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

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31 views

Method for defining a number of connected components of real algebraic surface

The question is simple: given the concrete polynomial $f(x,y,z)$ ($x$,$\,$ $y$ and $z$ are real numbers), is there any method for answering this question for a surface $f(x,y,z) = 0$? I'm interested ...
1
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1answer
36 views

Contractible CW-complex

Let Z be a CW complex so that for all $n \in \mathbb{N}$ every continuous $f:S^n\rightarrow Z$ is homotopic to a constant map, where $S^n:=\{x \in \mathbb{R}^{n+1}$ | |x|=1}. Then there is a ...
2
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2answers
90 views

Computing $\pi_4(S^3)$ using Serre spectral sequence

I'm following Davis & Kirk's computation that $\pi_4(S^3)=\mathbb Z/2$ using the Serre spectral sequence but I'm having problems at the very end. We consider a homotopy fibration $X\to S^3 \to ...
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1answer
47 views

definition of a $\Delta$ - complex

I've been given this image from hatchers algebraic topology as an example of a $\Delta$ complex but with the explicit definition as follows A $\Delta$-complex structure on a space X is a collection ...
0
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1answer
24 views

Group of deck transformations cyclic

Given a pointed topological space $(X,x_0)$, let $p\colon (\tilde{X}, \tilde{x}_0)\to (X,x_0)$ be a covering of that space. Write $p^{-1}(x_0)= \{\tilde{x}_0, \tilde{x}_1,\ldots,\tilde{x}_n\}$. I'd ...
1
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1answer
41 views

Fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$. ...
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0answers
37 views

Metrizability of the symmetric product of a metric space

The (infinite) symmetric product of a based topological space $(X,e)$, denoted by $SP(X,e)$, can be viewed as the topological space of ''multisets'' in $X$ containing the base point $e$ infinitely ...
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2answers
72 views

pushout of topological Hausdorff spaces is not Hausdorff

$A$, $X$, $Y$ are topological Hausdorff spaces, $f:A\to X$, $g:A\to Y$ continuous maps. I search an example where the pushout $Z$ of the morphisms $f$ and $g$ is not Hausdorff. I thought if I take ...
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1answer
33 views

Show that the inclusion $V \to U$ is nullhomotopic.

Show that if a space $X$ is deformation retract to a point $x \in X$, then for each neighbourhood $U$ of $x$ $\exists$ a neighbourhood $V \subset U$ of $x$ s.t the inclusion $V \to U$ is ...
5
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1answer
131 views

Question on proof of Lefschetz Fixed Point Theorem (from Hatcher Theorem 2C.3)

In Hatcher's statement of the Lefschetz Fixed Point Theorem (2C.3), he has a hypothesis that the space $X$ in question must be a retract of a finite simplicial complex. The first part of the proof ...
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0answers
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 ...
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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 ...
0
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1answer
35 views

Stiefel Whitney class and embedding problem.

I have a problem, reading the book "Characteristic classes". In the page 120, Corollary 11.4 is that If $M = M^n$ is smoothly embedded as a closed subset of the Euclidean space $\mathbb{R}^{n+k}$, ...
3
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1answer
74 views

line bundles over the circle

I read in various places that up to isomorphism there are only two line bundles ( 1-d vector bundles) over a circle, the trivial one and the mobius strip. On the other hand, when I make a mobius strip ...
2
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0answers
49 views

Homology of wegde sum, exact sequence

I am solving an exercise: showing that for two CW complexes $X$ and $Y$ the homology of the wedge sum $X \vee Y$ can be expressed as direct sum. This has already been solved elsewhere here. My ...
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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 ...
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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}) ...
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 ...
0
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1answer
58 views

Homology class and Betti number for a compact manifold with boundaries

If I take $Q=\mathbb{Z}_N \equiv \mathbb{Z}/(N \mathbb{Z})$, for a genus-g 2-dimensional Riemann surface $\Sigma$, I should have $$H_1(\Sigma; \mathbb{Z}_N)=\prod^{2g}_1 \mathbb{Z}_N,$$ So, ...
3
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1answer
47 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 ...
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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? ...
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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 ...
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 ...
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 ...
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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 ...
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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 ...
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 ...
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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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 ...
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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
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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 ...
3
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1answer
187 views

What is $\mathbb{P}^{\infty}$?

Can we look at a complex projective space $\mathbb{P}^{\infty}$? I am curious to know what would it be. What is the right intuition to think about it? I know $\mathbb{P}^{n}$ is a space of ...
3
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1answer
84 views

Meaning of a long exact sequence

Edit: The setting for the question is some abelian category. From this question I learned that one way to view a short exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ is as ...
5
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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 ...
4
votes
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 ...
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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 ...
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]? ...
5
votes
3answers
101 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. ...
6
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1answer
96 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 $k\neq n$. ...
2
votes
2answers
68 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 ...
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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
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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 ...
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0answers
23 views

Basic question about abelianization of Homotopy Groups and Homology [duplicate]

When precisely, is the homology group: $$H_n(T)$$ of a topological space, $T$, isomorphic to the abelianiation of the corresponding homotopy group $\pi_n$? Does this only occur when $n=1$, or is ...
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1answer
53 views

The relative homology group $H_{1}(\mathbb{R},\mathbb{Q})$

This problem is an extract of Hatcher's book. Show that for the subspace $\mathbb{Q}\subset \mathbb{R}$, the relative homology group $H_{1}(\mathbb{R},\mathbb{Q})$ is free abelian and find a ...
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0answers
30 views

Rotman, Algebraic Topology, Lemma $4.22$

Lemma 4.22. Let $X$ be a space and, for $i=0,1$, let $\lambda _i:X\rightarrow X\times I$ be defined by $x\mapsto (x,i)$. If $H_n (\lambda _0)=H_n(\lambda _1)$, then $H_n(f)=H_n(g)$ whenever ...
2
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1answer
66 views

Moore space, induced map in homology

Let $A$ be a finitely generated abelian group and $n$ a positive integer. I have built a connected space $M(A,n)$ such that all its reduced homology groups are zero but the i-th reduced homology group ...
2
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0answers
46 views

mapping cone and cylinder

Given a map of spaces $f:X \to Y$, the mapping cylinder is the adjunction space $$cyl(f)=(X \times [0,1]) \cup_f Y$$ where we regard $f$ as a map $f: X \times \{1\} \to Y$.\ On the other hand the ...
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0answers
35 views

Correspondence between first homology group and deck transformations.

Let $M$ be a connected topological manifold with universal covering $\pi: \widetilde{M} \rightarrow M$ and let $p \in \widetilde{M}$ be a point. Let $\alpha,\beta : \widetilde{M} \rightarrow ...
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1answer
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 ...
0
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0answers
36 views

Does the classifying map of a fibre bundle only depend on the transition functions?

Does the classifying map of a fibre bundle only depend on the transition functions? Precisely, Let $\xi$ and $\eta$ be two fibre bundles over $B$, whose transition functions are same, both of their ...