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

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5
votes
2answers
274 views

Winding number question.

In class we defined the winding number as follows: If $\gamma$ is a loop on $\mathbb{R}^2$ that does not pass through a point $p$, the winding number $W( \gamma, p)$ is an integer $n$ that $\gamma$ ...
3
votes
1answer
446 views

Rank of first homology group for surface with punctures?

I feel like this question will be a head-slapper once I figure out the answer, but for the moment I'm having trouble! Let $M$ be a compact, connected, orientable 2-manifold of genus $g$ with $b$ ...
3
votes
0answers
119 views

Multiple Dehn twists and minimal position

I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit. Let $a$ be a simple closed curve in a compact surface $S$ (possibly with marked points ...
2
votes
1answer
166 views

Exact 2-forms in $R^n$

A closed $1$-form in a simply connected set in $R^n$ is exact. I would like a similar condition (with a reference) on sets in $R^n$ that closed $2$-forms are exact. De Rham cohomology gives an ...
2
votes
0answers
699 views

Free Homotopy Class

Just a silly question here - what's the difference between free homotopy classes and homotopy classes? Pre-thanks for your response!
2
votes
0answers
80 views

Monodromy groups and the choice of a base point

For a path connected topological space $X$, the fundamental groups at two different base points are isomorphic. Does the same hold for the monodromy group of a covering map $Y \rightarrow X$, when ...
2
votes
0answers
314 views

Monodromy Theorem and Homotopy Lifting Theorem

I've just come across this proof of the following theorem that I can't convince myself is true. Any ideas whether it's correct? Suppose $\gamma$ and $\lambda$ are homotopic paths starting at $x$ in a ...
2
votes
0answers
175 views

How to prove the commutative diagram

Let $p_2:\mathbb S^1 \to \mathbb S^1$ be the two-sheeted covering map $p(z)=z^2$.If $f$ is odd($f(-z)=-f(z)$),show that there exists a continuous map $g:\mathbb S^1 \to \mathbb S^1$ such that $\deg ...
5
votes
1answer
393 views

Proving two torus maps are homotopic

I have the following problem: Given two maps $\varphi , \psi :T^2 \rightarrow T^2$, with $\varphi(p)=\psi(p)=p$ such that $\varphi_*=\psi_*$ (the induced homomorphisms on the fundamental groups based ...
4
votes
1answer
228 views

injective map in cohomology theory

I have the following question, which I dont really know if its true: Let $g : X \rightarrow Y$ be a continous map between two closed, oriented $n-$dimensional manifolds such that $g^{*} : H^{n}(Y, ...
12
votes
1answer
1k views

Contractible vs. Deformation retract to a point.

I have a quick question about the difference between the two concepts in the title. The question is basically ex.6 (b) in Hatcher's book titled "Algebraic Topology". Let $X$ be the subspace of $R^2$ ...
6
votes
1answer
213 views

The A^1-localization in the unstable motivic category

I am currently trying to study $\mathbb{A}^1$-homotopy theory and I have a question about the construction of the unstable motivic category. Here is roughly the construction I try to understand : ...
1
vote
1answer
81 views

Retraction and deformation of P2

Let $\mathbb{P}^2$ denote the projective plane. Given (no need to prove) that $H_1(\mathbb{P}^2) \cong \mathbb{Z}_2$ ,$H_2(\mathbb{P}^2) \cong 0$ and the open Möbius band $M$ is homotopy equivalent ...
2
votes
0answers
121 views

Spectral Sequence and Stiefel Manifold

Let $Spin(3)$ be embedded in $Spin(5)$ by the spin embedding then we have a fibration: $$Spin(3) \rightarrow Spin(5) \rightarrow Spin(5)/Spin(3)$$ Where $Spin(5)/Spin(3)$ is $V_{5,2}$, the Stiefel ...
3
votes
1answer
483 views

Problem from Hatcher's book (3.3.25)

I have begun to read in Hatcher's book "Algebraic topology", about cohomology. In doing so, I have tried to solve some problems. I have difficulties with problem 3.3.25: Show that if a closed ...
2
votes
0answers
128 views

Question about covering spaces

Suppose $\pi:M_1 \to M_2$ is a $C^\infty$ map of one connected differentiable manifold to another.And suppose for each $p\in M_1$,the differential $\pi_*:T_p M_1 \to T_{\pi(p)}M_2$ is a vector space ...
2
votes
2answers
506 views

Homotopic to a Constant

I'm having a little trouble understanding several topics from algebraic topology. This question covers a range of topics I have been looking at. Can anyone help? Thanks! Suppose $X$ and $Y$ are ...
0
votes
1answer
125 views

Fundamental Group Isomorphism

We've been learning about fundamental groups and various properties of $SU(2)$, and I want to improve my understanding by working on interesting problems I've come across. I have the following that I ...
2
votes
1answer
2k views

Why are the Möbius strip and the boundary of a Klein bottle homotopy equivalent to $S^1$

I'm trying to calculate the fundamental group of two Möbius strips which have been identified along their boundary (which is a Klein bottle, I think). I've chosen an NDR pair $A,B$ where $A$ and $B$ ...
2
votes
0answers
432 views

$X,Y$ are locally path connected and path connected, with universal covers $\tilde{X}, \tilde{Y}$. If $X \simeq Y$ then $\tilde{X} \simeq \tilde{Y}$

Suppose $X,Y$ are locally path connected and path connected, with universal covers $\tilde{X}, \tilde{Y}$. I'd like to prove that if $X \simeq Y$ then $\tilde{X} \simeq \tilde{Y}$. I've had the ...
2
votes
0answers
90 views

The Two Eilenberg-Moores

So, there is the Eilenberg-Moore spectral sequence, and there is (for any monad $(T,\mu,\eta)$ on a category $C$) the Eilenberg-Moore Category $C^T$ of $T$-algebras. The silly question, is the ...
1
vote
1answer
262 views

$f:X \to Y$ is a homotopy equivalence if there exists $g,h : Y \to X$ with $fg$ and $hf$ homotopy equivalences

Let $X$ and $Y$ be topological spaces, and let $f: X \to Y$. I'd like to show that if there are maps $g,h : Y \to X$ such that $fg$ and $hf$ are homotopy equivalences, then $f$ is a homotopy ...
3
votes
0answers
173 views

tensor product of two chain homotopic maps are again chain homotopic?

Let $C$,$C'$, $D$, $D'$ be chain complexes, $f$, $f'\colon C\to C'$ and $g$, $g'\colon D \to D'$ two pairs of homotopic chain maps.How to show $f \otimes g$ and $f' \otimes g' \colon C\otimes D\to ...
4
votes
3answers
2k views

Calculating the fundamental group of $\mathbb R^3 \setminus A$, for $A$ a circle

Let $ X = \mathbb R^3 \setminus A$, where $A$ is a circle. I'd like to calculate $\pi_1(X)$, using van Kampen. I don't know how to approach this at all - I can't see an open/NDR pair $C,D$ such that ...
4
votes
2answers
2k views

Fundamental group of the torus

I'm trying to get my head round the following calculation of the fundamental group of the torus, using Seifert Van-Kampen (I know it's easier to do this by considering covering spaces, but I'm trying ...
1
vote
1answer
204 views

Proof about universal covers

I have some Algebraic Topology notes that prove the following Lemma: If $X$ is simply connected and locally path connected, then every covering projection $p:Y \to X$ is trivial (i.e. the whole ...
5
votes
1answer
397 views

Homotopy groups of compact topological manifold

I am looking for a proof of the following fact: Every compact topological $n$-manifold $M$ has a continuous and not nullhomotopic map $f: S^k \rightarrow M$ for some sphere $S^k$ with $1 \leq k \leq ...
1
vote
2answers
218 views

Hatcher covering space proof seems wrong?

So I'm trying to understand the proof on page 63 http://www.math.cornell.edu/~hatcher/AT/AT.pdf In the proof he says that if $\tilde{f}_{1}(y) \not = \tilde{f}_2 (y)$, then $\tilde{U}_{1} \not = ...
0
votes
1answer
152 views

Covering projection is monomorphism

Let $p:(E,e_0) \rightarrow(X,x_0)$ be a covering projection. Show that $p \sharp: \pi_{1}(E,e_0) \rightarrow \pi_{1}(X,x_0)$ is a monomorphism. I was wondering here do I need to prove this is a ...
2
votes
1answer
288 views

Universal coefficient theorem of relative homology

In Hatcher, Corollary 3A.4 stated a universal coefficient theorem for relative homology, i.e. the following short exact sequence splits: $0 \rightarrow H_n(X,A) \otimes_\mathbb{Z} G \rightarrow ...
2
votes
2answers
318 views

Logarithms of formal group laws

I have some questions on formal group laws and their logarithms: Let $R$ be a graded commutative ring and $F \in R[[X, Y]]$ be a formal group law over $R$ that admits a logarithm. Can you tell me, ...
3
votes
3answers
267 views

Show the defined map onto $S^1$ is not a covering map.

Let the map $f:S^1 \times \mathbb{N} \to S^1$ defined by $f(z,n):=z^n$ is continuous and onto, but f is not a covering map from $S^1 \times \mathbb{N}$ onto $S^1$.
6
votes
0answers
391 views

Mayer-Vietoris implies Excision

Assume $H_n$ is a covariant homotopy functor on the category of locally compact Hausdorff spaces which has the Mayer-Vietoris property: whenever $X$ is the union of two closed subspaces $A$ and $B$ ...
2
votes
2answers
308 views

Homology of a sphere quotient points? [duplicate]

Possible Duplicate: Finding the homology group of $H_n (X,A)$ when $A$ is a finite set of points I want to work out the homology of a sphere $S^2$ quotient a set of finite points(say p ...
7
votes
1answer
284 views

homotopy direct limits

$X$ is said to be the homotopy direct limit of the sequence of subsets $X_1\subset X_2\subset ...$ if the projection $\cup_i X_i\times [i,i+1] \rightarrow X$ is a homotopy equivalence. The ...
12
votes
2answers
1k views

composition of covering maps

The origin of my question arose from a problem: Let $q: X \to Y$ and $r: Y \to Z$ be covering maps, let $p= r \circ q$. Show that if $r^{-1}(z)$ is finite for each $z \in Z$, then $p$ is a covering ...
8
votes
1answer
5k views

Calculating fundamental group of the Klein bottle

I want to calculate the Klein bottle. So I did it by Van Kampen Theorem. However, when I'm stuck at this bit. So I remove a point from the Klein bottle to get $\mathbb{Z}\langle a,b\rangle$ where ...
5
votes
0answers
348 views

Definition of Reshetikhin-Turaev TQFT

I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
5
votes
1answer
307 views

Surgery link for lens spaces

Let $p$ and $ q$ be a relatively prime integers. I want to know how to prove that a Hopf link with framing $-p$ and $-q$ is a surgery link for a lens space $L(p,q)$. The lens space is first a result ...
1
vote
0answers
144 views

Terminology in an Exercise of Hatcher

I am trying to solve an exercise in Hatcher "Algebraic Topology" but am a little confused by the terminology he is using (just so you know, it is exercise 5 in chapter 1.1). He writes that in the ...
2
votes
1answer
134 views

Question about Tamafumi Kaneyama's Paper: “On Equivariant Vector Bundles On An Almost Homogeneous Variety”

My reference: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.nmj/1118795362&page=record I have two question about Proposition 3.3.: Proposition3.3. ...
3
votes
0answers
470 views

Fundamental group of $\mathbb{R}^{3}\setminus \{ \mbox{2 linked circles }\}$

Calculate the fundamental group of the complement in $\mathbb{R}^3$ of $$\{ (x,y,z) \ | \ y = 0 , \ x^{2} + z^{2} = 1\} \cup \{ (x,y,z) \ | \ z = 0 , \ (x-1)^{2} + y^{2} = 1\}.$$ Note: this space ...
2
votes
1answer
218 views

Pullback of a locally constant sheaf by a function whose domain is simply connected

Let $\mathcal{A}$ be a locally constant sheaf on a topological space $X$ and let $\sigma:\Delta_p\to X$ denote a singular $p$-simplex. Writing the pullback of $\mathcal A$ by $\sigma$ as ...
1
vote
1answer
219 views

orientability of the möbius strip using homology

I read in Hatcher's "Algebraic topology" book about orientability of topological maifolds using homology. now I would like to know how one can apply this to show that the möbius strip is not ...
2
votes
3answers
356 views

Showing higher homotopy groups of $S^1$ are trivial

I'm trying to prove $\pi_{i} (S^1) \cong 0$ if $i>1$. Is this correct. You have a short exact sequence, $\mathbb{Z} \rightarrow \mathbb{R} \rightarrow S^1$ (from the fiber bundle of the covering ...
1
vote
1answer
1k views

Path lifting theorem

http://www.maths.manchester.ac.uk/~jelena/teaching/AlgebraicTopology/PathLifting.pdf I'm trying to generalize this theorem. But, was wondering in the proof given here and similarly in Hatchers. Can ...
1
vote
1answer
206 views

Question on covering spaces

Let $q:X\to Y$ and $r:Y \to Z$ be covering maps;let $p=r\circ q$. Show that if $r^{-1}(z)$ is finite for each $z\in Z$, then $p$ is a covering space. I'm confused that $p=r\circ q$ is obvious ...
2
votes
0answers
421 views

Calculating Homology Groups of $S^1\times X$

Given that $H_n(X)$ is free abelian, I'm trying to find the homology groups of $Y=S^1\times X$ using the Mayer-Vietoris theorem. My first attempt decomposed $Y$ as $A \cup B$ where $A=\{*\}\times X$ ...
3
votes
0answers
341 views

Double Coverings of the Double Torus

I'm trying to count all the double coverings of the double torus. I know that the fundamental group of the double torus is $$\pi_1(X)=\langle a,b,c,d;[a,b][c,d]\rangle $$ where ...
-1
votes
1answer
129 views

Are fibers of a fiber bundle the same as fibers of a covering space?

I was wondering is there any difference between them? As all fibers are fiber bundles, so surely the fibers are the same. But, then couldn't there be some special thing about a fiber of fiber bundle ...