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

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2
votes
1answer
98 views

The first Kirby move and $\mathbb{C}P^2$

A surgery on a link in $S^3$ can be regared as 2-hadnle attachements to $D^4$ (resulting 4-manifold $W$ whose boundary is the result of the surgery). I would like to know how the first Kirby move ...
2
votes
1answer
34 views

Difference between retract and deformation retract

I have a trouble with distinguishing retraction and deformation retraction intuitively. That is, deformation retraction is informally an operation on a space which continuously deform(for an ...
1
vote
0answers
27 views

Bundle map is isomorphism iff it covers a homeomorphism

Consider $P_0$ and $P_1$ principal G-bundles with projection maps $\pi_0, \pi_1$, respectively; $f:P_0 \rightarrow P_1$ a continuous G-equivariant map (i.e. a bundle map) and $g:X_0 \rightarrow X_1$ ...
0
votes
1answer
35 views

Mistake in Gabriel-Zisman regarding change-of-base of topological spaces?

In III.2.2 of Gabriel-Zisman, a Proposition is asserted which says that the base of change functor sending $X \to B$ to $X \times_{B} B'$, for any $B' \to B$ commutes with colimits in the $X$ ...
0
votes
0answers
50 views

$\Bbb R^5$ without two circles and line

How to prove that $\Bbb R^5$ without two $S^1$ and one $\Bbb R^1$ is homotopiclly equivalent to wedge of $(\Bbb R^5\backslash \Bbb R^1)$ and two copies of $(\Bbb R^5\backslash S^1)$? The only idea ...
1
vote
1answer
38 views

How do I prove that $R^n\setminus R^k$ is homeomorphic to $S^{n-k-1}\times R^{k+1}$?

Let $k,n$ be positive integers such that $k<n$. How do I prove that $\mathbb{R}^n\setminus \mathbb{R}^k$ is homeomorphic to $S^{n-k-1}\times \mathbb{R}^{k+1}$? I tried to put specific integers in ...
1
vote
2answers
49 views

Pullbacks and homotopy equivalences

Say I have a map between pullback squares $(Y \rightarrow Z \leftarrow X) \to (Y' \rightarrow Z' \leftarrow X')$. If the maps $X \to X'$, $Y \to Y'$ and $Z \to Z'$ are homotopy equivalences, does it ...
3
votes
0answers
53 views

Vector Bundles over Spheres

I would like to understand how to construct a vector bundle over the n sphere give a map of its equatorial $(n-1)$-sphere into the general linear group $GL_n(\mathbb{R})$. My thought is that one ...
0
votes
0answers
27 views

Is the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ isomorphic to $\frac{SO(3)}{H}$?

I have heard many times that the homotopy group of the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ and of the space $\frac{SO(3)}{H}$ are identical. I.e., $\frac{SO(3) \times Z_2}{H \times Z_2} ...
5
votes
1answer
86 views

What's wrong in my thinking about Bézout's theorem?

First, I know that every hypersurface of degree $d$ defined in $\mathbb{CP}^n$ is diffeomorphic. By using this fact, I wanted to calculate the Euler characteristic of hypersurface of degree $d$. To ...
2
votes
0answers
77 views

Show that $\mathbb{Z}_4\rightarrow \mathbb{Z}_8\oplus \mathbb{Z}_2\rightarrow \mathbb{Z}_4$ is exact [duplicate]

I want to know whether $0\rightarrow \mathbb{Z}_4\stackrel{f}\rightarrow \mathbb{Z}_8\oplus \mathbb{Z}_2\rightarrow \mathbb{Z}_4\rightarrow 0$ is exact wrt group homomorphism under addition. Since ...
5
votes
1answer
55 views

Cofibration necessarily has closed image?

I know how to show that if $i: A \to X$ is a cofibration, then $i$ is injective, and in fact a homeomorphism onto its image. My question is, must the image necessarily be closed? I've tried ...
0
votes
0answers
23 views

Confused with notations about Leray's theorem for singular cohomology

The following theorem is copied from Bott's book Differential Forms in Algebraic Topology in Page 192: Theorem 15,11 {Leray's theorem for singular cohomology with coefficients in a commutative ...
2
votes
2answers
106 views

Hatcher Exercise 2.2.38

I'm struggling to show exactness at $C_n\oplus D_n$. Let's take $(x,y)\in C_n\oplus D_n$ in the kernel of $C_n\oplus D_n\to E_n$, i.e. the pushforwards $x', y'$ into $E_n$ resp. satisfy $x' + y' = ...
2
votes
1answer
41 views

Higher homotopy groups of wedge of circles.

Using van-kampen theorem, Fundamental group of wedge of n-circles is free group on n-generator. But I don't know how to calculate higher homotopy groups of wedge of spaces, in particular circles. I ...
4
votes
1answer
51 views

Fiber bundle with null-homotopic fiber inclusion

It is an exercise from Hatcher (exercise 31, page 392): For a fiber bundle $F \to E \xrightarrow{p} B$ such that the inclusion $F \hookrightarrow E$ is homotopic to a constant map, show that the long ...
14
votes
2answers
198 views

Geometric reason as to why $H^2$ of the Klein bottle is $\mathbb{Z}/2\mathbb{Z}$?

I was reading this document when I came across the following: Recall that $H^2(K; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$. Here $K$ denotes the Klein bottle. Is there a good geometric ...
4
votes
1answer
91 views

Fundamental group of open subsets of $\mathbb{R}^n$

Suppose that $U$ is an open subset of $\mathbb{R}^n$. What can be said about its fundamental group? I'm sure that the answer should be well known, since this is rather natural question.
2
votes
1answer
58 views

Orientability of Surfaces and the Fundamental Group

Let $(M,g)$ be a compact riemannian 3-manifold and $\Sigma \subset M$ an embedded compact surface homeomorphic to the projective plane. Consider the application $i_\#:\pi_1(\Sigma)\to \pi_1(M)$ given ...
1
vote
0answers
34 views

Show that there exist no retraction from $RP^n$ to $RP^k$ if n>k.

I am trying this problem from Hatcher's algebraic topology book(4.2.1). If r:X$\rightarrow$A is retraction then I know that this induces injective map in the fundamental group level through inclusion ...
0
votes
0answers
25 views

how to calculate relative homotopy groups?

I am studying nth relative homotopy groups from Hather.For a pair (X,A) where A$\subset$X nth-relative homotopy groups is defined by homotopy class of maps$(I^n,\delta I^n,J^{n-1})$ ...
5
votes
1answer
61 views

Example of a surjective local homeomorphism that is not a covering? [duplicate]

Let $X$ and $Y$ be connected, locally path connected, and Hausdorff topological spaces. Can someone give me an example of a surjective local homeomorphism that is not a covering? I don't think this ...
4
votes
1answer
51 views

Determine when $T(S^n \times S^k)$ is a trivial tangent bundle.

My partial answer is that , to make $T(S^n \times S^k)$ trivial, a neccessary condition is that $n$ or $k$ is odd. My argument is as follows: Suppose $T(S^n \times S^k)$ is trivial, that is, $S^n ...
3
votes
0answers
42 views

Computing some fundamental groups [duplicate]

I'm studying algebraic topology and got stuck. a. $X_n\in \mathbb{R}^3$ is the union of $n$ distinct lines through the origin. Find $\pi_1(\mathbb{R}^3-X_n)$ for each $n$. b. Let $X$ be the sum of ...
1
vote
0answers
23 views

Under what conditions is the homology of a dg coalgebra a graded coalgebra?

I'm trying to get a feel for some differential graded (dg) structures. Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct ...
3
votes
1answer
20 views

Homology of a pair of simplicial complexes

Let $K=\{a, b, c, ab, bc\}$ (in graph-theoretic terms, a path $a,ab,b,bc,c$, where $a, b,$ and $c$ are vertices and $ab$ and $bc$ are edges) be a simplicial complex and let $L=\{a, b, c\}$ be a ...
2
votes
0answers
34 views

definition of homology via spectra

Let $K(\mathbb{Z}, n)$ denote a Eilenberg-Mac Lane space, characterized by $H^n(X, \mathbb{Z})=[X, K(\mathbb{Z}, n)]$ for all spaces $X$. In stable homotopy theory, the corresponding homology theory, ...
2
votes
1answer
95 views

Question about the fundamental group of a connected, open subset of $\mathbb{R}^2$

Let $U \subset \mathbb{R}^2$ be open and connected. Suppose $f: I \to U$ is a loop with $a = f(0) = f(1)$ such that $f$ doesn't wind around any $p \in \mathbb{R}^2 \setminus U$. a) Is it true that ...
2
votes
1answer
85 views

Where does the “CW” in CW-complex come from?

I've heard people say that the "CW" in CW-complex comes from the "CW" in JHC Whitehead, though nobody has ever given me a reference for this. Does anyone know where the "CW" in CW-complex comes from?
8
votes
1answer
66 views

A manifold such that its boundary is a deformation retract of the manifold itself.

If we have a compact orientable manifold $M$, we know that $\partial M$ is not a deformation retract of $M$. This follows from Poincaré Duality or Stokes Theorem. If we take away compactness, this is ...
0
votes
1answer
31 views

Hatcher algebraic topology book prop. 2.29

I am studying algebraic topology from Hatcher book and i don't understand the first sentence of proof of proposition 2.29. on page 135 , Proposition 2.29. $ \mathbb{Z}_2$ is the only nontrivial ...
1
vote
1answer
24 views

configuration-spaces and iterated loop-spaces

In the paper Configuration-Spaces and Iterated Loop-Spaces. Graeme Segal, page 213-214, it is obtained that the labelled configuration space $C_n$ is homotopy equivalent to a topological monoid ...
1
vote
1answer
32 views

Homotopy equivalence re-definition

Homotopy equivalence is defined thus: Two spaces $X$ and $Y$ are homotopy equivalent if there are continuous maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$ with $gf \sim \textbf{1}_X$ and $fg \sim ...
3
votes
1answer
53 views

classify all surface in which $K_{3,3}$ and $K_5$ can be embedded.

Using the fact that $K_{3,3}$ and $K_5$ are not planar, classify all surface in which $K_{3,3}$ and $K_5$ can be embedded. I know $K_5$ can be embedded into a torus. Can anyone give a hint for the ...
1
vote
0answers
28 views

Cup product: binary internal or external operation

Having a look at the definition of cup product https://en.wikipedia.org/wiki/Cup_product I harbour some doubts as to whether it is a binary internal operation (or function) or rather an external one. ...
0
votes
1answer
88 views

Questions on CW-complexes

I am trying to proof the following two statements. If $X_1 \subset \dots \subset X_i \subset \dots$ is a infinite sequence of CW-complexes, then $X = \bigcup X_i$ is a CW-complex and each $X_i$ is a ...
5
votes
2answers
76 views

actions of $\mathbb{Z}_2$ on spheres

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. Why $F(S^m,2)/\mathbb{Z}_2$ is ...
1
vote
0answers
58 views

short exact sequence of algebras over a field

Let $A,B,C$ be algebras over a field $F$ ($F=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime). The height of $A$ is defined to be $$ \mathrm{height}(A)=\sup_{a\in A}\inf\{n(a)\in \mathbb{N}\mid a^{n(a)+1}=0 ...
1
vote
2answers
69 views

Euler characteristic: dependence on coefficients

Let $X$ be a finite CW complex and $\chi(X)$ its Euler characteristic (defined using integer coefficients). When is it true that $\chi(X)=\sum (-1)^i \dim H_i(X;F)$, where $F$ is a field? I thought ...
2
votes
3answers
94 views

Show that the letters X and I (thought of as topological spaces) are not homeomorphic [duplicate]

I am reading Hajime Sato's: Algebraic Topology, an Intuitive Approach. His Sample Problem 1.3 is: Show that the topological spaces X and I are not homeomorphic. (Note that this requires a font where ...
3
votes
0answers
66 views

$R^2\setminus K, K$ compact, is not simply connected

So, I think I'm aware of the general idea of how to do this. $K$ is compact, hence closed and bounded, so there is some circle of finite radius, say $r$, that wraps around $K$. This loop isn't null ...
0
votes
0answers
76 views

Precisely what is meant by “$\pi_1(M)$ is torsion”?

I am reading a paper where one of the conditions for a Theorem to hold is "the group $\pi_1(M)$ is torsion", where here $M$ is a compact differentiable manifold. What is meant by the first homotopy ...
2
votes
3answers
99 views

Homeomorphic manifolds have the same dimension

So I want to prove: If two manifolds $M$ and $N$ are homeomorphic then $dim(M) = m = n = dim(N)$. My idea was to use the property of the manifolds that they are locally homeomorphic to the ...
3
votes
1answer
42 views

If $X \supseteq B \supseteq A$, then $B/A$ subspace of $X/A$

Problem 4.3.3 in Ronnie Brown's Topology and Groupoids asks Let $A, B$ be subsets of $X$ such that $A$ is closed and $A \subseteq B$. Show that $B/A$ is a subspace of $X/A$. I think I have ...
2
votes
0answers
66 views

Homotopic maps between connected spaces inducing the same homomorphism between the fundamental groups

This is Problem 7-9 in Lee's Introduction to Topological Manifolds: Suppose $X$ and $Y$ are connected topological spaces, and the fundamental group of $Y$ is abelian. Show that if $F,G: X ...
4
votes
0answers
41 views

Is wedge sum for finite CW complexes cancellative in the homotopy category?

Let $X,Y,Z$ be finite pointed CW complexes. Is it possible that $X\vee Z$ and $Y\vee Z$ are homotopy equivalent, but $X$ and $Y$ are not? Remark 1: Without the finiteness assumption on $Z$, there are ...
0
votes
0answers
17 views

Homology groups of $S^1\times (S^1 \vee S^1)$ [duplicate]

I'm trying to calculate the homology groups of $S^1\times (S^1 \vee S^1)$. This complex has two 2-cells, three 1-cells and one 0-cell, so using cellular homology, I have deduced that $H_2 = ...
0
votes
1answer
43 views

Notation: determinant of Jacobian matrix

Given a function $f:\Delta^n \to Y$ from a simplex into a riemmannian manifold. Furthermore given a point $x \in \Delta^n$ we can send an orthonormal basis at $x$ using $D_x f$ to a set of vectors ...
5
votes
0answers
63 views

“Toys” spaces in algebraic topology

I did follow a course of algebraic topology last semester and I still want to continue to do some computations. But in many books it's all the time the same examples which comes back for computing ...
5
votes
1answer
122 views

Non-orientable one dimensional manifold.

I was trying to solve a question from Hatcher's book in section 3.3. Question is: Show that there exist a non-orientable 1-dimensional manifold if Hausdroff condition is droped from the definition ...