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

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1
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2answers
31 views

Homotopy between any loop in an annulus and a loop in a circle contained in the annulus

Show that any loop in the annulus $\textbf{A}=\{(x,y) \big | 1 \le x^2 + y^2 \le 9 \} \subset \mathbb{R}^2$ with base point $(2,0)$ is homotopic to a loop whose image lies in the circle $A=\{(x,y) ...
8
votes
1answer
257 views

Lie groups as manifolds

In Weinberg's Classical Solutions in Quantum Field Theory, he states Lie groups, such as $SU(2)$ or $SO(3)$, may be viewed as a manifold. My questions are, If we can interpret, e.g. $SU(2)$ as a ...
1
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2answers
113 views

lifting a product of commutators of standard generators on 2-manifolds

I have a problem with understand the proof http://www.ams.org/journals/proc/1972-032-01/S0002-9939-1972-0295352-2/S0002-9939-1972-0295352-2.pdf I don't understand this part: "(...) we can easily ...
1
vote
4answers
353 views

Homology group of Real projective plane

I know the homology group of Real Projective plane $\mathbb{RP}^2$ $H_i(\mathbb{RP}^2) = 0$ for $i>2$, $\mathbb{Z}$ for $i=0$ , $\mathbb{Z}/2\mathbb{Z}$ for $i=1$ (non-reduced case). Proving when ...
0
votes
1answer
51 views

Equivalent statements to fixed-point theorem

I'm trying to show that they're equivalent statements: 1) $1_{S^1}$ is not homotopic to a constant map. 2) $S^1$ is not a retract of $D^2$ ($D^2$ is the closed unit ball). 3) Every continuous map ...
2
votes
0answers
78 views

References for Local Orientations and Fundamental Class

I am looking for references with examples about computing induced orientation given a self homeomorphism of a closed orientable n-topological manifold. I have in mind only A.Hatchers Algebraic ...
4
votes
1answer
214 views

Universal coefficient theorem with ring coefficients

The universal coefficient theorem for cohomology reads: $$0 \to Ext(H_{n-1}(C), R) \to H^n(C;R) \to Hom(H_n(C), R) \to 0,$$ where $C$ is a chain complex of free abelian groups and $R$ is a ring. It ...
2
votes
1answer
130 views

How to compute the fundamental group from first homology group?

I have been reading about the fundamental group and its connection to the first homology group. In fact, there is an isomorphism $$\pi_1^{ab}(X,x_0) \to H_1(X)$$ for every path-connected topological ...
2
votes
1answer
103 views

The fundamental group of $U(n)/O(n)$

Since $O(n)$ is a subgroup of $U(n)$, we can consider the quotient space $L_n=U(n)/O(n)$. The quotient space is homotopic to the ($n$-th) Lagrangian Grassmannian, and it is known that its fundamental ...
1
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1answer
61 views

regular values of PL maps

In the category of smooth manifolds and maps, $y$ is a regular value of $f$ iff the tangent map $df(x)$ is surjective for any $x\in f^{-1}(y)$. Then the preimage $f^{-1}(y)$ is a smooth submanifold. ...
2
votes
1answer
57 views

Is there any simple example that $lim^1$ terms appear?

limit of cohomology does not behave well in the sense that there will be $lim^1$ term. Is there any simple example that $lim^1$ terms appear? Thanks!
0
votes
0answers
61 views

Boundary of Mobius band

Let $M$ be a homeomorphic copy of the Mobius band in $\mathbb{R}^3$, let $S$ be its "boundary circle". Let $i:H_1(\mathbb{R}^3\setminus M)\to H_1(\mathbb{R}^3\setminus S)$ be the map induced by the ...
1
vote
2answers
118 views

Can $S^1 \times Y$ be homeomorphic to $\mathbb{R}P^2$ or $S^2$?

Does there exist a topological space $Y$ such that $S^1 \times Y$ is homeomorphic to $\mathbb{R}P^2$ or to $S^2$?
2
votes
1answer
38 views

Proof that if $S^{k}\rightarrow S^{m}\rightarrow S^{n}$ is a fiber bundle, then $k=n-1$ and $m=2n-1$.

I was reading a proof online (7a) here: http://www.math.wisc.edu/~dummit/sets/752-fs.pdf that if $S^{k}\rightarrow S^{m}\rightarrow S^{n}$ is a fiber bundle, then $k=n-1$ and $m=2n-1$ but I'm ...
6
votes
1answer
136 views

Suggestion about Algebraic Topology talk

following the content of the title I am writing here to ask some suggestions concerning a talk I will be presenting at my university in a week or two. The main topic I chose is the fundamental ...
3
votes
2answers
181 views

What is the suspension used in the Freudenthal suspension theorem?

The theorem states: The suspension map $\pi_{i}( S^{n})\rightarrow \pi_{i}(S^{n+1})$ is an isomorphism when $i<2n-1$ and a surjection when $i=2n-1$. In the case where $X$ is an ...
0
votes
0answers
21 views

Proving $\alpha*\alpha^{-1}*p*\alpha*\alpha^{-1}$ and $p$ are homotopic

For two points $x_1$ and $x_2$ in a path-connected component, we know that their fundamental groups are isomorphic. In other words $$\pi_1(X,x_1)\cong\pi_1(X,x_2)$$ Let $\alpha$ be a path from $x_1$ ...
2
votes
1answer
47 views

Number of cells in a minimal cell structure for a non-simply connected manifold?

I have obtained a cell structure of a connected (but not simply connected) manifold using Morse theory. Is there any way for me to know whether this cell structure is minimal?
1
vote
1answer
79 views

Naturality condition for connecting homomorphisms?

I've been reading about the Mayer-Vietoris sequence, but I don't follow a certain naturality condition. Suppose two spaces can be written as $X=X_1^\circ\cup X_2^\circ$ and $Y=Y_1^\circ\cup ...
0
votes
2answers
49 views

Why is the product of path homotopy classes not defined sometimes?

Munkres says on pg. 346 that the set of path homotopy classes does not aways form a group under the operation $*$ because the product of two path homotopy classes is not always defined. What does ...
1
vote
1answer
101 views

Is there a cayley graph for the Klein bottle?

When studying algebraic topology we learned about the fundamental group of the $2$-torus $T^2$ which is isomorphic to $$\langle a, b \mid aba^{-1}b^{-1} \rangle$$ (the free abelian group on two ...
2
votes
1answer
73 views

Cohomology after Dehn surgery

For $f:S\to M$ a knot in a 3-manifold, we can construct a 3-manifold $N$ by a $0/1$-type Dehn surgery along $f$: First remove from $M$ a solid torus which is a tubular neighbourhood of the knot $f$; ...
0
votes
0answers
26 views

How to prove homomorphisms with 'lifting'

for topology i just started a chapter called lifting and i'm having trouble using this concept to prove a homomorphism of a lifting correspondance. This is my following question: let $m \in ...
4
votes
1answer
145 views

Is the Hopf fibration the only fibration with total space $S^{3}$?

Let $S^{1}$ bundle over $S^{2}$ is homeomorphic (diffeomorphic) to $S^{3}$. Is the Chern class of the fibration 1, i.e. is the Hopf fibration up to isomorphism the only fibre bundle with total space ...
1
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0answers
67 views

A doubt in Munkres' Topology.

On pg. 343 of Munkres' Topology (Second Edition), there is a diagram given that I am having problems understanding. This diagram is Figure 51.7. Why are the paths $i$ and $e_0$ defined from $I$ to ...
0
votes
3answers
45 views

A problem regarding $k\circ (f*g)=(k\circ f)*(k\circ g)$.

My Algebraic Topology book states the following: Let $k:X\to Y$ be continuous path. If $f$ and $g$ are two paths in $X$ with $f(1)=g(0)$, then $$k\circ(f*g)=(k\circ f)*(k\circ g)$$ I'm trying ...
2
votes
0answers
56 views

Where have I gone wrong in understanding of CW complex and Cell homology?

I seem to have wrong understanding of CW complex and it would be nice if someone could help me out. The definition I have for Cell complex is the usual one I think. We define glueing of cells and ...
4
votes
2answers
147 views

Homeomorphism Compact Subsets

Are there compact subsets $A,B \subset \mathbb{R^2}$ with $A$ not homeomorphic to $B$ but $A \times [0,1]$ homeomorphic to $B \times [0,1]$?
7
votes
1answer
123 views

Isomorphism Finite Topological Space

Does there exist a finite topological space with fundamental group isomorphic to $\mathbb{Z_2}$?
1
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3answers
128 views

Cohomology groups of real projective space

My question concerns the cohomology groups $H^k(RP^n,\mathbb{Z}_2)$. We know that $H_k(RP^n,\mathbb{Z}_2) = \mathbb{Z}_2$ if $0 \leq k \leq n$ and is trivial otherwise. I looked up the solution and it ...
0
votes
1answer
46 views

Question on Hamming distance

Let V_n be n-dimentional vector space over GF(q). E is k-dimentional vector subspace which is a linear q-ary (n,m,d) code and also consider the radius e = [(d-1)/2]. Assume that E is not a perfect ...
3
votes
2answers
126 views

the mapping class group of the disk is trivial proof

Proof : Identify $D^2$ with the closed unit disk in $\mathbb{R}^2$. Let $\phi : D^2 \rightarrow D^2$ be a homeomorphism with $\phi_{\partial D^2}$ equal to the identity. We define, $F(x,t) = ...
7
votes
3answers
149 views

let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
3
votes
2answers
124 views

tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
1
vote
1answer
54 views

Is it true there exists $f:S^{2n}\longrightarrow S^{2n}$ making the diagram commutative?

Let $g:\mathbb R\mathbb P^{2n}\longrightarrow \mathbb R\mathbb P^{2n}$ be a continuous map where $\mathbb R\mathbb P^{2n}=\mathbb S^{2n}/\{\pm x\}$. Is it true there exists $f:\mathbb ...
2
votes
1answer
62 views

Covering map of $\mathbb R \mathbb P^2$

The question I am trying to answer is: Does the quotient map $ q:[0,1] \times [0,1] \to \mathbb R \mathbb P^2$ extend to a covering map $\mathbb R^2 \to \mathbb R \mathbb P^2$ I know that the ...
11
votes
1answer
193 views

$\mathbb{R}\text{P}^{n-1}$ is not retract of $\mathbb{R}\text{P}^n$

I have to solve the following: Show that $\mathbb{R}\text{P}^{n-1}$ is not retract of $\mathbb{R}\text{P}^n$ for $n\geq 2$. I have done this with knowledge of homotopy-groups, by showing that ...
0
votes
0answers
27 views

Euler characteristic and free action

If $K$ is a finite simplicial complex, and $G$ acts simplicialy on $K$ with no fixed points, show $\chi(K) = |G|\cdot\chi(K^2/G)$. Could I have a hint for how to start this question? I was told to ...
0
votes
1answer
32 views

Show that $\tilde{X} \rightarrow X$ is a covering map.

Let $\tilde{X}=\{(x.y)\in \mathcal{R}^2; \text{x or y is an integer}\}$ Let X=$\{(z_1, z_2) \in S^1\times S^1; z_1=1$ or $z_2=1\}$ and let $p:\tilde{X}\rightarrow X$ be defined by $p(x,y)=(exp(2\pi ...
0
votes
1answer
16 views

Show that a finite graph product of finitely generated groups is finitely generated, and similarly for finitely presented groups.

Show that a finite graph product of finitely generated groups is finitely generated, and similarly for finitely presented groups. I think when we have a finitely generated groups,the graph product of ...
1
vote
1answer
53 views

What is the second homotopy group of $R^3 \setminus \{ (0,0,0) \}$

I was told that it was $\mathbb{Z}$, and I can imagine a subgroup isomorphic to $\mathbb{Z}$ of 'wrappings' of the sphere around the point, but I am still convinced there are more homotopy classes. I ...
0
votes
0answers
36 views

Equivalence of Cohomology groups

Suppose $n=i+j,$ with $n, i,j$ positive integers. Let $I^k$ denote the $k$-dimensional unit square. It is claimed (in Hatcher's Algebraic Topology text) that $H^i(\mathbb{R}^n, \mathbb{R}^n \setminus ...
1
vote
1answer
34 views

In homology, when we operate the boundary twice we get zero, that is, $\partial^2=0$. Need help understanding proof.

Proof for $S=\Delta_n=(v_0 ... \hat{v_i} ...v_n)=d_i$ $\partial=\displaystyle\sum_i^n(-1)^id_i.$ Thus, $\partial^2=[\displaystyle\sum_i^n(-1)^id_i][\displaystyle\sum_j^n(-1)^jd_j] $ ...
3
votes
0answers
40 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
2
votes
0answers
63 views

Cross product of cohomology classes: intuition

Let $X$ and $Y$ be topological spaces and consider cohomology over a ring $R.$ Hatcher (in his standard Algebraic Topology text) defines the cross product of cohomology classes $$H^k(X) \times ...
0
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0answers
12 views

is simplicial approximation of $gof$ equal to the combination of simplicial approximation of $g$ with simplicial approximation of $f$?

suppose $M,L,K$ are complex and $f:|K| \rightarrow |L|$ and $g:|L| \rightarrow |M|$ are continues maps,can we consider combination of simplicial approximation of $f$ and $g$ as a simplicial ...
1
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0answers
29 views

Explain the terms k-simplex and simplical complex geometrically?

I m new to algebraic topology .so confused with these terms pls suggest simple books
1
vote
2answers
72 views

Group acts freely on a closed surface

My question is as follows: Let G be a finite group which acts freely as a group of homeomorphisms of a closed surface S (so the only element with fixed points is the identity) Then: Show the orbit ...
1
vote
1answer
44 views

Let $p$ be a covering space and $X, Y$ be path connected. Show there exists a map $q$ such that $q\circ p=1_{X}$ iff $p$ is a homeomorphism.

Let $p\colon X\rightarrow Y$ be a covering map where $X$ and $Y$ are path connected. Show that there exists a map $q\colon Y\rightarrow X$, such that $q\circ p=1_{X}$ if and only if $p$ is a ...
0
votes
1answer
52 views

Mayer-Vietoris sequence confusion

In the Mayer-Vietoris exact sequence $$... \rightarrow H_n(A) \oplus H_n(B) \rightarrow H_n(X) \rightarrow H_{n-1}(A \cap B) \rightarrow H_{n-1}(A) \oplus H_{n-1}(B) \rightarrow...$$ I am confused ...