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

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7
votes
1answer
159 views

A bundle which is not associated to a vector bundle.

Let $F\rightarrow E\stackrel{\pi}{\rightarrow} B$ be a fiber bundle with structure group $G$. We know that if we can reduce the structure group to a subgroup of $GL_n$ for some $n$ ...
3
votes
1answer
170 views

A little question about homotopy equivalence

We know the definition of homotopy equivalence: Let $f,g:X \to Y$ be continuous between topological spaces. We say $f$ is homotopic to $g$ if there is a continuous map $H : X \times I \to Y \ \ $ ...
0
votes
1answer
106 views

Presentation of $\mathbb{P}^2 \#\mathbb{T}$

how do I find the presentation of the fundamental group of $\mathbb{P}^2\#\mathbb{T}$? I only know that it is a quotient of the free group of rank 4 by the least normal subgroup containing the ...
1
vote
2answers
86 views

looking for an imbedding of the Torus in 3-dimensional euclidean space

does anyone know an explicit imbedding $h\colon T^2 \to \mathbb{R}^3$ of the torus $T^2=\mathbb{S}^1 \times \mathbb{S}^1$ into $\mathbb{R}^3$ ? Thanks in advance !!! Cheers...
13
votes
4answers
2k views

What algebraic topology book to read after Hatcher's?

I've currently finished chapter 2 of his book and done all the exercises of in chapter 0, 1 and 2. Was wondering when I finished reading this book what book do I read next in algebraic topology?
3
votes
0answers
67 views

Is there are a name for a simplicial complex that is homotopic to the clique complex of its 1-skeleton?

A hollow octahedron is a nice triangulation of the sphere, because once you know the edges, you know everything. The vertices are obviously the ends of the edges, and the faces are any collection of ...
0
votes
1answer
71 views

classifying vortices whose base space is $S^{3}$ or $S^{7}$

On $S^{1}$ there are three choices for a vector pointed inward to the same angle at the tangent line at every point. The vector is either in the opposite direction as the normal, or to the left of ...
5
votes
2answers
236 views

Universal covering of $SO(3,\mathbb{R})$

How do you prove that the universal covering of $SO(3, \mathbb{R})$ is $S^3$ ? Or equivalently, that it is diffeomorphic to $P_3\mathbb{R}$ ? Thank you for your answers.
2
votes
1answer
170 views

Intersection of the Irreducible Components of Intersections of Schubert Varieties

Let $K$ be an algebraically closed field and $G$ be the Grassmannian of $k$ planes in some $l$ dimensional vector space $V$ over $K$. Let $V_1\subsetneq ... \subsetneq V_l$ be a flag for $V$. A ...
0
votes
1answer
80 views

Representation of a fundamental group.

Consider the fundamental group $\pi_1(\mathbb{CP}^1\backslash \{a_1, \ldots, a_n\})$. It is said that there is a representation: $\pi_1(\mathbb{CP}^1\backslash \{a_1, \ldots, a_n\}) \to GL(n, ...
1
vote
1answer
261 views

What is the fundamental group of $\mathbb{CP}^1$ minus a finite set of points?

Let $\mathbb{C}\mathbb{P}^1$ be the projective space. Let $a_1, \ldots, a_n \in \mathbb{C}$. What is the fundamental group $\pi_1(\mathbb{CP}^1\backslash \{a_1, \ldots, a_n\})$?
3
votes
2answers
481 views

Hatcher chapter 0 exercise.

Show that $f:X \rightarrow Y$ is a homotopy equivalence if there exist maps $g,h:Y \rightarrow X$ such that $fg \simeq \mathbb{1}$ and $hf \simeq \mathbb{1}$. Why isn't this trivial. Surely if f is a ...
1
vote
1answer
132 views

Free Product of Groups

I need help with understanding the proof of the following Lemma(68.1, page 414) in Munkres' Topology, 2nd edition. I will be grateful if someone could take me through the proof or provide a different ...
5
votes
2answers
718 views

Intersection Pairing and Poincaré Duality

Let $M$ be an $n$-dimensional compact and oriented manifold. Then one can define the intersection pairing $H_k(M,\mathbb Z) \times H_{n-k}(M,\mathbb Z) \to \mathbb Z$. One possible formulation of the ...
3
votes
1answer
99 views

Singular Homology- Question regarding a theorem

I'm currently trying to learn Singular Homology from Munkres' book- "Elements of Algebraic Topology" . On page 173, theorem 30.7 appears: " If $f,g:(X,A) \to (Y,B) $ are homotopic, then $f _ * = g_* ...
3
votes
1answer
274 views

Surgery, framing and Dehn twist

Let $L$ be a framed knot in $S^3$. Let $U$ be a closed regular neighborhood of $L$ in $S^3$. How can I interpretate the following sentence? "We identify $U$ with $S^1 \times B^2$ so that $L$ is ...
1
vote
2answers
285 views

A Problem Reading Like “Snake Lemma” meets Cohomology

When studying homology, I have been told that there is a result called the Snake Lemma that via induction affords us with a long exact sequence of homology groups. I am wondering if the ...
4
votes
1answer
234 views

a map with contractible domain must be nullhomotopic

Let $f:X\to Y$ be a map between top spaces $X$ and $Y$. Is the following true? If $X$ is contractible then $f$ must be nullhomotopic. Here is an argument: Since $X$ is contractile then $id_X\simeq ...
3
votes
1answer
143 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 ...
3
votes
0answers
76 views

How to do this surgery?

Let $L$ be a $0$-framed trivial knot in $S^3 \subset B^4$. Take $B^3 \subset B^4$ such that $B^3$ splits $B^4$ into two and $\partial B^3$ intersects $L$ only two points. Take a neighborhood $U$ of ...
2
votes
1answer
174 views

Approaching a Cohomology Computation

If $U \subseteq \mathbb{R}^2$ is the complement of $d > 0$ points in the plane, and $H^k$ denotes the $k$-th cohomology group, how should I verify that $H^k (U)$ equals $\mathbb{R}$, ...
12
votes
2answers
558 views

For every $k \in {\mathbb Z}$ construct a continuous map $f: S^n \to S^n$ with $\deg(f) = k$.

Suppose $S^n$ is an $n$-dimensional sphere. Definition of the degree of a map: Let $f:S^n \to S^n$ be a continuous map. Then $f$ induces a homomorphism $f_{*}:H_n(S^n) \to H_n(S^n)$ . Considering the ...
1
vote
0answers
62 views

Surgery and boundary

Let $L$ be a framed link in $S^3$ with $m$ components and let $U$ be a closed regular neighborhood of $L$ in $S^3$. Let $B^4$ be a closed 4-ball bounded by $S^3$ so that $U \subset S^3$. Gluing $m$ ...
0
votes
1answer
125 views

Definition of a 4-cobordism with boundary

Definition of a 3-cobordism (in my context) is a pair $(M, \partial_{-} M, \partial_{+}M)$, where $M$ is a closed orientable topological 3-manifold and $\partial M$ is a disjoint union of ...
6
votes
1answer
164 views

Show that if $X$ is a compact metric space and an ANE, such that $H_n (X) \neq 0$, then $X$ cannot be embedded in $\mathbb{R}^n$.

An absolute neighborhood extensor (ANE) is a space $Y$ such that for every metric space $X$, $A$ - a closed subset of $X$, and a map $f:A \to Y$, there exists an open set $U$ containing $A$ such that ...
3
votes
0answers
283 views

question about the short exact sequence arising from a fibration

Let $F\hookrightarrow Y\stackrel{f}{\longrightarrow} B$ be a fibration. If $F$ is contractible in $Y$ via some homotopy $H: F\times I\rightarrow Y$, we get split short exact sequences: $0\rightarrow ...
6
votes
0answers
262 views

Fixed Points of a Reflection

This question is about problem 2.C.5 from Allen Hatcher's Topology. The statement of the problem is as follows: Let $M$ be a closed orientable surface embedded in $\mathbb{R}^3$ in such a way that ...
2
votes
0answers
191 views

The Cartesian product of two simplices

Here is a result similar in flavor to the barycentric subdivision post I wrote: If $\sigma$ and $\tau$ are two simplices of dimension at least 1, the product $\sigma \times \tau$ is generally not a ...
3
votes
1answer
268 views

Can someone explain induced homomorphism to me in the context of simplicial homology?

I have two simplicial complexes A and B, and A is a subcomplex of B. We know that there is an inclusion map from A to B, and I understand how to get the simplicial homology groups of each individual ...
3
votes
0answers
197 views

Simplicial complexes and deformation retracts

I spent a couple of hours today trying to prove the following: Let $L$ be a subcomplex of a simplicial complex $K$. Let $U_L$ be the union of the relative interiors of the relative interiors of all ...
1
vote
1answer
335 views

Computing homology of $S^1$ as in J.W. Vick's Homology Theory book and other questions

I am reading the book Homology Theory (GTM 145) by James W. Vick. On pages 25-26, the first homology of the circle is computed. However, in the computation, there is a sentence I do not understand. ...
4
votes
1answer
629 views

Barycentric subdivisions of simplices yield a simplicial complex

The following interesting result (in particular parts (b) and (d)) is stated either as a obvious fact or as an exercise in several books on algebraic topology: The barycenter $b_\sigma$ of an ...
4
votes
2answers
109 views

Finding a simplicial complex with a special homological feature

I have seen the following result in a few algebraic topology texts (such as Spanier), but only as an exercise: For any sequence $m_1, \ldots, m_n$ of nonnegative integers, there is a connected ...
5
votes
3answers
1k views

Good book on homology

I am looking for a good book on homology theory. I have taken topology classes up to fundamental groups and covering spaces (Munkers book). I have a very good background in algebra up to categories, ...
1
vote
1answer
159 views

Galois covers of Riemann surfaces

Let $G$ be a finite abelian group, and $C$ a compact Riemann surface (algebraic curve) of genus $g$. I am interested in topological Galois $G$-covers $X \to C$, aka \'etale $G$-principal bundles over ...
2
votes
1answer
499 views

Why is the winding number homotopy invariant?

It many sources it's stated that the winding number is invariant under homotopy, but I've yet to actually see why. Suppose you have the formal definition of the winding number. So for a continuous ...
3
votes
2answers
136 views

zeroth hompotopy set of a topological space

let $X$ be a topological space. we define an equivalence class on $X$ by $x\sim y$ if there exists a path $\gamma:I\to X$ that joins $x$ to $y$. now the zeroth homotopy set is the quotient ...
3
votes
0answers
173 views

Deformation retracts of CW complexes

I'm tearing my hair out trying to prove that a contractible subcomplex ,$K$, of a contractible CW complex, $L$, is a strong deformation retract of $L$.   What I have so far: I can show that the ...
4
votes
0answers
436 views

Hatcher 1.3. problem 16

Given maps $X\to Y\to Z$ such that both $Y\to Z$ and the composition $X\to Z$ are covering spaces, show that $X\to Y$ is a covering space if $Z$ is locally path-connected.
3
votes
3answers
436 views

Orientation reversing diffeomorphism

Why each sphere admits an orientation reversing diffeomorphism onto itself? (For even dimensional ones can we take conjugation map?) And why complex projective spaces do not admit? Is there a ...
0
votes
2answers
115 views

diffeomorphism invariance of characteristic classes

I read everywhere :"By definition, the characteristic classes of smooth manifolds are invariant under diffeomorphisms." Does it follow from de Rham cohomology? If this is so, then what about ...
0
votes
1answer
427 views

Degree 1 and orientation-preserving homeomorphism

When I read a text book, I encountered the sentence "The modular group of genus $n$ is the group of isotopy classes of degree $1$ self-homeomorphism of a closed oriented surface of genus $n$". ...
7
votes
1answer
329 views

Very basic homotopy question

I'm brand new to homotopy theory so I'm sure this question is utterly stupid. But anyway I'm trying to understand a proof in the book "Topology and Geometry" by Glen Bredon. This is the proposition: ...
7
votes
1answer
394 views

The union of growing circles is not homeomorphic to wedge sum of circles

Let $X$ be the subspace of $\mathbb{R}^2$ that is the union of the circles $C_n$ of radius $n$ and center $(n,0)$ for $n \in \mathbb{N}$. Show that $X$ and $\bigvee_\infty S^1$ are homotopy ...
12
votes
1answer
383 views

Why isn't $H^*(\mathbb{R}P^\infty,\mathbb{F}_2)\cong \mathbb{F}_2[[x]]$?

We just computed in class a few days ago that $$H^*(\mathbb{R}P^n,\mathbb{F}_2)\cong\mathbb{F}_2[x]/(x^{n+1}),$$ and it was mentioned that $H^*(\mathbb{R}P^\infty,\mathbb{F}_2)\cong \mathbb{F}_2[x]$, ...
1
vote
0answers
26 views

Isomorphims on homologies induced by cylindrical structure

This question is related to my previous question Cylindrical structure and homology. Let $S$ be a oriented compact (topological) 2-manifold. We consider a cylinder $M=S\times I$ over $S$, here $I=[0, ...
1
vote
0answers
87 views

Computing homology of the boundary of two “bonded” 2-simplices

The following is an exercise in computing homology: Let $K$ be the union of the boundaries of two 2-simplices, joined along one edge. Compute the homology of $K$. Since the standard 2-simplex is a ...
0
votes
2answers
102 views

A interesting problem concerning differential forms

The following is an exercise that I have attempted to work through but whose statement I find difficult to parse through: For an open set $U$ in $\mathbb{R}^n$, let $R$ be the ring of $C^\infty$ ...
1
vote
0answers
32 views

Cylindrical structure and homology

Let us consider a cobordism $(M, \partial_{-} M, \partial_{+}M)$, where $M$ is homeomorphic to $T \times I$, here $T$ is a torus $S^1 \times S^1$ and $I=[0, 1]$. I encountered the statement ...
1
vote
1answer
199 views

An exercise concerning simplicial complexes

The following exercise is drawn from Ch.24 of Fulton's "Algebraic Topology: A First Course." Exercise 24.37 (a) and (b) (a) If $U = \{U_v: v \in V\}$ is a finite collection of open sets whose union ...