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

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30 views

Retraction and intersection

Let $X$ be a topological space, and consider two open subsets $U$, $V$ of $X$ such that there exist two continuous maps $r_{U}: X\longrightarrow U$, $r_{V}:X\longrightarrow V$ which are homotopically ...
2
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0answers
26 views

cohomology of orbit space by a free group action

Let $G$ be a group. Let a principal $G$-bundle $G\to E\to B$. Then we have a fiber sequence $G\to E\to B\to BG$. Let $k$ be a field. Suppose $H^*(BG;k)$ and $H^*(E,k)$ are known. How to get ...
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0answers
32 views

cohomology of orbit space

Let $p$ be an odd prime. Let $T^p=S^1\times\cdots \times S^1$ be the $p$-dimensional torus. Then $$H^*(T^p;\mathbb{Z}_p)=\otimes_pH^*(S^1;\mathbb{Z}_p)=\otimes_p\Lambda_{\mathbb{Z}_p}[a].$$ Here ...
3
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0answers
39 views

Fundamental groups of the projective plane

I'm practicing for my topology final. Munkres' Topology, Theorem 74.4 states that: Let $X$ denote the $m$-fold projective plane, that is the space obtained by taking the connected sum of the ...
3
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1answer
31 views

Practice Problem Fundamental Group of 7-figured polygon

The question is from Munkres: Consider the space $X$ obtained from a seven-sided polygonal region by means of the labelling scheme $abaaab^{-1}a^{-1}$. Show that the fundamental group of $X$ is the ...
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0answers
43 views

Punctured plane

What does one point compactification of singly, doubly, triply punctured plane $\mathbb{R}^2$ look like? What would their fundamental groups look like? I'm trying to visualize but can't seem to draw ...
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2answers
58 views

Join of topological spaces; Mayer-Vietoris

let $X$ and $Y$ be topological spaces, $X\star Y:=\frac{X\times Y\times [0,1]}{\sim}$, where $\sim$ is genereted by $(x,y_1,1)\sim (x,y_2,1)$ and $(x_1,y,0)\sim (x_2,y,0)$ for $x, x_1,x_2\in X$, $y, ...
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0answers
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 ...
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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 ...
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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
42 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 ...
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 ...
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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
50 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
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 ...
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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 ...
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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
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 ...
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
votes
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
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 ...
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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 ...
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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
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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
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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. ...
6
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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 $k\neq n$. ...
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 ...
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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 ...