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

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real Grassmann manifolds, Schubert cells, and boundary maps for computing homology

The problem: Compute $H_\ast (G(n,k); \mathbb{Z}/m\mathbb{Z})$. Define $G(n,k)$ to be the space of $k$-dimensional vector subspaces of $\mathbb{R}^n$. Define the Schubert cell $e(m_1,...,m_s)$, so ...
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155 views

The definition of normal covering in Hatcher book

In page 70 of Hatcher's book, in the section Deck Transformations and Group Actions, the author defines a normal covering in the following way: A covering space $p:\tilde X\to X$ is called normal ...
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124 views

Complement of deformation retract

Let $X$ be a topological space and $V$ and $N$ are subspaces of $X$ such that $N$ deformation retracts onto $V$. I want to show that $X-V$ deformation retracts onto $X-N$. So i need to construct a ...
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162 views

Does the reduced Mapping cylinder have the same homotopy type of unreduced Mapping cylinder?

Let $ f:X\to Y $ be a map in the pointed category of topological spaces $ Top_* $. And let $ U:Top_*\to Top $ be the "forgetful" functor (which "forgets" the basepoint). We can look at the reduced ...
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83 views

Thom space 2 definitions

For a vector bundle thom space $T$ is defined as $T=E/A$, where $E$ is the total space and $A$ is the set of vectors in $E$ of length $\geq 1$. Alternatively, $T$ is the mapping cone of the associated ...
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114 views

contractible open sets

If $U\subset\mathbb{R}^n$ is an open and contractible subset such that there is a continuous function $f\colon U\to\mathbb{R}$ with only one minimum and the level curves of $f$ are connected by paths, ...
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76 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 ...
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75 views

Surgery and boundary

Let $L$ be a framed link in $S^3$ with $m$ components and let $U$ be a closed regular neighborhood of $L$ in $S^3$. Let $B^4$ be a closed 4-ball bounded by $S^3$ so that $U \subset S^3$. Gluing $m$ ...
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32 views

Is the stable homotopy group of sphere a commutative ring? If not, are there easy examples?

Is the stable homotopy group of spheres a commutative ring? If not, are there easy examples? In the Adams spectral sequence converging to the stable homotopy group of spheres, it seems that any page ...
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14 views

If the reduced homology of K is nonzero, k is evasive.

Where can I find a proof for this theorem of Kahn, Saks and Sturtevant? K is a simplicial complex. Theorem $\textbf{10.1}.$ If $\tilde H_*(K)\neq 0$, where $\tilde H_*(K)$ denotes the reduced ...
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29 views

Homotopy Invariance: Cone Construction and Prisms Operators

I'm looking at different approaches to proving the homotopy invariance of homology. Rotman and Dieck both mention "the cone construction", but hatcher only introduces the prism operators and does not ...
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33 views

On Hopf invariant

I didn't understand following expression from Hatcher. Let $f: S^{2n-1} \to S^{n}$. If $f$ is a constant map, then $Cf=S^{2n} \lor S^{n}$ and $H(f)=0$ since $Cf$ retracts onto $S^n$.
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23 views

Does Alexander Duality commute with inclusion?

This is a follow up to this question I asked previously: Alexander Duality for Relative Homology I am working with two compact pairs of spaces $(A,B)$ and $(A',B')$, where $A'\subset A$ and ...
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49 views

A question regarding singular homology (The geometrical interpretation of cycles in singular homology)

From Hatcher's Algebraic topology : Cycles in singular homology are defined algebraically, but they can be given a somewhat more geometric interpretation in terms of maps from finite Δ-complexes. To ...
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52 views

How can I show $G_0$ and $G_1$ are conjugate subgroups?

Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the induced homomorphism from the fundamental group of $E$ to the fundamental group of $B$. Let $e_o$ and $e_1$ be points in E ...
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37 views

mapping cylinder contractible iff Hn(f):Hn(X)->Hn(Y) is an isomorphism

The mapping cylinder will be defined as $Z_f=X\times[0,1]\coprod Y/\sim$, where $\sim$ is defined by $(x,1)\sim f(x)$. Let $f:X\to Y$ a continuous map between topological spaces and the map ...
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60 views

Does an odd degree map on $S^n$ descend to an odd degree maps on $\mathbb{R}P^n$?

Suppose there is a map $f:S^n\to S^n$ that induces non-trivial on $\mathbb{Z}/2$ homology group homomorphisms, further suppose $f$ descends to $f':\mathbb{R}P^n\to\mathbb{R}P^n$. Does it then follows ...
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33 views

What is a thin loop?

I read one definition of a thin loop: $\gamma$ is a thin loop if there exists a homotopy of $\gamma$ to the trivial loop with the image of the homotopy lying entirely within the image of $\gamma$. ...
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94 views

Creating topological spaces with portals

I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension. I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
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19 views

Proving these spaces are homeomorphic

As part of a proof I am reading it states that the part of the sphere given by: $S$ = $\{(x,y,z) \in \mathbb{R} : x^{2} + y^{2} + z^{2}=1, x \geq 0, y \geq0, z\geq0\} $ is homeomorphic to the closed ...
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31 views

Prove that $C_1$ and $C_2$ are homotopic fixing endpoints.

Let $C_1$ and $C_2$ be two great circles in $S^2$, intersecting at the points $p,q$. If we consider $C_1$ and $C_2$ as curves starting and ending at $p$. Prove that $C_1$ and $C_2$ are homotopic ...
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30 views

Signature of a finite covering space

Suppose $\tilde{M}\rightarrow M$ is a finite (k-fold) covering of the smooth, oriented, compact 4-manifold $M$. Is there a relation between the signatures ...
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37 views

continuity of a map

let B be the closed unit ball & D the open unit ball. If g is a continuous function from B$\rightarrow R$ can one find always a continuous function from $R^2 \rightarrow R$ such that f=g on B?The ...
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20 views

rigorously defining homotopy inverses between the wedge sum and union of growing circles

I'm trying to solve the exercise 1.2.20 of Hatcher algebraic topology and stuck with the homotopy equivalence part. I can't construct explicit homotopy inverses. Could anyone show me what the ...
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41 views

$\mathbb{R}P^2$ and its fundamental group by identification of edges of unity square

Suppose we identify edges of the the unity square $[0, 1] \times [0, 1]$, as in the picture: http://de.wikipedia.org/wiki/Datei:ProjectivePlaneAsSquare.svg Now to compute the fundamental group, for ...
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59 views

Showing that there is no base-point preserving homotopy

I'm working on this problem and showed that X is contractible. In fact I showed that X has the origin (0,0) as its deformation retract. However, I'm stuck at the second part. It seems intuitively ...
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44 views

What is the map $\Sigma K(G,n) \to K(G,n+1)$?

Since $\Omega K(G, n+1)$ is a $K(G,n)$, we have a CW approximation/homotopy equivalence $K(G,n) \xrightarrow{\sim} \Omega K(G,n+1)$. The adjoint of this map is a map $\Sigma K(G,n) \to K(G,n+1)$. ...
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20 views

How do you compute the simplicial homology of an $n$-gon with all edges and vertices identified?

Suppose you have an $n$-gon with all vertices identified, and all edges identified. I think the optimal way to compute the homology groups would be to view this as a cell complex consisting of a ...
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27 views

Could anyone suggest me a counter example about liftings?

A book reads: Refer to the proof of the following assertion: Given a map $F\colon Y \times I \to S^1$ and a map $\tilde{F}\colon Y \times\{0\} \to \mathbb{R}$ lifting $F| Y\times\{0\}$, there is a ...
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28 views

How does one compute a group modulo a torsion group?

Let's say I have some group $G$ and a subgroup $H$ such that $H$ is a torsion group (i.e. $\forall h \in H$, $h$ has finite order. How do I compute the factor group $\frac{G}{H}$? What effect does the ...
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17 views

If $D^1\cup_f D^1=S^1$?

Suppose $f\colon S^0\to S^0$, so we can form the attaching space $D^1\cup_f D^1$. Is my intuition correct that this space is just $S^1$? Since $S^0=\{1,-1\}$, $f$ is either the identity, or swaps the ...
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62 views

Is the pullback of two covering spaces $\tilde X$ and $\hat X$ a covering space?

Suppose we have two covering spaces $p:\tilde X \rightarrow X$ and $q:\hat X \rightarrow X$ of the same space. Is the pullback $\tilde X \times_X \hat X$ also a covering space of $X$? If yes, what ...
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37 views

A question about winding numbers.

This is a question from Needham's "Visual Complex Analysis". Kindly refer to the photo below. Let $K$ be a line moving downwards. The book says that if we move a point $r$ from the left to the ...
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88 views

Cardinality of fibres of covering maps of connected spaces

If I have a covering map $p:E \rightarrow B$ and some connected set $U$, that is evenly covered, then $p^{-1}(U)$ as a partition into slices is unique. Now, if I assume that $B$ is connected, then I ...
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71 views

Reduced suspension and unreduced suspension

In May's "A concise course in Algebraic Topology" Chap 14 section 1, the author says $\Sigma (X_+)$ is $\Sigma X\vee S^1$ where $X$ is an unbased space and $X_+$ is the union of a disjoint basepoint ...
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44 views

Empty set in a simplicial complex

Should the empty set be considered a simplex in a simplicial complex? Which justifications exist for the answer? I guess it is somewhat comparable to $1$ not being a prime number.
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45 views

a topological space is the union of its irreducible components

if we define irreducible component of a topological space $X$ as the maximal closed irreducible subset of $X$,prove that we can write $X$ as the union of its irreducible components. how to approach ...
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26 views

a region homeomorphic with klein bottle

prove that if we consider this shape in the picture below with the equivalency relation that : a & b are in one class if they are antipoles in inner or outer circles, then the induced quotient ...
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36 views

CW complex with no cells in dimension $n$

Hi need some help with the following problem: if $X$ is a CW complex with no cells of dimension $n$ then $\tilde{H}^n(X,G)=0$. where $G$ is any group. thanx.
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54 views

Lifting property of a covering space

A common example of a covering space is $p:\mathbb R \rightarrow S^1 $, $p(t)= e^{2 \pi it}$, which can be looked at as an infinitely long spiral placed over a circle. Consider the double loop ...
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31 views

Finding lifted paths, homotopy lifting

I am given a covering map $p: \mathbb{R}^+ \times \mathbb{R} \to \mathbb{R}^2 \setminus \{0,0\}$ defined by $p(r, \theta)=(r \cos 2 \theta,r \sin 2 \theta)$ Let $\alpha: [0,1] \to \mathbb{R}^2 ...
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22 views

Prove that $h$ and $p*q$ are homotopic relative to {$0,1$}

Let $0<s<1$. Given paths $p$ and $q$ with $p(1)=q(0)$, define $h$ by the formula $$h(t) = \begin{cases} p(t/s),& \text{if} \quad 0 \leq t \leq s \\ q((t-s)/(1-s)), &\text{if} \quad ...
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92 views

Show that an inclusion is an isomorphism in homology

I'm struggling a bit with an exercise from a book, in a chapter about the Jordan-Brouwer separation theorem. It goes as follows: (note: $s_{n-1}$ is a topological space homeomorphic to ...
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29 views

Necessary and sufficient condition for existence of a deck transformation

I am considering the following problem: $\tilde X$ path connected, $X$ path connected, locally path connected, $P:\tilde X \to X$ covering map, $x_0 \in X, \tilde x_0, \tilde x_1 \in p^{-1}(x_0).$ I ...
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46 views

The exact sequence of a pair - something fishy going on here!

This question is related to my previous question. After knowing that $X \cong S^3$ and $A \cong S^1$, with $X/A \cong S^2$, I attempt to construct the long exact sequence of a pair. I need to use the ...
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76 views

Liftings in Covering Map Closed

Let $p:\tilde{X} \mapsto X$ be a covering map with $\tilde{X}$ path connected. Why are all liftings of a closed path $f$ in $X$ either closed or not closed? If $\omega$ is a path from $\tilde{x_0}$ ...
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18 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 ...
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62 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 ...
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65 views

Question about degrees of maps from $S^1 \rightarrow S^1$

Note: Since the degree of a map is independent of the base-point I'll speak loosely and just say $\pi_1(S^1)$. One definition of the degree of a map $f$ from the circle to itself is the number $k$ ...
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30 views

Showing that a map can be deformed into the identity.

Suppose $F((a,b), k) = (ae^{\pi i k}, be^{\pi i k})$ where $0 \leq k \leq 1$. Now would $g(a,b)$ = $(-a,-b)$ if $g : S^{1} \rightarrow S^{1}$?