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

learn more… | top users | synonyms (1)

0
votes
0answers
19 views

Basic question about lifting maps to covering spaces

Any continuous map $f: X_1 \to X_2$ "lifts" to a map $\tilde f: \tilde X_1 \to \tilde X_2$ (provided that $X_1$ and $X_2$ have universal covers). The space $\tilde X_1$ is certainly ...
1
vote
1answer
19 views

$X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point

Im trying to show that: for $X,Y$ topological spaces $X$ is contractible and $Y$ is path connected then $[X,Y]$ has a single point while $[X,Y]$ denote the set of homotopy classes of maps of $X$ ...
0
votes
3answers
32 views

A proof of compactness, connectedness of real projective space

I need a reference for a complete proof of the below theorem: Let $RP^n$ be $n$-dimensional real projective space. Then $RP^n$ is a compact, connected manifold. (Consider the standard topology over ...
1
vote
1answer
38 views

Is the constant sheaf $\mathbb{Q}$ injective?

Let $X$ be a topological space, and let $\mathbb{Q}$ be the constant sheaf of abelian groups on $X$ associated to the group of rational numbers under addition. Is $\mathbb{Q}$ an injective object in ...
0
votes
0answers
24 views

lifting loops on surface with abelian fundamental group for decrease their self-intersection number

I found a proof of following statement (which is available here ), but I'm not sure if we must assume that fundamental group of surface $M$ is non-abelian: Lemma: Let $M$ be a compact orientable ...
1
vote
1answer
30 views

Finding generators of homology groups

Take the simplicial complex with vertices {a,b,c} and edges {ab, ac, bc}. In other words, a circle. If I build a chain complex, and make the matrix of my differential, I get that the kernel of ...
1
vote
1answer
26 views

spin structure definition

Suppose we have a principal $SO(n)$-bundle $E$ over $B$, with projection map $p$. We say that it admits a spin structure if there is a prinicipal $spin(n)$-bundle $E'$ over B, with projection map ...
-3
votes
0answers
38 views

A homotopy of functions which isn't a path homotopy [on hold]

The definitions are: A homotopy between two continuous functions $f$ and $g$ from the topological space $[0,1]$ to a topological space X is defined to be a continuous function $H : [0,1] \times [0,1] ...
1
vote
1answer
19 views

Homotopy colimit of a 3x3

Hi I am wondering how you calculate homotopy colimits of a 3x3 diagram. In particular if we have (sorry not sure how to Tex these) Top/bottom row: * <-- * --> * Middle row: * <-- X --> ...
6
votes
1answer
94 views

Deck transformations on $S^1\times \mathbb{R}P^2$

I'm studying for qualifying exams and stuck on the following problem: Suppose that $S^1\times \mathbb{R}P^2$ covers a space, and let $h$ be a deck transformation of the covering. Show that the ...
1
vote
0answers
26 views

Meyer-vietoris sequence to compute the compact cohomology for Möbius strip

How do you use Meyer-vietoris sequence to compute the compact cohomology for Möbius strip without the bounding edge? Please give detail math. In particular explain how inclusion map is used. On page ...
0
votes
1answer
25 views

Classical presentation of fundamental group of surface with boundary

It is well known fact about fundamental group of orientable compact surface: Letting $g$ be the genus and $b$ the number of boundary components of surface $M$. There is a generating set ...
1
vote
0answers
24 views

boundary in homology group

f is a reflection on a sphere $S^{n}$, $\sigma_{1}$ is a diffeomorphism from $D^{n}\subset \mathbb{R}^{n}$ to one of the two caps of the sphere, separated by the plane of the reflection and ...
1
vote
0answers
49 views

The cohomology of $S^3/D^*_k$

I have tried to compute the de Rham cohomology and the homology over the integers of the space $S^3/D^*_k$, where $D^*_k$ is the binary dihedral group of order $4k$ and I would like to know if ...
3
votes
2answers
24 views

Complement of a solid genus-2-handlebody in $S^3$

I'm not sure if this is a stupid question or not but is the complement of a solid genus-2-handlebody in $S^3$ also a solid genus-2-handlebody? Thanks!
2
votes
1answer
45 views

Quotient map not nullhomotopic

I have the following qual problem: Let $M$ be a connected closed surface, not necessarily orientable, with an embedded closed disk $D$. Let $Q$ be the quotient space of $M$ by $\overline{M\setminus ...
2
votes
1answer
139 views

Homotopy invariance in homology

i have this from Hatcher's book "Algebric topology" And i don't understand why we have $i-1$ in $(-1)^{i-1}$ and strict inequality in $P\partial(\sigma)$ ? Please. Thank you.
2
votes
0answers
23 views

An exercise in homology computation / What is the geometric fixed points of an Eilenberg Maclane Spectrum?

The question I want to ask has a reasonably elementary formulation and I think there is a good chance it can be answered in this form (by someone more computationally skilled than me, or perhaps by ...
2
votes
0answers
66 views

$C(X)$ is finite dimensional iff $X$ is finite [duplicate]

If $X$ is compact Hausdorff space and $C(X)$ is the set of all continuous complex valued functions on $X$,then prove that $C(X)$ is finite dimensional if and only if $X$ is finite. My problem:If we ...
2
votes
1answer
26 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
0answers
36 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
28 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
42 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
37 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
29 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
64 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
176 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
52 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
46 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
44 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
100 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
43 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
42 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
50 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
61 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
43 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
21 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
64 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
49 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
40 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 ...
1
vote
0answers
34 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
34 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
45 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
31 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
40 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
19 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
37 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
37 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 ...