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

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2
votes
1answer
40 views

How do we obtain the following identification

I don't understand geometrically why the identification below let us generate the shape on the right can someone explain or give me some intuition ?
1
vote
0answers
33 views

Question about the Hessian Criterion on a curve with singularity

So in class we have this theorem we call the Hessian Criterion: If we have a singular point in an affine curve in $\mathbb{C}^2$. Then $\frac{ \partial ^2 f}{\partial x^2}\frac{ \partial ^2 ...
3
votes
0answers
38 views

Correspondence defines embedding of $G_n(\mathbb{R}^m)$ into $G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = G_{n+1}(\mathbb{R}^{m+1})$? [closed]

Does the correspondence$$X \overset{f}{\to} \mathbb{R}^1 \oplus X$$defines an embedding of the Grassmann manifold $G_n(\mathbb{R}^m)$ into $$G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = ...
0
votes
0answers
16 views

Question Regarding Variaties finding the coresponding polynomial [closed]

Hi guys I have a general question. Say we have a set and we suspect it is a variety. How does one find the polynomial corresponding to it. I am thinking say the set ${(a^2,a^3+1)}$ where a is a ...
-1
votes
0answers
44 views

There is a theorem analogous to the Brouwer fixed-point for the 2-dimensional sphere?

Intuitively I think that for $f$: $\mathbb{S^2}\rightarrow\mathbb{S^2}$ a continuous function exist at least two fixed points. Using the same reasoning the statement of Brouwer fixed-point theorem I ...
2
votes
1answer
44 views

Given a column vector, can we add columns, continuously dependent on the given one, to get an invertible matrix?

Given a vector $x$ in the $n=6$-dimensional Euclidian space $\mathbb{R}^n$, do there exist $n-1$ continuous functions $f_1$ to $f_{n-1}$ such that the matrix $$(x,f_1(x), … ,f_{n-1}(x))$$ is ...
2
votes
1answer
38 views

How is identification is done in the definition of CW complexes

Consider the definition of CW complex from hatcher I am trying to understand the issue with the identification, because I feel there is something I don't understand. I decided to do an example and do ...
1
vote
1answer
25 views

Free group action on $S^n$ proof in Hatcher

Theorem: :$\mathbb{Z}_2$ is the only nontrivial group that can act freely on $S^n$ if $n$ is even. Proof: Since the degree of a homeomorphism must be $\pm 1$, an action of a group on $S^n$ determines ...
1
vote
0answers
47 views

Defining a continuous complex logarithm on open set $U \subset \mathbb{C}$

Suppose you are given an open set $U \subset \mathbb{C}$ and a continuous function $f: U \rightarrow \mathbb{C}-\{0\}$. And $f$ has the next property: For every closed loop $ c: I \rightarrow U$ ...
0
votes
0answers
52 views

Question about the induced Hurewicz isomorphism

In my notes it's claimed that the group homomorphism $$\Phi: \pi_{1}(X,x_{0}) \to H_{1}(X), \space \{f\} \mapsto[f]$$ clearly induces a group homomorphism $\Phi_{*}: \pi_{1}(X,x_{0})^{ab} \to ...
1
vote
0answers
51 views

Algebraic Topology problem on continuous functions over disks on sphere

I have come across the next problem, and I would like a little hint. Everything I'm thinking is not working or is a dead end. Let $f,g : D^2 \rightarrow S^2$ be continous maps such that $(x,y) \in ...
0
votes
0answers
30 views

A. A. Markov's paper on insolubility of the homeorphy problem [duplicate]

I am aware that this has been asked before, but the paper is nowhere to be found online, the provided link in the old thread leads to nowhere, and I'm really at wits end to find this paper, can anyone ...
6
votes
0answers
50 views

Constructing $\mathbb{P^n}$ “bundle” with different $n$

Can we find a complex/smooth manifold and map $f\colon X\to\mathbb{C}^3$ such that it is a $\mathbb{P}_\mathbb{C}^m$ bundle over linear subspace $\mathbb{C}^1$ and $\mathbb{P}_\mathbb{C}^n$ bundle ...
0
votes
1answer
40 views

Show that Affine Curves are not compact

Hi guys I have a question and not sure how to connect the dots. I am suppose to show that over a algebraically closed field $K=\mathbb{C}$. The affine variety in $K \times K$ is never compact. There ...
0
votes
0answers
27 views

A question in Hatcher's proof of homotopy lifting property [duplicate]

In Hatcher's Algebraic Topology p.30, in the last but one paragraph, he said: After replacing N by a smaller neighborhood of $y_0$, we may assume that $\tilde{F}(N\times t_i)$ is contained in ...
2
votes
2answers
60 views

Elementary geometric characterization of spheres?

I've read the following two theorems. Theorem. A compact connected metric space whose points are cuts points with the exception of at most two is homeomorphic to the unit interval. Theorem. A ...
0
votes
1answer
41 views

Can we get a torus by identifying surface with removed disc and mobius strip?

If we take a surface and remove a disc, then identify this resulting circle with the boundary circle.. does this produce a torus?
0
votes
0answers
63 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 ...
3
votes
1answer
81 views

Homology of knot exterior in general manifold for not null-homologus knot

I was trying to figure out the homology of the knot complement when $K$ is not a rational null-homologous knot ($[K]\neq 0\in H_1(X_K,\mathbb Q)$). We then know by half-lives half dies that the ...
3
votes
1answer
61 views

Compatibility between Unreduced Suspension Iso and Reduced Suspension Iso

I need some clarifications on these two "basic" things because I realised I was using them carelessly and now I want to know once and for all the relation between the two. Let us assume working with ...
0
votes
2answers
59 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 $H:I\times S^{\infty}\to S^{\infty}$ ...
4
votes
1answer
69 views

Complement of a knot that *isn't* rationally null-homologous

Let $K$ be a knot in a closed, oriented 3-manifold $Y$. It is a standard fact that if $K$ is (at least rationally) null-homologous, then $H_1(Y-K;\mathbb{Z})$ is isomorphic to $H_1(Y;\mathbb{Z})\oplus ...
0
votes
1answer
57 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
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
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 $n=2$. ...
1
vote
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 + ...
0
votes
0answers
12 views

Topological nets and triangulations

How does one construct a net and triangulation for a space? For example the identification space of the unit square with these identifications $(0,y)$~$(1, 1-y)$ for all $0 \leq y \leq 1$ ...
0
votes
2answers
48 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 ...
0
votes
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 ...
-1
votes
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 ...
3
votes
1answer
54 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
votes
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 ...
1
vote
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 ...
4
votes
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
votes
0answers
43 views

Compact 1-manifolds

I want to find all compact 1-manifolds. I think these will be lines, with fixed endpoints, and also the case where the endpoints meet. Hence I think all compact 1-manifolds are homeomorphic to the ...
0
votes
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
votes
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..
3
votes
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?
1
vote
0answers
36 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
votes
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 ...
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 ...
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 ...
1
vote
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
votes
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
42 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 ...