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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1answer
25 views

Relative homology of disk and any of its subspace is isomorphic to reduced homology of the subspace?

Consider the pair of topological space $(\mathbb{D}^n,X)$, where $X \subset \mathbb{D}^n$ is a subspace. We know that there is a long exact sequence of reduced homology groups, $$\cdots \to ...
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0answers
21 views

Complex (topological) K-theory of $S \Sigma$ for a surface $\Sigma$.

In the following question here: What's the K-group of a surface? The $K$-theory of a compact orientable surface is computed. I was curious if it was also possible to compute the "higher" ...
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0answers
12 views

“Visual” interpretation of the Bott Periodicity for complex vector bundles

I've just read the Bott Periodicity proof in Hatcher book about K-theory. At first it seems that using this theorem I could conclude that every complex vector bundles over $S^{2n+1}$ is trivial, but ...
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0answers
21 views

Prove that reduced homology is relative homology to a point.

I am trying to follow the proof here that reduced homology $\tilde{H}_n(X)$ is the same as $H(X,x_0)$, the relative homology of $X$ to a point $x_0 \in X$. What I don't get is the very last step, ...
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0answers
15 views

Attaching space, help on visualization

Let $X$ be a topological space, $A\subset X$ a closed subspace. $CA$ means the cone of $A$, and by $SA$ I'll denote the suspension of $A$. I need to prove that $$ \left( \left( (X \cup CA) \cup ...
2
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2answers
48 views

Homology groups equal when $\cdots \rightarrow 0 \rightarrow H_n(\mathbb{S}^m) \rightarrow H_{n-1}(\mathbb{S}^m) \rightarrow 0 \rightarrow \cdots$

I'm reading a set of notes but I don't understand the following concept. We have a long exact sequence $\cdots \rightarrow 0 \rightarrow H_n(\mathbb{S}^m) \rightarrow H_{n-1}(\mathbb{S}^m) ...
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0answers
21 views

Two sheeted disconnected cover of a connected topological space has exactly two components

Let $\tilde{X}$ be a two-sheeted cover of a connected topological space X. If $\tilde{X}$ is disconnected then this has exactly two components. Further each component is homeomorphic to X by the ...
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0answers
38 views

question about Hopf fibration.

Let's consider the Hopf map: $$ \eta: S^3 \longrightarrow S^2$$ $$ S^3\ni q=(a,b,c,d)=(z,w) \longmapsto \psi(q)(p)=\begin{pmatrix} a^2+b^2-c^2-d^2& \\ 2(bc+ad) & \\ 2(bd-ac)& ...
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0answers
24 views

$\pi_0$ of $M(2) \wedge M(2)$

My motivation is trying to understand Tom Goodwillie's argument here: http://mathoverflow.net/questions/87919/difficulties-with-the-mod-2-moore-spectrum and the only thing I don't get is why ...
3
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1answer
28 views

Equivalence of relative and (reduced) homology for arbitrary pairs

I could not find my mistake in the following argument, though I know it is wrong. This is more like a "Q&A", since there is nothing to "prove" in the positive sense. Here it goes: For an ...
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0answers
33 views

How to describe a homomorphism from a fundamental group to a finite group?

Let $S$ be a connected locally noetherian scheme with $s$ a geometric point of $S$. I read something like this: to give a surjective continuous homomorphism from $\pi_1(S, \bar{s})$ to a finite group ...
3
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1answer
29 views

$\tilde{H}_i(S^n-X)$, $X$ a Finite Graph

I came across this question. Prove that $\tilde{H}_i(S^n-X)\cong H_{n-i-1}(X)$ if $X$ is a finite connected graph embedded in $S^n$. By Alexander Duality, this is true if the group on the right is a ...
2
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2answers
32 views

Expressing homotopy groups of spaces of (unpointed) maps $S^1\to M$ in terms of homotopy groups of spaces of pointed maps.

I came across the following problem while studying for a topology exam: Let $M$ be a topological space, let $\Lambda(M)=M^{S^1}$, the space of continuous maps $S^1\to M$ with the compact-open ...
2
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0answers
24 views

Coboundary map problem

I have to show that if $A$ and $B$ are compact connected subsets in the plane such that $A\cap B$ is not connected (and not empty), then $\Bbb R^2\setminus(A\cup B)$ is not connected. The tool I must ...
13
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1answer
258 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
0
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1answer
30 views

Suspension of a CW complex

I want to prove that the suspension $\Sigma X$ of a CW-complex $X$ is a CW-complex, buy I'm starting with CW-complexes and I don't have a clue of how start, so I'd appreciate any help. Thanks. ...
1
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1answer
43 views

Why notion of fundamental group is defined only over a connected scheme?

I went to different references on fundamental group on schemes. It is quite strange for me that the notion of fundamental group is only defined on connected scheme. Does anybody know why?
1
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1answer
29 views

Is $H^0(S^0;G)\simeq G\oplus G$ or $G$?

In the article on topospaces for the (co)homology of spheres, it says $H^0(S^n,G)\simeq H^n(S^n,G)\simeq G$. Is this true when $n=0$? I think not, for if we view $S^0$ as the union of two ...
0
votes
1answer
17 views

Degree of an induced map on $\mathbb{CP}^n$

Let $r :\mathbb{C}^{n+1} \rightarrow \mathbb{C}^{n+1} $ be the map $r(z_0, z_1,\ldots, z_n)=(-z_0, z_1,\ldots, z_n)$. $r$ induces a map $\bar r : \mathbb{CP}^n \rightarrow \mathbb{CP}^n $. What is the ...
0
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1answer
41 views

Why is $H^2(\mathbb{R}P^2,\mathbb{Z})\simeq\mathbb{Z}_2$?

Why is the second cohomology group of $X=\mathbb{R}P^2$ with $\mathbb{Z}$-coefficients $\mathbb{Z}_2$? We can put the usual $\Delta$-structure on $X$ with two vertices, three $1$-simplices, say $a$, ...
6
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1answer
123 views

Mathematical background for TQFT

I am physicist. I`ve started studying Topological QFT. What would you recommend to read in mathematical field for understanding Witten’s old articles of 80s-90s? What books/articles could help form ...
0
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1answer
43 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
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1answer
24 views

Fixed point equivalence [duplicate]

Let $\Bbb D^n$ be n-dimensional ball and $S^{n-1}$ the $n-1$ dimensional sphere realized as boundary of $\Bbb D^n$. Prove that following are equivalent. There is no retraction $\Bbb D^n\to S^{n-1}$ ...
0
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0answers
28 views

Fixed point of simplicial map is subcomplex

If $s:|K|\to |K|$ is a simplicial map, prove that the set of fixed points of s is the polyhedron of a subcomplex of $K^1$, though not necessarily a subcomplex of $K$. I know s at least fixes ...
3
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1answer
213 views

Classification of fundamental groups of non-orientable surfaces

I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$. I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...
0
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0answers
33 views

How to compute the Lefschetz number

Given a continuous function $f: X \to X$ how do you compute: $$ \Lambda_f = \sum_{k \geq 0} (-1)^k \mathrm{Tr}(f_*|H_k(X,\mathbb{Q})) $$ which is known as the Lefschetz number. For instance let $X: ...
2
votes
2answers
43 views

Is there an isomorphism $\mathrm{Hom}(H_1(X),G)\simeq\mathrm{Hom}(\pi_1(X),G)$ when $X$ is path connected?

In Hatcher 3.1.5 on pg. 205, one proves that if $\varphi\in C^1(X;G)$ is a cocycle, where $X$ a space and $G$ an abelian group, then for paths $f$ and $g$ one has various properties $\varphi(f\cdot ...
14
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1answer
344 views

What are the attaching maps for the real Grassmannian?

The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition. The study of characteristic classes tells us that these ...
3
votes
2answers
42 views

How to show that $f_* (\sigma)=\sigma$ where $f$ is mapping between projective spaces $\mathbb{R}\text{P}^3$

Suppose that $f:\mathbb{R}\text{P}^3 \to \mathbb{R}\text{P}^3$ is continuous mapping without fix points and let $\sigma$ be (some) generator of group $H_3(\mathbb{R}\text{P}^3)$. Prove that ...
0
votes
1answer
35 views

Composition of simplicial approximation is again simplicial approximation

If $s:|K^m|\to |L|$ is a simplicial approximation to $f:|K^m|\to |L|$ and $t:|L^n|\to |M|$ is a simplicial approximation to $g:|L^n|\to |M|$. Is $t\circ s: |K^{m+n}|\to |M|$ necessarily a simplicial ...
5
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2answers
81 views

Explicit expression for homeomorphism and homotopy equivalence

In many cases of topology when one needs to show two spaces $X$ and $Y$ are homeomorphic or homotopy equivalent, one uses some description instead of constructing an explicit homeomorphism or homotopy ...
4
votes
1answer
78 views

Topological degree of a complex valued map defined over a circle

Given a continuous map $f \colon S^n \to S^n$, it induces a map $f_{*} \colon \tilde{H}_n(S^n) \to \tilde{H}_n(S^n)$ of the form $f_{*}(z)=k*z$, where $k$ is an integer. Define the degree of $f$ as ...
0
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0answers
27 views

Homeomorphic homogeneous 3-simplicial complexes

I have a simple question on homogeneous (i.e. made only of tetrahedron) 3-simplicial complexes. Suppose we have an homogeneous 3-simplicial complexes. Suppose we choose any couple of simplices having ...
0
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1answer
40 views

Relative cohomology versus cohomology.

Let $S$ be a closed oriented surface, $X$ a finite set of points on $S$. Is it true that $$ H^1(S \setminus X, \mathbb{C}) \simeq H^1(S,X,\mathbb{C}) $$
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0answers
23 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
0
votes
2answers
35 views

Null homotopic by simplicial approximation

If $m<n$ use the simplicial approximation theorem to prove that any map $f:S^m\to S^n$ is null homotopic. Deduce that $\pi_1(S^n)$ is trivial if $n>1$. we have not covered lot on simplicial ...
1
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1answer
97 views

How an empty set is collapsed to a point?

In the original book of Conley Index Theory: Isolated Invariant Sets and the Morse Index chp3.3, p6, Charles Conley mentioned that ...
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0answers
40 views

simplicial approximation of $f(x)=4x^2-1$ [on hold]

I have a question on practice sheet: Find simplicial approximations to $f:|K|\to |L|$ and $f:|K|\to |M|$ where $f(x)=4x^2-1$ I doubt this question. Do you have any idea to start with.
2
votes
0answers
28 views

Filling the details of a construction via clutching function of a Vector Bundle

Let $(E,\pi,X)$ a complex vector bundle over X (which we assume to be Compact-Hausdorff) Let $$f \colon E \times S^1 \to E \times S^1$$ an automorphism of the product bundle $E \times S^1$. We define: ...
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2answers
28 views

Fundamental Group of the special Euclidean matrix group of the plane

How do you do this? Compute the fundamental group of the special Euclidean group of the plane, that is, all matrices of the form: $ \left( \begin{array}{ccc} \cos(z) & \sin(z) & x \\ ...
2
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0answers
13 views

Euler class of quotient bundle of real projective space

Let $\gamma$ be the real tautological line bundle over the real projective space $\mathbb{R}P^n$, $V$ be the trivial bundle of rank $n+1$, and $Q$ be the quotient bundle $V/\gamma$. What is the Euler ...
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0answers
46 views

Ring structure of $K(X)$ - definition of multiplication

Maybe it's a silly question, but I can't find a satisfactory answer to it. Hatcher defines the multiplication of two arbitrary elements in $K(X)$ as $$(E_1-E_1')(E_2-E_2') := E_1 \otimes E_2 - E_1 ...
4
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0answers
54 views

A question on fixed point property

Assume that $0<k<n-1$, Note that $\mathbb{C}P^{k}$ can be considered as a closed subset of $\mathbb{C}P^{n}$, in a natural way. We collapse $\mathbb{C}P^{k}$ to a point. The resulting space is ...
0
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1answer
17 views

Isotropy groups of tetrahedron after identifying its sides

If we identify the 4 sides of a regular tetrahedron in $\mathbb{R}^3$ by letting the group of all isometries of the tetrahedron act on it, what would the resulting space look like? The resulting ...
2
votes
1answer
50 views

Determining the induced map on homology $\tilde{H}_n(\mathbb{R}^n-\{0\})$ of $f\colon \mathbb{R}^n\to\mathbb{R}^n$ based on sign of $\det(f)$.

I'm having difficulty understanding the following. It appears as Exercise 7, p. 155 in Hatcher's Algebraic Topology: (this is not homework, by the way) For an invertible linear transformation ...
1
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1answer
432 views

Group action and covering spaces

Let $X$ be a path-connected and locally path-connected topological space. The action of a topolgical group $G$ on $X$ is a covering space action. For any subgroup $H < G$, we have a composition ...
1
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1answer
15 views

An equivalence class is open in the basis of a covering map.

Let $p:\tilde X\to X$ be a covering. Define an equivalence class on $X$ as follows $$x\sim z \Leftrightarrow |p^{-1}(x)|=|p^{-1}(z)| $$ where $||$ means the cardinality of the fiber. Take any $x\in ...
3
votes
2answers
70 views

Topological/homotopical classification for 1-dim CW-complexes?

It's a common exercise to classify a collection of 1-dim objects, say the figures of 0-9, or A-Z, up to homeomorphism or homotopy equivalence. I suddenly raise a question in general: Is there any ...
6
votes
2answers
126 views

How do Homology Groups work

How do homology groups work? Looking at the wikipedia article, it lists, for example, $H_k(S^1) = \mathbb Z$ for $k = 0,1$ and ${0}$ otherwise. It also says that $H_k(X)$ is the k-dimensional holes in ...
2
votes
1answer
36 views

Calculating fundamental group of adjunction space with linear transformation.

$X = D^{2} \times S^{1} \cup_{f} S^{1} \times D^{2}$, where $f : S^{1} \times S^{1} \to S^{1} \times S^{1}$ is a map induced by the linear map on $\mathbf{R}^{2}$ given by the matrix $$\left( ...