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

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Fundamental group of cylinder

I calculated the fundamental group of the cylinder, $C$, using the following method: triangulate $C$ find max contractable subspace realise generators on remaining 1-simplices I found the ...
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“Representability” of $\pi_1:\mathsf{hTop}_\ast \longrightarrow \mathsf{Grp}$?

In this MO question, Qiaochu Yuan asks about limit preservation of "representable" functors which are not $\mathsf{Set}$-valued. The answer gives a simple sufficient condition, possessed by monadic ...
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An example of infinite Betti number refers to G-covering, and the motivation of G-covering

I saw from a book that if G is infinite, then when considering G-covering: $p:X'\to X$, the P-th Betti number of $X'$ can be infinite. Can you give me an example of that. Also I am curious why we need ...
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Connected sum $S_1$ # $S_2$ is commutative and associative

The connected sum of two surfaces $S_1$ and $S_2$ is formed by removing a circular hole from each surface and identifying the boundaries together Show that the connected sum $S_1$ # $S_2$ is ...
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1answer
45 views

Reference for: simple closed curves generate the fundamental group

In a 2-complex $X$, it is "obvious" that the simple closed curves through the $0$-cell $v$ generate the fundamental group $\pi_1(X, v)$. (By a "simple closed curve" I mean a loop which does not self-...
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deformation retraction as mapping cylinder

In Hatcher's Algebraic Topology, the mapping cylinder is defined as the quotient space of the disjoint union $(X\times I)\sqcup Y$ (where $I$ is the unit interval) of a continuous $f:X\to Y$, where $(...
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2answers
37 views

Is this the projective plane or the Klein bottle? (Fundamental polygon)

I am trying to identify the topological type of this fundamental polygon and I think it is the projective plane or the Klein bottle If we treat the top green and red arrows a single arrow then we ...
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106 views

What is wrong with this proof that the identity map of $S^1$ is nullhomotopic?

I have read that the identity map of the unit circle $S^1$ is not nullhomotopic. In fact, I am very new to the subject, so I wonder what is wrong with the following reasoning (that seems to suggest ...
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I don't get it, does “augmented chain complex” actually mean anything?

If I understand correctly, chain complexes make sense in any category enriched in the world of pointed sets. In practice, there's also a notion of an augmented chain complex, where we have an extra ...
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What does “carried by” a subcomplex mean?

Elements of Algebraic Topology by Munkres says the following on pg. 32 Given a $1$-chain $c$, it is homologous to a chain $c_3$ that is carried by the subcomplex $M$... What does carried by a ...
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Compact 3-manifolds with boundary an orientable surface

The usual embedding of the closed orientable surface $M_g$ of genus $g$ in $\mathbb{R}^3$ bounds a compact orientable $3$-manifold which is homotopically equivalent to a bouquet of $g$ circles. If a ...
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39 views

Link between $\mathbb{Z}$ and the fundamental group's of 'common' topological spaces

I have noticed that some of the most common topological space have fundamental groups related to $\mathbb{Z}$, as we can see below: Why is this the case? Is it is because they are all realted ...
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1answer
43 views

Homotopy 'diagrams' for Klein bottle and projective plane

Background: I recently discovered that the complement to the circle and vertical axis shown below is homotopy equivalent to a torus Also complement to three infinite straight non-intersecting ...
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2answers
45 views

Cohomology of a mapping torus

How does the monodromy in a mapping torus $K_{\phi}$ affect the de Rham cohomology, if at all? Maybe this is naive, but I don't see how twisting one of the ends of $K\times I$ via the diffeomorphism $...
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1answer
51 views

Homotopy of two circles contained in an open ball.

The following question is on my homework assignment and I have no idea how to even start answering it: Are any two distinct $S^1$ → $B(0,r)$ maps Homotopic? You can assume the circles are simple-...
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1answer
34 views

Representing Covering Spaces by PErmutations

I am having trouble understanding the exposition in the subsection titled Representing Covering Spaces by Permutations in Section 1.3 of the book Algebraic Topology by Hatcher. Hatcher starts by ...
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1answer
46 views

Computing homology group using Mayer-Vietoris sequence

Suppose I am given an exact sequence: $$0\to G\xrightarrow{f} \mathbb{Z} \xrightarrow{g} \mathbb{Z} \xrightarrow{h} H\to 0 $$ where the first $\mathbb{Z}=H_3(A\cup B)$ and the second $\mathbb{Z}=H_2(...
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1answer
34 views

Low torsion in orientable manifolds?

The final sentence on page 170 of Stillwell's Classical Topology an Combinatorial Group Theory is: Poincaré justified the term "torsion" by showing that $(m-1)$-dimensional torsion is present only ...
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33 views

Cylinder as trivial fiber bundle with fiber $S^1$?

In prepartion for the example of a Mobius strip, a cylinder is often taken as a first example of a trivial fiber bundle with fiber $I$ and total space $S^1\times I$. However, it seems to me that we ...
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1answer
34 views

Definition of G-crossed complex.

I was reading about crossed complexes following R.Brown. I was wondering how one define G-crossed complexes for a topological group G? Is it just dimension wise action of the group?
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1answer
79 views

Is Whitehead's manifold with a point removed homotopy equivalent to a sphere?

A contractible open subset of $\mathbb{R}^n$ need not be homeomorphic to $\mathbb{R}^n$. The Whitehead manifold is an open subset of $\mathbb{R}^3$ which is contractible but not homeomorphic to $\...
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adjoint functor of inverse image functor

$f: U\hookrightarrow X$ an open immersion of two complex manifolds. $f^{-1}$ is inverse image functor, in usual sense, from category of sheaves of abelian groups $\mathcal{Ab}(X)$ over $X$ to category ...
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28 views

Fundamental group of simple graphs

Find the fundamental groups of graphs A, B and C as shown: They look simple but I am unsure what their fundamental groups would be. I was thinking that $A$ and $B$ are generated by just one ...
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1answer
25 views

The space of connections is affine thus contractible?

In Ralph Cohen's notes on the topology of fiber bundles pp.62 he states that, since the space of connections $\mathcal{A}(P)$ (where $P$ is a principal $G$-bundle is affine) it is contractible. I ...
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20 views

How does a boundary operator act on a 2-simplex?

Let $A $ be a 2-simplex with vertices $\{0, 1, 2\} $. I want to show that $\rho_1\circ\rho_2 (A)=0$, where $\rho $ is the boundary operator. How do I go about doing that? The major problem that I ...
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Nonzero-homologous simple loop in Mobius band only winds once

I have a question as follows: Let $C$ be a closed curve in the Mobius band without self intersections. Prove that if $C$ is of non-zero homology, i.e., $C$ does not bound any face, then $C$ winds only ...
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1answer
43 views

Can the K-theory of a space be a field?

If $X$ is a compact Hausdorff topological space, is it possible to $K(X)$ be a field considering the operations over vector bundles, $\oplus$ and $\otimes$? It is known that $K(X)$ has a ring ...
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Barycentric subdivision proof

In the proof of barycentric subdivision in singular homology, we take the subdivision operator $b: C_q(X) \to C_q(X)$, and do some algebra to define an operator $b^\infty$, which applies $b$ "as many ...
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1answer
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Is a simplex with permuted vertices $\pm$homologous to the original?

Take a singular $n$-simplex $\sigma: \Delta^n \to X$, where $\Delta^n\subset \mathbb{R}^{n+1}$ is the convex hull of the standard basis, with the obvious vertex ordering. Then one can obtain $(n+1)!$ ...
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1answer
42 views

Is there a projective morphism from the quadric surface to the projective plane with degree 1?

Is there a projective morphism from the quadric surface $\mathbb{P}^1\times\mathbb{P}^1$ to the projective plane $\mathbb{P}^2$, with degree $1$?
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Prove that the induced map is of degree n

Let $P(z)$ be a complex polynomial of degree n. $$P:S^2 \rightarrow S^2 $$ $S^2 - p_0 \cong C$ (stereographic projection) and $P(\infty)=\infty$. I'd like to prove that $P_{*}:H_2(S^2) \rightarrow ...
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51 views

Do compact connected smooth manifolds admit the structure of a CW complex with a single 1-cell? [closed]

This seems intuitive to me, since they admit a CW decomposition with finitely many cells. But I can't see how to prove it.
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Homology: Why isn't this a Klein Bottle?

The question in context is: What familiar space is the quotient $\Delta$-complex of a 2-simplex $[v_0,v_1,v_2]$ obtained by identifying the edges $[v_0,v_1]$ and $[v_1,v_2]$, preserving the ordering ...
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1answer
35 views

Fundamental group of cylinder quotient

Let $X = S^1 \times [0,1]$ be the cylinder, and define an equivalence relation on $X$ by $(z,1) \sim (iz,1)$. What is the fundamental group of $X/\sim$? Is $X$ the same as the mapping cylinder of $(...
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1answer
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Identifying the antipodal points of one boundary of the cylinder gives the Möbius band.

In Example 1.35, Hatcher writes in his book Algebraic Topology, the following (not paraphrased): Let $X=S^1\times I$, and let $A$ be the quotient space obtained by defining the relation $(z, 0)\...
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1answer
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How to prove a bundle map is a bundle isomorphism?

Example #1 In proving pullback bundle is homotopy invariant Ralph Cohen's notes on the topology of fiber bundles use the following proof based on the covering homotopy theorem (pp.47): Let $p: E \to ...
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The universal property of pullback bundle

The universal property says the following: for any pair of maps $i: Z \to X$ and $j: Z \to E$ fitting into the following commuting diagram where $p: E \to B$ is a fiber bundle (by the way this is true ...
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142 views

Detail in the proof that sheaf cohomology = singular cohomology

Theorem: If $X$ is locally contractible, then the singular cohomology $H^k(X,\mathbb{Z})$ is isomorphic to the sheaf cohomology $H^k(X, \underline{\mathbb{Z}})$ of the locally constant sheaf of ...
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1answer
59 views

Reference request: Inclusion of smooth maps into continuous maps between smooth manifolds is a weak homotopy equivalence.

Let $M,N$ be smooth manifolds. It seems to be well known that if the sets $C^0(M,N)$ and $C^\infty(M,N)$ are equipped with the appropriate topologies (I suppose the weak/strong Whitney topology), then ...
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1answer
22 views

Trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$

I am trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$ I know that $RP^2$#$T^2$~$RP^2$#$K^2$ and that $X(M$#$N)$=$X(M)+X(N) -2$ where X is the ...
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1answer
99 views

Proving equality of homology of a product

I'd like to prove the following equality: $$H_i(X \times S^{n}) = H_i(X) \times H_{i-n}(X) $$ For $n=0$ it's pretty obvious, hence I'll use induction.Dividing $S^{n}$ into two hemipsheres $D^{n}_1,D^{...
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1answer
30 views

Induced homomorphism from homology group of circle to homology group of $\mathbb{R^2-}0$ is trivial

Let $C_r$ be a circle of radius $r$ in complex plane, and let $f:C_r\to\mathbb{R^2}-0$ defined by $f(z)=z^n+a_{n-1}z^{n-1}+...+a_0$ and suppose that it has no zero on and inside the circle $C_r$. ...
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64 views

A Natural Question When Reading Van Kampen Theorem

Let $A$ and $B$ be path connected open subspaces of a topological space $X$ and assume that $A\cap B\neq \emptyset$ is simply-connected. Let $x$ and $y$ be two points in $A\cap B$. Let $\gamma$ and $\...
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1answer
51 views

Trying to Understand Van Kampen Theorem

Theorem. Let $X$ be the union of two path-connected open sets $A$ and $B$ and assume that $A\cap B\neq \emptyset$ is simply-connected. Let $x_0$ be a point in $A\cap B$ and all fundamental groups will ...
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Structure space of a commutative ring with unity

I was reading the topic for the discussion of Stone-Cech compactification but stocked at some point: Suppose that $(R,+,.)$ is a commutative ring with unity. Let $\mathcal{M}(R)$ denotes the ...
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Are all subgroups of the fundamental group of a compact smooth manifold finitely generated?

And if not, is there a way to assign a size to a subgroup by considering the compactness of the corresponding covering space?
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48 views

Fundamental class for tangent bundle

For closed orientable manifolds $X$ we can define the fundamental class $[X]$ which is just a choice of generator of the top homology group $H_n(X; \mathbb{Z})$. However, in the context of the ...
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degree 1 map $f : M \to S^n$

Let us consider $M$ a closed and connected n-manifold which is also orientable. An exercise in Hatcher claims that for any such $M$ there is a continuous map $f: M \to S^n$ such that it's degree is 1, ...
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Every space $X$ can be identified with the closed subspace of the reduced cone $\operatorname{C}(X)$ of $X$.

For any pointed space $(X,x_0)$, we define the cone $(\operatorname{C}(X),*)$ of $X$ to be the smash product $(X\wedge I, *)$ where the base point of $I$ is assumed to be $0$. The map $\mathbf{i} : X\...