi(x,y)=(\frac{x}{2^n},2^ny)$. I am trying to decide if $\Gamma$ defines a properly discontinuous action on $X=(0,\infty)\times(0,\infty)$. I ...
0
votes
1answer
41 views

A counter example for the homeomorphism between quotient product of coproduct and the space itself

I need an example that: For sets $X,Y$ in $\mathbb{R}$, s.t $X\cup Y=\mathbb{R}$, and $X\sqcup Y/\sim$ is not homeomorphic to $\mathbb{R}$, where $\sim$ means identifying the $x\in X$ and $y\in Y$ if ...
4
votes
1answer
46 views

Homology with local coefficients as a functor from pointed, path-connected spaces and $\pi_1$-modules.

A local system of coefficients on a space $X$ is a functor $F\colon \Pi(X)\rightarrow Ab$ from the fundamental groupoid to the category of abelian groups. From this, one can define the homology groups ...
1
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2answers
40 views

Homotopy equivalent but not deformation retraction [closed]

Can somebody come up with an example where $X \subset Y$, the inclusion gives a homotopy equivalence between $X$ and $Y$, but there is no deformation retraction from $Y$ onto $X$?
0
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2answers
46 views

When is the $i$-th homology group of the $p$-skeleton of a complex isomorphic to the $i$-th homology group of that complex?

For what values of $i$ is it true that $H_i(K^{(p)})\simeq H_i(K)$? My guess is that this is true for $i>dim K$. Otherwise, we can use $n$-simplex for a counterexample.
0
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1answer
43 views

Local diffeomorpism is a covering?

$D:\tilde{M}\rightarrow \Omega$ is a local isometry (developing map) onto a connected manifold $\Omega$. $\tilde{M}$ is simply connected, has constant sectional curvature and there exists a deck group ...
3
votes
1answer
44 views

Constructing an explicit non-contractible path in $\text{GL}_n(\mathbb{R})$

As can be seen here, the fundamental group of $\text{GL}_n(\mathbb{R})$ is $\mathbb{Z}/2\mathbb{Z}$ (for $n \ge 3$). (For $n=2$ it is $\mathbb{Z}$). Is there a way to find an explicit representing ...
2
votes
1answer
49 views

$n \times n$ invertible matrix defining diffeomorphism

I was reading the proof of the Hairy Ball Theorem in Madsen and Tornehave's book "From calculus to cohomology", and at some point they refer to the Lemma 6.14, which says the following: An invertible ...
4
votes
2answers
128 views

Calculate the Wu class from the Stiefel-Whitney class

The total Stiefel-Whitney class $w=1+w_1+w_2+\cdots$ is related to the total Wu class $u=1+u_1+u_2+\cdots$: The total Stiefel-Whitney class $w$ is the Steenrod square of the Wu class $u$: ...
4
votes
2answers
64 views

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$?

If $\mathbb{RP}^n$ can be immersed in $\mathbb{R}^{n+1}$, how do I see that $n$ must be of the form $2^r - 1$ or $2^r - 2$? For starters, I know that if the $n$-dimensional $M$ can be immersed in ...
6
votes
3answers
209 views

Fundamental groups of codimension 1 manifold complements

Let $M$ be a smooth manifold of dimension at most $3$ and $S \subset M$ a smoothly embedded compact connected codimension $1$ manifold, separating $M$ into two components, $M_1$ and $M_2$. I wonder ...
1
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0answers
45 views

normal bundles are stably isotopic

I am searching for a reference about the following theorem: Let $M \xrightarrow{i_{0}} \mathbb{R}^{k},\,\,$ $M \xrightarrow{i_{1}} \mathbb{R}^{l}$ be two embeddings of the manifold $M$. We know by ...
0
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0answers
25 views

Relation between compactly supported cohomology and locally finite homology

Building up on a previous question, I am currently investigating in the properties of locally finite homology. Suppose that $X$ is a reasonably well-behaved space. I want to find out whether there is ...
0
votes
1answer
52 views

What does “linear and injective on each fiber” really mean?

The question is about the proof of the following result: For a paracompact space $B$, the map $[B, \operatorname{Gr}_k] \to \operatorname{Vect}^k(B)$, $[f] \mapsto f^*(\gamma_k)$ is a bijection, ...
1
vote
1answer
51 views

Hatcher Exercise 0.19 Attaching $2$-cells to $S^2$

Hatcher has the following exercise in chapter $0$: Show that the space obtained from $S^2$ by attaching $n$ 2 cells along any collection of $n$ circles in $S^2$ is homotopy equivalent to the wedge ...
3
votes
1answer
32 views

On the boundary map of a locally finite chain complex

I am just learning about locally finite homology and I'm having a bit trouble understanding some of its concepts. There doesn't seem to be a whole lot of (non-advanced) literature on this topic, so I ...
4
votes
1answer
40 views

Basic question on cohomology ring

To show (1) $S^2\vee S^1\vee S^1$ is not homotopy equivalent to $S^1\times S^1$ (2) $S^1\vee S^2\vee S^3$ is not homotopy equivalent to $S^1\times S^2$ I use the same method: For (1) the ...
1
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0answers
30 views

Fibration and induced mapping

I need help in such a problem, suppose it's rather simple one, but I have no skill in a field.. If $p: E\rightarrow B$ - fibration with a fibre F, then for every locally compact space $X$ an induced ...
2
votes
1answer
34 views

How is this “Stiefel manifold”-like fiber bundle called? How to characterize it?

Let $M$ be a smooth manifold of dimension $n$. Consider the space $V_k(M)$ of pairs $(x,\alpha)$ where $x \in M$ and $\alpha$ is a linear embedding $\mathbb{R}^k \hookrightarrow T_x M$ (or ...
1
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1answer
27 views

Properly discontinuous action on homology

Let $\Gamma$ be a finite group with a properly discontinuous action on $X$. How can I show that $\Gamma$ acts on $H_k(X)_{\mathbb{Q}}$? It's not clear to me why I need to take rational coefficients.
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2answers
42 views

Is there any expression to calculate the homology groups of a quotient space?

Let $B \subset A$ where $A$ is a topological space and $A/B$ the space obtained from $A$ via collapsing $B$ to a single point. I was wondering if there is any expression for $H_k(A/B)$ in terms of ...
0
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2answers
45 views

How to show a straight line homotopy is continuous?

Given $f$ and $g$ continuous maps from $X$ into $\mathbb{R}^{2}$, how to show that the straight line homotopy map $F(x,t)=(1-t)f(x)+tg(x)$ is continuous?
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0answers
28 views

Homology of SO(3)

In Tohru Eguchi, Peter B. Gilkey, and Andrew J. Hanson's review paper "Gravitation, gauge theory and differential geometry," I came across the following claim about the Homology of SO(3): I cannot ...
0
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1answer
35 views

Cohomology of Eilenberg Maclane space

In a book on spectral sequences that I am reading, it is stated, without proof, that $H^i(K(\mathbb{Z},2);H^0(K(\mathbb{Z},1);\mathbb{Z}))$ is isomorphic to $\mathbb{Z}$ for even $i$ and $0$ for odd ...
1
vote
1answer
28 views

Definition of orientable as given in Hatcher's Algebraic Topology

A $\textbf{local orientation}$ of a manifold $M$ at a point $x$ is a choice of generator $\mu_x$ of the infinite cyclic group $H_n(M, M- \{x\} )$. For example, in the case of $M= \mathbb{R}^n$, ...
0
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0answers
47 views

Deformation Retract of Complement of Two Linked Circles in $\mathbf R^3$

On pg. 47 of Hathcer's Algebraic Topology, the author discusses the fundamental group of $\mathbf R^n-(A\cup B)$, where $A$ and $B$ are circles in $\mathbf R^3$ which are linked. The author writes ...