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
56 views

Some Quotient Space $(\prod_{i=1}^n S^2/\bigvee_{i=1}^n S^2 )^4= \bigvee_{1\leq i< j\leq n } S^4$

I want to know a 4-skelecton of quotient space $(\prod_{i=1}^n S^2/\bigvee_{i=1}^n S^2 )^4 = \bigvee_{1\leq i < j\leq n} S^4$ If $n=2$, then I can realize. But in $ n=3$ case I have no idea. ...
4
votes
1answer
65 views

Aspherical homology class

I am completely stuck on the following algebraic topology exercise: Let $X$ and $Y$ be CW complexes and $\alpha \in H_p(X)$, $\beta \in H_q(Y)$, $p, q > 0$, homology classes such that the homology ...
3
votes
2answers
525 views

Why is the infinite sphere contractible?

Why is the infinite sphere contractible? I know a proof from Hatcher p. 88, but I don't understand how this is possible. I really understand the statement and the proof, but in my imagination this is ...
3
votes
1answer
166 views

Global Section for Hopf Fibration

I want to know the existence of global section of $\pi : M\rightarrow M/G$, where $M$ is a Riemannian manifold with $G$-action. For instance in case of $M=S^2$ and $G={\bf Z}_2$ there exists no ...
2
votes
1answer
84 views

Why can an orientation on $X$ be written as a sum of cycles on this open cover?

I'm reading a proof of the following theorem Let $X$ and $Y$ be compact connected oriented $n$-manifolds, and let $f:X\to Y$ be continuous. Let $y\in Y$, and assume that $f^{-1}(y)$ is finite. ...
1
vote
1answer
170 views

Help understanding CW-complex construction.

From p.5 of Hatcher's Algebraic Topology: I thought attachment maps would be from $n$-cell to $n$-cell since it says "by attaching $n$-cells", but in that link they're from $S^{n-1}$ to ...
4
votes
0answers
105 views

deck transformations and covering spaces

Let $p:\tilde X\rightarrow X$ be a universal covering space, and let $H\leq G$ where $G$ is the group of covering transformations. Let $q:\tilde X \rightarrow \tilde X/G$ be the quotient map which is ...
5
votes
1answer
98 views

How hard is it to endow a $\textit{Spin}^{c}$ structure on four-dimensional manifolds?

I am in a certain math conference and we came across Seiberg-Witten equations. Since I am really novice in the field, I asked if all "reasonable" four manifolds carry a $\textit{spin}^{c}$ structure. ...
2
votes
1answer
127 views

Can't follow a proof in Hatcher's book

I have trouble in following one proof in Hatcher's Algebraic topology. On Page 210, Section 3.2 cup product, in Example 3.11 ($n$-Torus) it is claimed that the sequence $$ 0\to H^n(I\times Y; R)\to ...
5
votes
2answers
165 views

Textbooks with exposition done mostly in proof outlines or exercises?

As the title indicates, I'm trying to find books where the exposition of the main course of thought is done entirely or mostly in outlines of proofs, or as exercises with or without hints. I'm trying ...
2
votes
0answers
56 views

The Lefschetz Fixed Point Theorem in Coefficient Ring $G$

As you know Lefschetz number is $\tau (f)\doteq \sum_n (-1)^n {\rm tr}(f_\ast : H_n(X)\rightarrow H_n(X))$ where $X$ is a finite simplicial complex and $f : X\rightarrow X$. So LFPT is that if ...
2
votes
1answer
278 views

Alternative definition of covering spaces.

in a lecture I have seen a definition of a covering space, different from what I would call the usual one (e.g. the one in Munkres): A surjective continuous map $p:E\rightarrow B$ of spaces $E$ and ...
3
votes
1answer
110 views

Relative homology groups

The local homology of a manifold $X$ at a point $x$ is defined as the relative homology $H_n(X, X-{x};\ \mathbb Z)$. It holds true for relative homology that under certain conditions $H(X,A) = ...
1
vote
1answer
28 views

If $M=U\cup V$, $U$ and $V$ have finite dimensional cohomology then $U\cap V$ has finite dimensional cohomology..

I need some help with the following: Let $M$ be a differentiable manifold such that $$\textrm{dim}(H^k(M))<\infty$$ for every $k=0, \ldots, n$ where $H^k(M)$ is the $k$-th De Rham cohomology group ...
0
votes
1answer
203 views

About zeros of vector fields in compact surfaces

I'm studying compact surfaces and in particular the relationship between zeros of vector fields defined on them and Euler characteristic of the surface herself. Let be $S$ a compact (smooth) surface ...
3
votes
1answer
80 views

Cohomology calculation for maps to the 2-sphere.

Let $Y^3$ be a closed 3-manifold and $f\colon Y\to \operatorname{SO}(3)$, $g\colon Y\to S^2$ be smooth maps. Define $g'\colon Y\to S^2$ be the following composition: ...
2
votes
0answers
44 views

Acyclic Hurewicz fibrations

If I understand correctly, the claim below follows from some well-known facts about the Quillen and Hurewicz model structures on the category of all topological spaces: If $p : X \to Y$ is a ...
2
votes
0answers
117 views

Zero exponent sum w.r.t group words in knot group's presentation

I am reading, "Plane Curves Associated to Character Varieties of 3-Manifolds" by Cooper, Culler, Gillet, Long, and Shalen and on page 28 ( http://www.math.uic.edu/~culler/papers/PlaneCurves/curves.pdf ...
1
vote
1answer
73 views

Showing homotopy of two paths if they are homotopic after a delay

Let $X$ be a topological space and let $\gamma, \delta : [0,1] \rightarrow X$ be two paths from $x$ to $y$. Now define $\widehat{\gamma}: [0,2] \rightarrow X$ by $$\widehat{\gamma}(t) = \begin{cases} ...
3
votes
1answer
208 views

What is the cone over a simplicial set?

At the moment I'm reading through Edward B. Curtis, 'Simplicial Homotopy Theory' (Advances in Mathematics 6, 107-209 (1971)) in order to learn about simplicial sets and I run into a problem where the ...
4
votes
1answer
160 views

There is no continuous map from $D^2\to S^1$ such that…

Proposition: There is no continuous map from the unit disc $D^2$ to its boundary $S^1$ whose restriction to $S^1$ is the identity on $S^1$. My proof: Assume that there is such an $f$. Let $g: S^1\to ...
4
votes
0answers
193 views

Leray-Hirsch theorem

I'am studying the book "Bott, Tu Differential forms in algebraic topology." I don't understand the proof of Leray-Hirsch theorem via Cech-de Rham complex. Lets consider some bundle $\pi: E \mapsto ...
11
votes
1answer
172 views

Is there a nontrivial topological group that's isomorphic to its fundamental group?

All I know is that the topological group has to be Abelian. I have no idea how to prove or disprove this statement. Thanks in advance.
1
vote
1answer
163 views

Connectedness implies the equinumerosity of fibers

I need to show that if $X$ is a covering space of $Y$ with the covering map $p$ and $Y$ is connected, then $p^{-1}(y)$ have the same cardinality for every $y\in Y$. I have this hint: A function ...
3
votes
1answer
190 views

Orientation on manifold in terms of homology

One can define the orientation of a manifold $M$ in terms of relative homology groups $H_n (M, M-\{p\})$ by noting that $H_n (M, M-\{p\}) \cong \mathbb Z$ and then designating a generator. Is it ...
3
votes
1answer
119 views

Covering a genus g surface with n disks, and conversely

Suppose $X$ is a closed compact surface (i.e. a compact two-dimensional manifold without boundary). If $X$ has genus $g$, what is the minimum number of (homeomorphic images of) open balls required ...
3
votes
1answer
61 views

How does a group action on a space induce action on cohomology ring of the space?

This is mostly a question to make sure I have some signs correct. Let a group $G$ act on a space $X$. As notation, for $g \in G$ let $\phi_g : X \rightarrow X$ be the map induced by the action of ...
1
vote
0answers
25 views

Let $S^3 = \{(z_1, z_2) \in \mathbb{C}^2 | ||z_1||^2 + ||z_2||^2 = 1\}$, $D^2 = \{z \in \mathbb{C} | ||z|| \leq 1\}$.

Let $S^3 = \{(z_1, z_2) \in \mathbb{C}^2 | ||z_1||^2 + ||z_2||^2 = 1\}$, $D^2 = \{z \in \mathbb{C} | ||z|| \leq 1\}$. Let $X$ be the quotient of the disjoint union of $D^2$ and $S^3$ by the smallest ...
1
vote
1answer
66 views

Show that $Y$ is locally compact.

Recall that a space $X$ is locally compact if for any point $x$ in $X$, and any neighborhood $U$ of $x$ in $X$,there is a neighborhood $V$ of $x$ in $X$,and a compact subspace $C$ of $X$ such that $x ...
7
votes
2answers
892 views

map of arbitrary degree from compact oriented manifold into sphere

This is a question from a qualifying exam. Let $X$ be a compact, oriented $n$-dimensional manifold. Show that for any $k \in \mathbb{Z}$, there exists a continuous map $f: X \to S^n$ of degree $k$. I ...
4
votes
0answers
169 views

Cohomology of the pullback of a fiber bundle over the torus (generalised Eilenberg-Moore spectral sequence?)

I've found myself in the following situation and the tools that have been suggested to me to calculate the necessary invariants don't quite seem up to the job (I may be wrong in this as I have very ...
0
votes
1answer
60 views

A question about double cover of Lie group

If the fundamental group of a symplrctic Lie group be infinite cyclic, why it should has a unique connected double cover?
2
votes
1answer
291 views

Is there a definition of the transfer homomorphism (between cohomology of cover and base) without referring to chains?

Let $\pi: \tilde{X} \rightarrow X$ be an n-sheeted covering. Hatcher (section 3G), defines the transfer homomorphism, $\pi^*: H^k(\tilde{X}, Z) \rightarrow H^k(X, Z)$ on the chain level by sending ...
2
votes
1answer
73 views

Double cover of symplectic groups

What is the normal definition of double cover of Symplectic group? I couldn't find a simple and understandable definition
4
votes
1answer
181 views

Simplicial homology of a tetrahedron

I'm studying the basics of homology on Nakahara, Geometry, Topology and Physics and I'm trying to work out the (simplicial) homology group of the tetrahedron described by the complex ...
1
vote
1answer
65 views

Simple question concerning the properties of the fundamental group

I need to prove that every element of the fundamental group has an inverse. First we define a map $\phi:I\to I$ homotopic to $\operatorname{Id}_I$. If $\phi$ is the constant zero function isn't it ...
1
vote
1answer
74 views

Lifting homeomorphisms covering

Hello I had a question regarding a lemma from the paper: http://www.math.columbia.edu/~jb/bir-hilden-annals.pdf I don't understand the proof of Lemma 5.1. Notation: $T_{0,0}$ is the 2-sphere, ...
1
vote
0answers
35 views

Relative homology of interlevel set

Let us consider a function $f\colon \mathbb{R}^3\to\mathbb{R}$, $f(x,y,z) = x^3+y^3+z^3 - 5yz$. Can anybody drop a hint how to compute relative homology of interlevel sets with coefficients in ...
0
votes
1answer
117 views

Homeomorphism under subspace topology in Hausdorff space

Let $Y$ be a Hausdorff space, and $U,V \subset Y$ are homeomorphic under subspace topology. Does this imply if $U$ is open(or closed) then $V$ is open(or closed) under original topology? I can't ...
1
vote
0answers
329 views

Find all the covering spaces of the Klein Bottle

I am studying for a qualifying exam and it asks for all the covering spaces of the Klein Bottle. Does anyone have any suggestions? I am not sure how to go about finding ALL of them.
4
votes
2answers
73 views

How to use Mayer-Vietoris to show $\chi(X)=2\chi(M)-\chi(\partial M)$ where $X$ is the double of $M$?

I'm in trouble with the following problem: Let $M$ be a manifold with compact boundary $N$ and let $X$ be the double of $M$, that is, the manifold without boundary one gets by glueing $M$ with ...
3
votes
0answers
49 views

eilenberg-steenrod for pairs in any model category?

The Eilenberg Steenrod axioms are functors on the homotopy category of pairs of "spaces" $(X, A)$. Typically they are introduced when $X$ and $A$ are some sort of topological spaces. My question is ...
2
votes
0answers
93 views

Visualize a projective curve $X^3+Y^3=Z^3$ in $P_2(C)$ as a torus

Let $P_2(C)$ be the 2 dimensional complex projective space, I want to prove that the projective curve defined by $M=\{[X,Y,Z]\in P_2(C)|X^3+Y^3=Z^3\}$ is a torus. I know that there is a theorem saying ...
6
votes
1answer
335 views

To prove a vector bundle $\xi$ is orientable $\iff w_1(\xi)=0$

I went about it this way: $\xi$ an $n$-rank vector bundle over a topological space $M$ is orientable. $\iff$ its top exterior power $\bigwedge^n\xi$ is trivial $\iff$ the 1-frame bundle ...
0
votes
1answer
136 views

There is more then one Homology Theory for spaces, which are not Hausdorff

All of us know: if we have a CW-complex, then using an arbitrary homology theory (with the axioms of disjoint union and dimension axiom) we always get the same homology groups up to an isomorphism. ...
1
vote
1answer
88 views

Isomorphism relative homotopy groups

Suppose that we have $Y \subset X$ topological space such that $$ \pi_i(X,Y)=0$$ for all $0 \le i < k$. How can I prove that the homomorphism induced by inclusion $i: Y \hookrightarrow X$, say ...
3
votes
1answer
61 views

Why not extending to the whole disk implies have a zero

For any complex polynomial $p(z)$ of order $m$, we showed earlier that on a circle $S$ of sufficiently large radius $r$ in the plane, $$\frac{p(z)}{|p(z)|}\quad \text{and}\quad ...
2
votes
1answer
61 views

fundamental group of a graph is free

Let $X$ be a connected graph, and $T$ its maximal tree. Via covering spaces and deck-transformations, how one can prove that $\pi_1(X)= \pi_1(X/T)$?
1
vote
1answer
167 views

helix and covering space of the unit circle

Does a bounded helix; for instance $\{(\cos 2\pi t, \sin 2\pi t, t); -5\leq t\leq5\}$ in $\mathbb R^3$ with the projection map $(x,y,z)\mapsto (x,y)$ form a covering space for the unit circle ...
3
votes
0answers
93 views

About Linking Number

I'm looking for some references about Linking Number. I already know these http://mathworld.wolfram.com/LinkingNumber.html http://www.matapp.unimib.it/~ricca/publications/2011/JKTR11.pdf Anything ...