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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1answer
65 views

If $i$ is an inclusion why is the induced $i_*$ an epimorphism

Given the following exact homology sequence of a pair. This is in Example 2 (page 134) from Munkres. This is where I am always stuck computing homology using exact sequence. I cannot grab the last ...
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2answers
38 views

The number of intersection points between a trivial loop and a meridian in the torus

Let A and B be two closed curves intersect on the torus transversally at a point, the intersection index of the crossing point is defined to be positive if the tangent vectors to A and B form an ...
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1answer
48 views

Presentation of $\pi_1$ of compact orientable surface by induction?

I need to prove by induction $\pi_1(\Sigma_g)= \left\langle a_1,b_1,\dots ,a_g,b_g\mid \prod_i [a_i,b_i] \right\rangle$. For genus 1 this holds since $\pi_1(T^2)\cong \mathbb Z\times \mathbb Z$. For ...
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0answers
62 views

Why is the fundamental group of the plane with two holes non-abelian?

I know $\pi_1(\mathbb{R}^2\setminus\{x,y\}) = \mathbb{Z}\ast\mathbb{Z} = \langle a,b\rangle$, but it's non-abelian-ness isn't obvious to me. Specifically, I draw a box and two points to represent $x$...
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68 views

when a space is a product of eilenberg mclane spaces for $\pi_1(X)$ is not abelian

In When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$. , I discuss in my answer when an Abelian cw complex $X$ is a product of Eilenberg-Maclane spaces, and show that it ...
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1answer
46 views

Show $\mathbb{C}P^n$ is a $2n-$manifold [in singular homology theory]

There is a Theorem in the book that says: The space $\mathbb{C}P^n$ is CW complex of dimension $2n$. I wonder some questions: Is there any Theorem or result that if a space has CW complex structure,...
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1answer
24 views

A spectrum $I$ is $E$-injective iff the map $i:I\rightarrow I\wedge E$ is an inclusion of a retract.

I was reading some notes on stable homotopy theory and I came across the statement in the title of this question. "Suppose $E$ is a ring spectrum, then $I$ is $E$-injective if and only if the ...
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0answers
39 views

Does an isogeny always define a covering map?

Consider a map $f: G_1 \to G_2$ between two topological groups. If $f$ is an isogeny when viewing $G_1,G_2$ as algebraic groups does $f$ always define a covering map when viewing $G_1,G_2$ as ...
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0answers
15 views

Homology of triples

A triple of topological spaces $(X, A, B)$ consists of a topological space $X$ and two subspaces $A, B$ with $B \subseteq A \subseteq X$. A map of triples $f \colon (X, A, B) \rightarrow (Y, G, H)$ ...
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1answer
26 views

Induced homomorphism of a covering space

How can I determine what's the induced homology homomorphism of a covering $S^{n} \rightarrow RP^{n}$? I suppose that a Hurewicz homomorpism would be pretty effective, but since I know nothing about ...
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2answers
46 views

To show that product $Z=X×Y$ in the product topology is a CW complex

I would like to prove the following: If $X,Y$ are CW complexes, and either $X$ or $Y$ is locally compact then the product $Z=X×Y$ in the product topology is a CW complex. (-see here) In order to ...
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1answer
39 views

The Presentation Complex of $\mathbf Z\times \mathbf Z=\langle x, y|\ xyx^{-1}y^{-1}\rangle$ is the Torus

Let $G=\mathbf Z\times \mathbf Z$ and let $\langle x, y| \ xyx^{-1}y^{-1} \rangle$ be a presentation for $G$. In Example 1.46 of Hatcher's Algebraic Topology, the author mentions that the ...
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1answer
57 views

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products?

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products? I know it preserves products, but it seems to actually create them.
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1answer
89 views

Why do subvarieties correspond to Hodge classes?

Let $X$ be a smooth complex projective variety and define $$Hdg^k(X)=H^{2k}(X,\mathbb{Z})\cap H^{k,k}(X)$$ the group of integral $(k,k)$ cycles on $X$. Now it is a fact that we can associate to the ...
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0answers
52 views

Cohomology of local systems on spaces with the same homotopy type

Suppose we have a homotopy equivalence $f: X \to Y$ with homotopy inverse $g: Y \to X$, and a local system (i.e. a locally constant sheaf) $\mathcal{V}$ on $Y$. In this situation, are any of the ...
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1answer
24 views

Resources that explains “Cut and Glue” Technique for Delta Complex?

I am looking for any resources (book/online) that teaches and further elaborates on how the "cut and glue" technique works for $\Delta$-complexes. To be precise, I am looking for techniques and at ...
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1answer
408 views

Why is the homotopy category actually important?

Since the homotopy category (of whatever) generally has very few limits and colimits, and these don't coincide with homotopy limits and colimits in the lifted model structure, why do we care about the ...
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0answers
45 views

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 ...
3
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1answer
90 views

“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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0answers
32 views

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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0answers
23 views

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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0answers
14 views

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 ...
5
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2answers
104 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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3answers
68 views

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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0answers
25 views

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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2answers
108 views

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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1answer
38 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
32 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
42 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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0answers
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?
5
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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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0answers
55 views

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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1answer
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 ...
2
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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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1answer
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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1answer
53 views

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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0answers
70 views

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
40 views
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17 views

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 $(n+1)!$ ...
2
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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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36 views

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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1answer
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.