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

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0
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13 views

Barycentric subdivision

Could sb tell me the proof that making barycentric subdivision twice of a polyhedron always gives a regular triangulation with tetrahedon faces? (This is not true with just one barycentric ...
0
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0answers
32 views

Some problem regarding $S^{\infty}$…

I have some questions regarding $S^{\infty}$. First of all I am facing some some problem regarding the definition of $S^{\infty}$. So can anyone please explain how can we see $S^{\infty}$, any ...
7
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1answer
111 views

Why is this space aspherical?

Let $X = Y \cup Z$ be a connected, path-connected Hausdorff space. Suppose that $Y$, $Z$, and $Y\cap Z$ are all connected, path-connected, and aspherical, and that the homomorphism $\pi_1(Y\cap Z) ...
3
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1answer
48 views

Borel-Moore Homology and Kunneth Formula

Given two algebraic varieties $X$ and $Y$. It is true that $H^{BM}_n(X\times Y) \cong \bigoplus_{i+j=n} H^{BM}_i (X)\otimes_\mathbb{Q} H^{BM}_i(Y)$. I think that the proof is similar to the one in ...
1
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1answer
118 views

Group isomorphism for deck transformation in covering space.

When reading Lee's book, I encountered the following exercise: Let $\mathcal{P}\colon M\rightarrow G\backslash M$ be the covering arising from a free and proper discrete group action of $G$ on ...
0
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1answer
35 views

topological graph theory and the first Betti number

I am confused by a statement: in Wikipedia, In topological graph theory the first Betti number of a graph G with n vertices, m edges and k connected components equals $$m - n + k.$$ I am ...
1
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0answers
53 views

Hopf invariant and homotopy groups of spheres

I am trying to understand how to use Hopf invariant, to calculate $\pi_{4n-1}(S^{2n})$. I've started with defining a new space $X$ adjoining $D^{4n}$ to $S^{2n}$ via a map $\phi\in\pi_{4n-1}(S^{2n}) ...
1
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1answer
82 views

Betti number and the homology class - what determines the coefficient $Q$?

From Wikipedia: For a non-negative integer $k$, the $k$th Betti number $b_k(X)$ of the space $X$ is defined as the rank (number of generators) of the abelian group $H_k(X)$, the $k$th homology group ...
2
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0answers
43 views

Eilenberg–Steenrod axioms for homology without pairs of spaces

Say a functor $H\colon \mathrm{Top} → \mathrm{Ab}^ℤ$ satisfies the following set of axioms: Homotopy: If maps $f \colon X → Y$ and $g\colon X → Y$ are homotopic, then $H(f) = H(g)$. Excision’: If $T ...
1
vote
2answers
143 views

Show that $P\colon S^3\to SO(3)$ is a covering map.

Please help, anyone? This I have done so far: identify $S^3$ with the quaternions of unit length and identify $R^3$ with the pure quaternions, that is, those of the form $\{b_i, c_j, d_k\}$, for ...
2
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1answer
76 views

An example of why this homomorphism of homotopy groups is not injective.

For a given path-connected space $X$, I recently learned that one could construct a CW complex $X_{1}$ by considering each generator $j: S^{q} \rightarrow X$ of $\pi_{q}(X)$ and setting ...
2
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0answers
35 views

Sequence of cofibrations gives strict commutativity of given triangle

The problem is the following: Suppose $X_0\rightarrow X_1\rightarrow\cdots$ is a sequence of cofibrations. We will denote $f_n:X_n\rightarrow X_{n+1}$. Assume that we have also maps ...
3
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2answers
58 views

Question about the problem that $P^n$ admits a field of tangent $1$-planes if and only if $n$ is odd.

I want to ask the problem 4-C in the characteristic classes written by John W. Milnor. Problem [4-C]. A manifold $M$ is said to admit a field of tangent $k$-planes if its tangent bundle admits a ...
3
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1answer
108 views

comparing Betti numbers

My question is about what one could say about the Betti number of both spaces $X$ and $Y$ relative to one another if we have a map $f$ between them (e.g., a classical case is when $f$ is a covering ...
1
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0answers
42 views

The smallest $n> 0$ with the nonzero $n$th Stiefel-Whitney class is a power of 2 when total Stiefel-Whitney class is not trivial.

This is the Problem 8-B form the characteristic classes by John W. Milnor and James D. Stasheff. [Problem 8-B]. If the total Stiefel-Whitney class $w(\xi) \neq 1$, show that the smallest $n>0$ ...
2
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1answer
56 views

Relative $CW$-complexes and Serre fibrations

Suppose we have a (continuous) map $p:E\rightarrow X$. Assume that $p$ has the right lifting property with respect to any relative $CW$-complex $i:A\rightarrow B$. I want to show that $p$ is a Serre ...
6
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4answers
332 views

Klein bottle as two Möbius strips.

I read that glueing together two Möbius strips along their edges creates a surface that is equivalent to the so-called Klein bottle. The Möbius strip comes in two versions that are mirrored ...
0
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1answer
48 views

Homotopy Type of Surface of Genus g

Need help with the following exercise; "Let M be a compact orientable surface of genus g. Prove that M with a point removed has the same homotopy type as 2g circles with a point in common." I have ...
1
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0answers
37 views

Null-homotopic maps

Assume that $[\alpha]\in\pi_n(X,x_0)$. I want to prove the following: $[\alpha]=0$ if and only if $\alpha:S^n\rightarrow X$ extends to a map $D^n\rightarrow X$. Can someone help me with this proof? ...
1
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1answer
45 views

$\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$

I need help with this problem Let $\sum X =C^+X \cup_X C^-X$ be the union of two cones on $X$ with a common base. Show that $\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$. Dose this give an ...
3
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1answer
153 views

Homotopy groups of $n$-torus with a point removed.

Is there a simple way how to compute and present homotopy groups of $T^n=S^1\times \ldots\times S^1$ with a point (or several points) removed?
0
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1answer
43 views

Locally finite type space relate homology and cohomology

This is certainly an easy question... Why does a map of spaces $f:X\rightarrow Y$ which induces an isomorphism in cohomology $f^*:H^*(Y)\rightarrow H^*(X)$ induces an isomorphism in homology ...
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0answers
26 views

Is there a complete Link Invariant for links with N crossing.

Are there known examples of pairs $\left(f, N\right)$, where $f$ is a link invariant that is known to be complete when restricted to link diagrams that have at most $N$ crossings? (Ideally, f should ...
1
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0answers
23 views

group cohomology of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $F$ be a field. What is the cohomology $$ H^*(\Sigma_k;F)=H^*(K(\Sigma_k,1);F)=H^*(B\Sigma_k;F)? $$ For $F=\mathbb{Z}/p\mathbb{Z}$ for prime ...
5
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1answer
112 views

Leray-Hirsch Using Kunneth Formula from “Differential form in Algebraic Topology” by Bott and Tu

Kunneth Formula: Let M and F are manifolds. If M has a finite good cover then $H^n(M\times F)=\bigoplus _{p+q=n} H^p(M)\bigotimes H^q (F)$ Bott and tu says One can prove Leray-Hirsch theorem by the ...
6
votes
1answer
97 views

Is this morphism of spectra zero in the stable homotopy category?

Let $f\colon A\to B$ a morphism of spectra and suppose that both spectra $A$ and $B$ have only one non-zero stable homotopy group $\pi_n$, more precisely $$ \pi_k(A)=0=\pi_k(B) $$ for $k\neq n$. ...
5
votes
1answer
77 views

Comparing notions of degree of vector bundle

In this question, $X$ will be a smooth complex projective variety. This question will be about comparing two different ways of calculating the degree of a vector bundle on such an $X$. I understand ...
-2
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0answers
56 views

Orientability of a manifold

If $X$ is a $n$-manifold, the orientation of $x$ is defined with a choice of generator of $H_{n}(X,X\setminus x)$. 1/ Show that deleting a point from a manifold of dimension greater than $1$ does ...
4
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2answers
146 views

The Zariski density for two given sets.

Let $A$ and $B$ be two subsets of $\mathbb{C}^n$: $ A = \mathbb{Z}^n$, and $B=\{ (z_1,z_2, \dots , z_n) \in A \text{ such that } z_1>z_2>\cdots> z_n\}$. My questions: Are these two subsets ...
0
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1answer
43 views

How to calculate homotopic groups : $\pi_n(Z)$ and $\pi_n(S^0)$ .

While doing an exercise, I need to show that $Z$ and $S^0$ are not homotopically equivalent. To do so, I'd like to show that $\pi_n(Z) \neq \pi_n(S^0)$ for some $n$. But I can't figure out if to ...
2
votes
1answer
39 views

Reference for secondary cohomology operations

I am learning some homotopy theory and am currently reading Mosher and Tangora. I love the content of this book, it's very terse and comes straight to the point. At the same time I find it very ...
0
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0answers
57 views

What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent?

Suppose that X is a topological space and $x_0$, $x_1$ are points of $X$. What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent? I presume it's enough to just say the path ...
2
votes
0answers
57 views

ruling out non Pseudo-Anosov automorphisms

We are given a fibration $S\to M\to S^1$ where S is a compact hyperbolic surface, M a 3-manifold and $S^1$ the circle. Topologically speaking, it is clear that M has to be the mapping torus ...
1
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2answers
73 views

Covering space of $S^1 \vee S^1$?

Is this a covering space of $S^1 \vee S^1$? I'm not sure what the map from this space onto $S^1 \vee S^1$ does. What is mapped onto which $S^1$?
2
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1answer
65 views

Is this a covering space of $S^1 \vee S^1$?

Is the following a covering space of $S^1 \vee S^1$ ? It would appear so since there is no point that has more than 2 incoming or outgoing arrows. It seems that the potential covering map $p:Y\to ...
0
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0answers
40 views

Short exactness sequence of chain complexes (singular homology)

I want to prove this claim http://planetmath.org/longexactsequenceofhomologygroups that $$0\to C_k(A,B)\to C_k(X,B)\to C_k(X,A)\to 0$$ with the maps $C_k(i):C_k(A,B)\to C_k(X,B)\; [\sigma]\mapsto ...
5
votes
3answers
102 views

The fundamental group of the projective plane minus 2 points?

I'm trying to compute the fundamental group of $\mathbb{RP}^2$ minus 2 points. I'm using the presentation $\langle a\mid a^2\rangle$. Meaning that I'm taking the disk and identifying the two sides. ...
3
votes
2answers
75 views

cohomology of classifying space of cyclic group

(1). Let $p$ be a prime number. Let $B\mathbb{Z}_p$ be the classifying space of the discrete group $\mathbb{Z}_p$. How to obtain $$ H^*(B\mathbb{Z}_p;\mathbb{Z}_p)=\mathbb{Z}_p[t]\otimes \Lambda[e]? ...
3
votes
3answers
128 views

What space is this Homeomorphic to?

Let $X$ be a topological space and let $A ⊂ X$. Let $\sim$ be an equivalence relation on $X$ such that the equivalence classes are: $A$ itself and Singletons $\{x\}$ such that $x ∉ A$. Then ...
4
votes
2answers
109 views

How do I show that $S^1$ is the suspension of $S^0$?

How do I show that $S^1$ is the suspension of $S^0$? I have all the definitions here, I'm just bad at applying them. The suspension of a topological space $X$ is the quotient $CX / (X × ${$1$}$)$, ...
4
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1answer
149 views

Functors that are the homology of a chain complex

Is there an a priori reason why the singular homology and cohomology groups of a space should be computable as the homology of chain complexes? Certainly you can express any family of functors (say, ...
9
votes
1answer
147 views

Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
4
votes
1answer
69 views

The groups $[\Sigma^nX,Y]$ versus the homotopy groups

If $X,Y$ are pointed spaces, denote by $[X,Y]$ the pointed homotopy classes of pointed maps $X\to Y$. The sets $[\Sigma^nX,Y]$ actually have the structure of a group for $n\geq 1$. Here $\Sigma$ ...
3
votes
1answer
97 views

Almost complex structures on a manifold and its exotic copy

Consider a compact simply-connected smooth $4$-manifold $M$ and its exotic copy $M'$. Identify the underlying topological manifolds. Dold-Whitney theorem guaranties ($\mathbb R$-linear) isomorphism of ...
2
votes
2answers
69 views

K-theory formulation of the index theorem

The Atiyah-Singer index theorem is one of the most important results in twentieth century's mathematics. It states that for an elliptic differential operator $D$ on a smooth, oriented compact manifold ...
1
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1answer
41 views

Relative homology groups

I have to compute the homology groups $H_{n}(X,A)$ when $X$ is $S^{2}$ or $S^{1}\times S^{1}$ and $A$ is a finite set of points in $X$. So, I write the exact long sequence : $...\rightarrow ...
1
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1answer
58 views

constructing an explicit homotopy

I can see that the paths $(\cos(\pi s), \sin(\pi s))$ and $(\cos(\pi s), -\sin(\pi s))$ in $\mathbb{R}^2 \setminus \{0\}$ are 'homotopic' But can't construct an explicit homotopy between them. Could ...
0
votes
1answer
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 ...
1
vote
1answer
23 views

null homotopy of 1-skeleton of simply connected simplicial complex induced by homotopy identity

I am reading a book suggesting that the following is true: Let $X$ be a simply connected (maybe finite) simplicial complex. Then there exists a continous map $$g : X \to X,$$ homotopic to the ...
1
vote
2answers
44 views

How do I show that this map is path homotopic to a constant map?

Let $\alpha:[0,1]\rightarrow \mathbb{C}\setminus\{0\}$ be null-homotopic loop. Since $\mathbb{C}\setminus\{0\}$ is path connected, $\alpha:[0,1]\rightarrow \mathbb{C}\setminus\{0\}$ is homotopic to ...