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

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3
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1answer
70 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
3
votes
1answer
103 views

Homotopy equivalence -> element in outer automorphism of fundamental group

Let $X$ be a path-connected topological space. for $x \in X, G = \pi_1(X,x)$ Show that a homotopy equivalence $f : X \to X$ gives a well-defi ned element $g \in Out(G)$. How might one begin on this ...
3
votes
1answer
190 views

Application of Lefschetz duality to prove Lefschetz hyperplane theorem

I'm trying to understand the proof of the Lefschetz hyperplane theorem in Milnor's book "Morse Theory", page 41 but I can't understand his use of Lefschetz duality. At this point it has been proven ...
3
votes
1answer
129 views

Counting roots of polynomial inside $S^1$

I would like to ask for a hint to this problem: Let $p$ a polynomial function on $C$ with no root on $S^1$. Show that the number of roots of $p$ with $|z|<1$ is the degree of the map $q: S^1 \to ...
3
votes
1answer
54 views

Unknotting number formally?

I am reading Colin C. Adams's very nice but not always rigorous "The Knot Book" right now. How does one formalize the unknotting number? (For example, is some restriction on embeddings ...
3
votes
1answer
104 views

How many triangles are there in each “layer” of Poincaré disk?

Assume we grow from a single triangle layer by layer to get the whole disk. Every time a new ring of triangles makes all the vertices of the triangles already in the picture surround by seven ...
3
votes
1answer
113 views

A specific example of a CW complex and a few questions concerning it.

The question I am facing is this one: Construct a CW complex X with a 0-cell x(n) for each natural number $n \geq 0$ and a 1-cell $D_{n}^1, n \geq 1$ which is glued to $x(0)$ at one end and $x(n)$ at ...
3
votes
1answer
103 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 ...
3
votes
1answer
81 views

Misprint in Switzer's Algebraic Topology?

I am currently reading Switzer's book "Algebraic Topology: Homotopy and Homology". On page 50, the proof of 3.30 c), he claims that a certian composition is something I can't see how it possibly can ...
3
votes
1answer
91 views

Yoneda's lemma and $K$-theory.

The K-theoretic form of Bott periodicity is that $\tilde{K}(X)\cong \tilde{K}(\Sigma^2 X)$, i.e., $\langle X, BU\times \mathbb{Z} \rangle \cong \langle \Sigma^2 X, BU\times \mathbb{Z} \rangle$. The ...
3
votes
1answer
303 views

Tangent bundle of Grassmann manifold

I have to prove that the tangent bundle of Grassmann manifold $G_n(\mathbb{R}^{n+h})$ is isomorphic to $\operatorname{Hom}(\gamma^n(\mathbb{R}^{n+k}),\gamma^\perp)$, with $\gamma^{\perp}$ is the ...
3
votes
1answer
204 views

Hawaiian Earring

Let $X=[0,1]$ and $A=\{0\}\cup\{\frac{1}{n}|n\in\mathbb Z\}$. Note that $(X,A)$ is not a good pair. Show that $H_1(X,A)$ is not isomorphic to $H_1(X/A)$. I have a sequence of homology groups: ...
3
votes
1answer
139 views

Basic question about definition of Chern classes

Apologies if there is something I missed with a quick internet search, but why do we define Chern classes for complex vector bundles (instead of real vector bundles for example)? If we define chern ...
3
votes
1answer
66 views

condition on existence of quillenization

$X_{Ab}$ is the Quillenization of a path-connected space $X$ if $X_{Ab}$ has abelian fundamental group, and there exists a continuous map $X\rightarrow X_{Ab}$ inducing an isomorphism ...
3
votes
1answer
235 views

Sphere with three Möbius strips glued and sphere with a handle and a Möbius strip glued

I am reading the first chapter from Topology by Armstrong. There, after stating the classification theorem for closed surfaces, he has mentioned an example that a sphere with one handle and one Möbius ...
3
votes
1answer
163 views

Covering spaces!

If one wanted to find all connected covering spaces of a product of two spaces, say $S^1\times RP(3)$, how would you go about it? I'm thinking finding the fundamental group of $S^1\times RP(3)$, and ...
3
votes
1answer
93 views

What does this free quotient space look like?

Let $S^2=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2+z^2=1\}$ and $S^1=\{(s,t)\in \mathbb{R}^2|s^2+t^2=1\}$. Suppose that $\mathbb{Z}/2\mathbb{Z}$ acts on $S^2\times S^1$ in such a way that the generator of ...
3
votes
1answer
174 views

How does a simplicial map induce a map on chain complexes

I am working on the following exercise: Show that a simplicial map induces a map on chain complexes (hence maps on homology between simplicial complexes). Here is what I have so far... Let $K, L$ be ...
2
votes
1answer
45 views

Hatcher exercise 2.1.6 (Simplicial homology)

Compute the simplicial homology groups of the $\Delta$-complex obtained from $n+1$ $2$-simplices $\Delta_0^2,...,\Delta_n^2$ by identifying all three edges of $\Delta_0^2$ to a single edge, and for ...
2
votes
1answer
41 views

What exactly is meant by “an integer basis of the $\mathbb{Z}$-module $H^*(M)$”?

I thought I understood the concept of a cohomology ring, but am confused by the following statement found in a textbook. Context: M is a symplectic manifold of dimension $2n$. "Let us choose an ...
2
votes
1answer
36 views

Showing that identity and g are not homotopic (without Homology)

Question: Are the identity mapping on $S^1$ and the reflection about the $x$-axe homotopic? This is a question which I already know the answer. The objective is to find better answers and suggestions ...
2
votes
1answer
21 views

Why does this open cover of $T^n$ have intersection $T^{n-1}\sqcup T^{n-1}$?

When computing the de Rham cohomology of the $n$-torus $T^n$, usually one takes an open cover $T^n=A\cup B$, where $A=T^{n-1}\times S^1\setminus{N}$ and $B=T^{n-1}\times S^1\setminus\{S\}$, where ...
2
votes
1answer
56 views

Inclusion induces Isomorphism on Homology

I'm having some trouble figuring out exactly how to prove that an inclusion map induces an isomorphism on homology. Let X be the 1-skeleton of the Torus $T$, so a wedge of two circles. Let ...
2
votes
1answer
45 views

Connected Sum of Surfaces

I am trying to prove that the connected sum of surfaces is a surface. My definition of surface is: A topological space locally homeomorphic to $\mathbb{R}^2$, second countable, Hausdorff and ...
2
votes
1answer
46 views

Obstruction Theory: extending a map on CW complexes

I'm trying to read about obstruction theory from Davis & Kirk and trying to find a map $ g \colon X_n \rightarrow Y $ from the $ n $-skeleton of a relative CW complex $ (X,A) $ to a path-connected ...
2
votes
1answer
28 views

Generators of a finite CW complex

My question regards a simple application of Van Kampen's theorem. Suppose $X$ is a finite CW complex with a finite one-skeleton. I have to show that X is finitely generated. I know that this is just ...
2
votes
1answer
23 views

Representation of the fundamental class of a closed orientable $n-$manifold

Let $M$ be a closed orientable $n$-manifold with a ∆-complex structure. Let ${σ_1 . . . , σ_k}$ be the set of all $n$-simplices. How does one prove that the fundamental class $[M]$ can be represented ...
2
votes
1answer
28 views

Boundary maps of the projective plane as a $\Delta$-complex (homology)

Hi, very simple question here. In Hatcher's 'Algebraic Topology' the diagram above is used to describe the projective plane as a $\Delta$-complex(see p.102). Later the 2-boundary maps are given by ...
2
votes
1answer
55 views

Acyclic model type result: existence of a chain homotopy

If $\sigma: \Delta_n \to X$, define $\overline{\sigma}: \Delta_n \to X$ by$$\overline{\sigma}(t_0, \dots, t_n) := \sigma(t_n, \dots, t_0).$$Define a map $T: C_n(X) \to C_n(X)$ by $T(\sigma) := ...
2
votes
1answer
75 views

Homology of universal cover of $S^1 \vee S^1 \vee S^2$ is not the same as homology of $\Bbb R^2$

I want to show, that although the homology groups of $X :=S^1 \vee S^1 \vee S^2$ and the torus $T^2$ are isomorphic, the homology groups of their universal covers are not. Let $U_X$ be the universal ...
2
votes
1answer
33 views

Unit close disc to prove a matrix algebra identity?

I need to prove that every $3 \times 3$ matrix with real positive entries has one eigenvector with a positive eigenvalue. Now, how do I prove this using the fact that the set $B=\{x=(x_1,x_2,x_3)\in ...
2
votes
1answer
53 views

Question about two homeomorphic closed manifolds

I was studying about algebraic topology with my study group. So, there was a question that held all of the study members confused. If two closed manifolds are homeomorphic, they must have same ...
2
votes
1answer
40 views

$E_{\infty}$ spaces are $A_{\infty}$ spaces

While studying the well-known "Geometry of Iterated Loop Spaces", I found this corollary which is not completely clear to me. (By $\mathcal{M}$ is meant the operad given by $\mathcal{M}(j):=\Sigma_j$, ...
2
votes
1answer
31 views

Continuity, Smash product, etc.

Let $X,Y,K$ be pointed spaces and $K$ locally compact Hausdorff. Let $f:X\rightarrow Y^K$ and define $g:X\wedge K\rightarrow Y$ by $g(x\wedge k)=f(x)(k)$. I want to prove that $f$ is continuous iff ...
2
votes
1answer
69 views

Isomorphism of Fundamental Groups (arcwise connected)

In an arcwise connected topological space $X$, we can show that the two groups $\pi(X,x)$ and $\pi(X,y)$ are isomorphic for $x,y \in X$ by defining a mapping $u: \pi(X,x) \to \pi(X,y)$ by $\alpha ...
2
votes
1answer
92 views

Euler Classes, Chern Classes, $S^2$ Bundles, and $CP^1$ Bundles

I am just starting out learning about characteristic classes (Euler, Chern, etc.) from Bott and Tu's book, and I had the following question. Let $E$ be an oriented $S^2$ bundle over $M$ with ...
2
votes
1answer
30 views

Covering of Graph

Let $X$ be graph and $p:E\rightarrow X $ be covering map. It is followed by Theorem $83.4$, $E$ is graph. Now, assume that $v\in V(X)$ is a vertex with $deg(v)<\infty$ and $w\in p^{-1}(v)$. Can we ...
2
votes
1answer
86 views

Cell structure of $S^2 \times S^1$

Can anyone please provide the cell structure of $S^2 \times S^1$? I know that there are one cell in each dimension from 0 to 3 but I am not sure about the attaching maps. Thanks in advance.
2
votes
1answer
137 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 ...
2
votes
1answer
149 views

Why do these geometric assumptions imply these statements about relative homology?

I'm reading the paper Coverage in sensor networks via persistent homology. As in the paper, let $\mathcal{D}$ be a bounded domain in $\mathbf{R}^d$. We make the following assumptions: A5 The ...
2
votes
1answer
77 views

Homotopy group of Lens space minus point

I'm trying to solve the following exercise from Algebraic Topology by Hatcher, self-study. Let $ X $ be obtained from a lens space of dimension $ 2n+1 $ by deleting a point. Compute $ \pi_{2n}(X) ...
2
votes
1answer
80 views

A map $f: X\rightarrow Y$ is a homotopy equivalence if and only if $h\circ f,f\circ k$ are homotopy equivalences of $X,Y$ respectively.

Show that $f\colon X \rightarrow Y$ is a homotopy equivalence if and only if there exist maps $k,h\colon Y\rightarrow X$ such that $f\circ k$ is a homotopy equivalence of $Y$ to itself, and $h\circ f$ ...
2
votes
1answer
74 views

The space $\Delta^n$ with all faces of the same dimension.

If the space $A$ is obtained from $\Delta^n$ by identifying all faces of the same dimension; What is a $\Delta$-complex structure on the space $A$? And how can you compute the Simplicial Homology ...
2
votes
1answer
120 views

Equal winding number implies two paths are path homotopic?

Let $\alpha,\beta:[0,1]\rightarrow\mathbb{C}\setminus\{p\}$ be two (continuous) paths (not necessarily closed) with same endpoints ($\alpha(0)=\beta(0)$, $\alpha(1)=\beta(1)$), we know that if ...
2
votes
1answer
182 views

Show the existence of a certain subgroup of F2

This is a homework question: I need to find three subgroups of the free group with two generators, $F_2$, with certain properties. I have found the other two by constructing covering spaces of $S^1 ...
2
votes
1answer
38 views

isometric imbedding of the projective plane

How to isometrically imbed the projective plane (identifying antipodal points of the unit sphere) in $\mathbb{R^5}$? My textbook indicates that there is an isometric imbedding of the projective ...
2
votes
1answer
86 views

CW-complex with zero boundary operators

If I have a CW-complex, is it possible to find a homotopically equivalent one that will have zero boundary operators? It shouldn't be always possible to find such a triangulation for the initial ...
2
votes
1answer
72 views

List of most useful coverings and their applications?

I've heard that many problems may be simplified when looking at covering spaces, but I haven't been able to find a good list. What are the most common covering spaces one should understand by heart? ...
2
votes
1answer
225 views

Prove that the fundamental group of $X$ is Abelian

Let $X$ be a path-connected topological space. And there is a continuous map $F: X\times X \to X$ such that: $$F(x,x)=x \ \text{ and }F(x,y)=F(y,x).$$ Prove: The fundamental group of $X$ is Abelian. ...
2
votes
1answer
52 views

Cohomology of volume forms

If g and h are Riemannian metrics on the same manifold, say both of volume 1, then it follows (I guess from Poincaré duality) that their volume forms dvol_g and dvol_h are cohomologous. Question: is ...