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

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2
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1answer
31 views

How does one triangulate the mapping cylinder of a diffeomorphism?

The question is fairly self-explanatory. In particular, I would like to know how to triangulate the mapping cylinder arising from applying a Dehn twist to the torus. The reason is that I am thinking ...
3
votes
1answer
45 views

Some questions about homology with local coefficients.

If $F:\pi_1(X)\rightarrow Ab$ is a system of local coefficiens, on the topological space $X$, then we can define the homology of $X$ with coefficients $F$ by taking the homology of the chain complex ...
3
votes
0answers
31 views

Group bundle over a topological space

Suppose $p:\tilde X\rightarrow X$ is the universal cover of $X$. Take $G$ a group where $\pi_1(X,x)$ acts by isomorphisms. I read that if we consider $X\times G$ ($G$ with the discrete topology) and ...
1
vote
1answer
44 views

Property of Homology: group isomorphism

I have this proposition, but I don't understand how they use the axiom 5, since in the axiom 5; $f,g: (X,A)\rightarrow (Y,B)$ and in the theorem we have $f:(X,A)\rightarrow (Y,B)$, $g:(Y,B)\rightarrow ...
0
votes
1answer
41 views

A short exact sequence

I have this proposition, and I don't understand how to do to obtain the short exact sequence: where axiom 4 is:
2
votes
2answers
32 views

Homology of a finite graph follows from Mayer-Vietoris sequence?

Problem (Fulton's Algebraic Topology: A First Course, Exercise 10.15) If $X$ is a finite graph with $v$ vertices and $e$ edges, and $X$ has $k$ connected components, show that $H_1X$ is a free ...
2
votes
1answer
65 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 ...
6
votes
3answers
184 views

Klein bottle as two Möbius strips.

I read that glueing together two Möbius strips along their edges creates a surface that is equivalent to the so-called Klein bottle. The Möbius strip comes in two versions that are mirrored ...
2
votes
3answers
58 views

Hatcher, p.35, $\mathbb{R^{n}} - \{x\}$ is homeomorphic to $S^{n-1} \times \mathbb{R}$

Could some help me understand how $\mathbb{R^{n}} - \{x\}$ is homeomorphic to $S^{n-1} \times \mathbb{R}$. Also, if the one point set is replaced by any finite set, how does the argument work. This ...
1
vote
0answers
52 views

A sufficient condition for the composition of covering maps to be a covering map

Let $q:X \rightarrow Y$ and $r:Y \rightarrow Z$ be covering maps and $p= r \circ q$. If $r^{-1}(z)$ is finite for all $z \in Z$, then $p$ is a covering map. Now I found the following proof: ...
0
votes
1answer
46 views

Cardinality of fibres of covering maps of connected spaces

If I have a covering map $p:E \rightarrow B$ and some connected set $U$, that is evenly covered, then $p^{-1}(U)$ as a partition into slices is unique. Now, if I assume that $B$ is connected, then I ...
4
votes
2answers
121 views

Property of homology

I have this proposition, and I have two questions: 1) Why $H_k=\text{Im} i_*\oplus \ker r_*$ ? 2) Why $j_*: \ker r_*\rightarrow H_k(X,A)$ ? Edit: For the second, I try the 1th theorem of ...
-2
votes
1answer
45 views

A continuous map $f: S^2 \rightarrow S^2$, satisfying $f(x)\neq f(-x)$ for every $x\in S^2$. Prove that f is not surjective.

A continuous map $f: S^2 \rightarrow S^2$, satisfying $f(x)\neq f(-x)$ for every $x\in S^2$. Prove that f is not surjective.
0
votes
1answer
43 views

Manifolds as homology classes

I have found that a k-dimensional submanifold of a manifold M can be considered as a class in the homology group $H_{k}(M)$. Why ?
2
votes
1answer
54 views

Motivation for the proof of the associativity of multiplication of equivalence classes of paths

After having defined the equivalence classes of paths in a topological space in chapter two of the book A Basic Course in Algebraic Topology, William S. Massey proves the lemma The multiplication ...
1
vote
2answers
65 views

Property of excision of Homology

Please what is the difference between these two excision property: Let $X$ a topological space, $A$ a sub-space of $X$ and $U\subset A$ such that $\overline{U}\subset \stackrel{\circ}{A}$ . The ...
2
votes
0answers
52 views

Map between projective spaces induces trivial map on first homotopy groups

I have the following problem: Let $n>m>0$, show that every map $f:\mathbb{RP}^n\to\mathbb{RP}^m$ induces the trivial map on the fundamental groups. I paste the given solution below: Now, ...
1
vote
1answer
25 views

example of weakly homotopic sphere

There are spaces such as pseudocircles that are weakly homotopic to sphere but are not homotopic to spheres. But pseudo circles are non-hausdorff spaces. I need an example of a paracompact hausdorff ...
3
votes
0answers
36 views

Cycles non-homologous but with the same winding number at each point outside?

Problem (Fulton's Algebraic Topology: A First Course, Problem 6.28) Can we find a closed set $X\subseteq\mathbb R^2$, such that there's a cycle $\gamma$ in $X$ not homologous to zero but for each ...
2
votes
2answers
66 views

prove that the identity map $i:S^n\to S^n$ is not nullhomotopic.

Using the fact that: For all $n\in \mathbb{N}$ there is no retraction $r:B^{n+1} \to S^n$, prove that the identity map $i:S^n\to S^n$ is not nullhomotopic. This is a problem in section 56 of Munkres' ...
2
votes
4answers
26 views

$X$ is contained in an annulus containing a circle in the annulus.Show that $\pi_1(X)$ contains a subgroup isomorphic to $\mathbb{Z}$.

In $\mathbb{R^2}$ let $C=\{|x|=2\}$ and $A=\{1<|x|<3\}$, let $X$ be a path connected set such that $C \subset X \subset A$. Show that $\pi_1(X)$ contains a subgroup isomorphic to $\mathbb{Z}$. ...
2
votes
1answer
51 views

How to show the fundamental group of torus is abelian in a homotopic way?

I know the torus is homeomorphic to $S^1 \times S^1$ and the fundamental group is $ \mathbb{Z} \times \mathbb{Z} $, but in the real case, (let the generators of the torus's fundamental group be $a$ ...
1
vote
1answer
42 views

Showing the image of $H^j(X;\mathbb C^\times)$ lies in the torsion subgroup of $H^{j+1}(X;\mathbb Z)$

Let $X$ be a (compact, if necessary) topological space. Then from the short exact sequence of constant sheaves $$ 0 \to \mathbb Z \to \mathbb C \to \mathbb C^\times \to 0 $$ we have a connecting ...
2
votes
1answer
51 views

A technical step in the proof of Atiyah-Bott fixed point formula

From John Roe, Elliptic operators, topology and asymptotic methods, page 135. Here Roe tried to prove that $$ \DeclareMathOperator{\Tr}{Tr} \sum_{q}(-1)^{q} ...
1
vote
0answers
36 views

How to make the orbit space $T/G$ of torus $T$ homeomorphic to the Klein bottle?

Actually it is one of the exercises of Munkres. $G$ is a group of homeomorphisms of the torus having order $2$. How do I get $G$ in order to make $T/G$ homeomorphic to the Klein bottle? Can anybody ...
3
votes
1answer
46 views

A generalization of Jordan curve theorem to connected open sets in the plane

Problem (Fulton's Algebraic Topology: A First Course, Problem 5.23) Let $U\subseteq\mathbb R^2$ be any connected open set in the plane. If $X\subseteq U$ is homeomorphic to $[0,1]$, then ...
2
votes
1answer
33 views

Triangulation Definition Via Cell Partitions

There are two ways of defining a CW-complex. The first is to "inductively build" CW-complexes: you start with 0-cells as your 0-skeleton, attach 1-cells to that to get your 1-skeleton,... and so on. ...
3
votes
1answer
45 views

Cover a sphere by two closed subsets not containing a closed self-antipodal connected subset?

Question (Fulton's Algebraic Topology, A First Course, Problem 4.40) Suppose the sphere $S^2=A\cup B$ where $A,B\subseteq S^2$ are two closed subsets of $S^2$. Is it true that either $A$ or $B$ must ...
1
vote
1answer
21 views

A question regarding simplicial mappings.

Show that give any map $f_0$ from the set of vertices of $\sigma$ to the set of vertices of $\tau$, where $\sigma$ and $\tau$ are simplices, there is a unique simplicial map $f:\sigma\to\tau$ whose ...
2
votes
0answers
39 views

Proving that homotopic maps have the same degree

Let $M, N$ be compact, connected, oriented manifolds. The degree of a map $f:M \rightarrow N$ is defined as the integer $k$ which satisfies $\int_{M} f^{*}\omega = k\int_{N}\omega$. Using the fact ...
1
vote
1answer
40 views

Proposition 0.16 in Hatcher's AT

In the proof of the quoted proposition, it is mentioned that $D^n \times I$ retracts onto $D^n \times \left\{0\right\} \cup \partial D^n \times I$ and an example is given in a figure with $n=2$, which ...
1
vote
1answer
31 views

What is the induced functor of covering spaces to covering groupoids?

I'm reading May's book, 'A Concise Course in Algebraic Topology' and I'm confused about what he means by the induced functor from a covering space. First, here are some helpful/relevant definitions. ...
1
vote
0answers
21 views

Induced group action: homology vs cohomology

Let $M$ be a compact orientable manifold with finitely generated free abelian cohomology groups in even dimensions and $0$ otherwise. Conditions imposed on cohomology clearly imply that $H_i(M) \cong ...
2
votes
2answers
68 views

Homotopy direct limit versus direct limit

Let $X_1\to X_2\to \cdots$ be an infinite sequence of maps. Then, as I understand, the homotopy direct limit of this sequence is the space formed by taking the disjoint union of the $X_i\times I$ and ...
1
vote
0answers
17 views

Dold's proof of equivalence singular and cellular homology

I would like to ask for some help understanding a claim in Dold's proof of the equivalence of cellular and singular homology. The point is that I don't get why $\delta_n=j_*\delta_*$ where: ...
4
votes
1answer
88 views

Cohomology of wedge equals direct sum of cohomologies

I have seen the following fact used somewhere (for example to show that $\mathbb{RP}^3$ is not homotopy equivalent to $S^3\vee\mathbb{RP}^2$): Let $X,Y$ be two path connected pointed spaces such ...
3
votes
1answer
37 views

Does the $\Omega$-spectrum functor send exact triangles to homotopy cofiber sequences?

The functor $\Omega^\infty\colon Spectra\to Spaces$ which takes a spectrum, replaces it by the associated $\Omega$-spectrum and then takes its $0$th space sends exact triangles to homotopy fiber ...
3
votes
1answer
57 views

The nature of isomorphism between fundamental groups with different base points

New to algebraic topology. Munkres (Topology, 2 ed.) in the last paragraph on page 332 says that "If $X$ is path-connected, all the groups $\pi_1(X,x)$ are isomorphic, so it is tempting to try to ...
0
votes
1answer
28 views

homotopy class of maps in terms of homotopy groups of spectra

Given spectra $X$ and $Y$, the set $[X,Y]$ of homotopy classes of maps from $X$ to $Y$ can be endowed with an abelian group structure. Can the group $[X,Y]$ be expressed in terms of the homotopy ...
1
vote
1answer
26 views

Use the universal cover to prove $\gamma * \gamma$ is nullhomotopic if $\gamma$ is a loop in the projective plane

Use the universal cover to prove $\gamma * \gamma$ is nullhomotopic if $\gamma$ is a loop in the projective plane Well, If I was not asked to prove it this way, I could have argued like : ...
1
vote
1answer
25 views

Embedding a space in its cone

Let $X$ be a topological space, and $C(X)= (X \times [0 ,1])/(X \times {1} )$, define $f\colon X \to C(X)$ as $f(x)=[x,t]$ for some fixed $t$ s.t $\ 0\leq t <1$. I have to show this is a ...
2
votes
1answer
123 views

Homology and topological propeties

i have this theorem with it's proof but i don't understand the last part They use this proposition: My question is Why $\varphi^c\cap U_i$ is closed and pairwise disjoint ? where ...
2
votes
1answer
34 views

Proposition 1A.1 in Hatcher's Algebraic Topology

In the proof of the quoted Proposition, we have a connected graph $X$ and a sequence of subgraphs $X_0 \subset X_1 \subset \cdots$ such that $\cup_i X_i$ is both open and closed. Then Hatcher deduces ...
1
vote
1answer
52 views

What's the meaning of this about the cyclic fundamental group?

"You can also get a cyclic group of order p by attaching a disk to a circle by wrapping it around the circle p times (the fact that the fundamental group is Z/pZ follows from Van-Kampen’s theorem). " ...
2
votes
1answer
72 views

Some questions about cellular homology and cohomology

Consider the CW structure on $\mathbb{RP}^n$ given by one cell in every dimension. This gives rise to the cellular complex $C_\bullet(\mathbb{RP}^n)$ which is generated by a single element $c_i$ for ...
7
votes
1answer
129 views

Compact subspace of a covering space

I've been working through Massey's A Basic Course in Algebraic Topology and I've gotten stuck on the following exercise (V.8.4): Let $X$ be a regular topological space, and $(\tilde{X}, p)$ a ...
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vote
0answers
21 views

Extension to rational and real chains

In the paper on stable commutator length, D. Calegari says that generalized $\operatorname{scl}$ function can be extended by linearity to rational group $1$-chains and by continuity to real chains ...
2
votes
0answers
24 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 ...
1
vote
1answer
34 views

Does the Wirtinger presentation extend to compliments of graphs and links?

In a previous question I asked about a specific fundamental group problem, which was resolved via SVK but I was also interested in whether or not the Wirtinger presentation was valid in some way. In ...
1
vote
1answer
51 views

A continous map between the two torus and the torus

Let $\Sigma$ be the doubled torus (a compact oriented) surface of genus 2) and let $T$ be the torus. Suppose $f: \Sigma \rightarrow T$. Prove that $f$ is not a local homeomorphism. Attempt at ...