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
30 views

Prove that the fundamental group of $P^2$ the real projective plane is the group with two elements

Presumably the group will just contain $e$ and another $g$. My idea is to use the proof that $\pi_1(S^1)=\mathbb{Z}$ where the covering map from $S^2$ to $P^2$ maps to the line in $\mathbb{R}^3$ ...
0
votes
0answers
14 views

Open subsets of CW complex.

Consider an easy CW complex, for example, the figure $8$ where we will denote the attaching point by $x_0$. It is easy to draw covering spaces of the figure $8$. It seems that we take copies of our ...
0
votes
0answers
38 views

Group structure on $[X, \Omega^{2}(Y)]$ is abelian

I'm very begginer when it comes to algebraic topology. I have no idea how to prove (firstly see why it could be true or even start) this statement: Composition of loops gives the structure of group ...
0
votes
0answers
95 views

Showing integers with discrete topology homeomorphic (or not) to wedge sum of intervals

I am trying to learn basic topology, and have just started wedge sums and Seifert-van Kampen. There are two exercises that I believe should be fairly simple but I am struggling to understand how to ...
0
votes
0answers
54 views

Delta-complex structure on S^2

In Hatcher, it is stated that we can obatin a $\Delta$-complex structure on $S^2$ by taking two copies of $\Delta^2$ and identifying their boundaries via the identity map. However, there is no ...
0
votes
0answers
24 views

A problem of the definition of relative homology

This is a lemma on the textbook algebraic topology a first course (Greenberg). I supposed that they are equal, not isomorphism. Proof:$$H_q(X,A)\cong Z_q(X,A)/B_q(X,A)$$ I am wondering $Z_q(X,A)$ is ...
0
votes
0answers
38 views

What is dual to relative homology?

Let $(X,A)$ be a pair of manifolds, where $A \subseteq X$. Then we can define the relative cohomology of the pair $H_{\bullet}(X,A)$ to be the homology of the chain complex ...
0
votes
0answers
15 views

Set of beginning part and ending part of all loops based at a point is an open neighborhood

Problem: Suppose topological space $X$ is connected and locally path-connected, and the function $f:X\rightarrow S^1$ is continuous. Prove that if the induced homomorphism ...
0
votes
0answers
11 views

Find and sketch a lift for the covering $p:\mathbb{R}\to S^1$ given by $t \to e^{2\pi i t}$

Actually my question is is it the same as the path $f: I\to S^1$ beginning at $(1,0)$ given by $f(s)=cos\pi s, sin\pi s)$ lifted to $f'(s)=s/2$ beginning at $0$ and ending at $1/2$?
0
votes
0answers
37 views

Riemann - Hurwitz Formula for topology.

I am quite confused about the notion of branch points at infinity? and even in general the idea of branch points? I know branch points to be where points diverges to infinity. Could someone please ...
0
votes
0answers
33 views

Help with formalisation of a reasoning about simplicial sets

We briefly mention during lectures that the homotopy relation between two maps of simplicial sets is not necessarily symmetric. So I thought for a counterexample but I don't know how to formalise it ...
0
votes
0answers
36 views

Given $(X,A)$,$(Y,B)$ such that $X/A$ and $Y/B$ are homotopy equivalent,are their relative homology groups isomorphic?

Suppose that $(X,A)$,$(Y,B)$ are pairs of topological spaces. If $X/A$ and $Y/B$ are homotopy equivalent, are $H_*(X,A;\mathbb{Z})$ isomorphic to $H_*(Y,B;\mathbb{Z})$ ?
0
votes
0answers
30 views

fundamental group of a graph and graph homomorphisms

I am having trouble understanding what is called for in the following exercise in hatcher. Construct a connected graph $X$ and maps $f,g : X \to X$ such that $fg = {\bf 1}$ but $f$ and $g$ do not ...
0
votes
0answers
24 views

Fundamental group of suspension of three points

In Hatcher's Algebraic Topology, pg 53, he gave an example of path connectedness being necessary. However, I do not get the reason how come this example will serve the idea that one space will have ...
0
votes
0answers
73 views

Hatcher Exercise

One of Hatcher's exercises asks me to prove that if we have a polynomial with complex coefficients, and we view it as a continuous map of the Riemann Sphere into itself then the degree of the ...
0
votes
0answers
26 views

relationship between polyhedron and its polar set

Polyhedron is defined by intersection of finite half space in Euclidean space. Let $P$ be a polyhedron, denote $$P^*=\{u\in\mathbb R^n|<u,x>+1\geq0,x\in P\}$$ as polar set. Theorem $0\in P$ ...
0
votes
0answers
42 views

Comparing the proofs $58.2$ and $58.7$ of TOPOLOGY by Munkres

First , I give the photos of comparison . In the first one $j$ is continuous and homotopy equivalence to the map $r$ In the second case $f$ is continuous and homotopy equivalent to ...
0
votes
0answers
30 views

Relative cohomology of retract

Inspired by this question. If $A$ is a retract of $X$ then for all $n \ge 0$ $H^n(X) \simeq H^n(A) \oplus H^n(X,A)$? I know there is a link between homology and cohomology which is the Poincare ...
0
votes
0answers
24 views

Geometric picture of attaching a 2-cell to the 2-torus

Im currently reading about attaching spaces and I was wondering what is the geometric picture of attaching a 2-cell to the 2-torus via the map $f: S^1 \xrightarrow{(1,0)} S^1 \times S^1$. For me it ...
0
votes
0answers
24 views

Acyclicity and connectivity

For a topological space $X$, set: $$\text{acyclicity} (X) =\max\{k : \tilde{H_i}(X,\mathbb{Z}_2) = 0,\text{ for every }\,\, 0\leq i\leq k\} $$ Is there an example of triangulable topological space ...
0
votes
0answers
39 views

Fundmental group of cone

Let $K$ be a simplicial complex with an edge loop $\alpha=(a_0,a_1,a_2)$. Let $X$ be a simplicial complex consisting of $K$ together with another additional vertex $b$ in the center, connected to ...
0
votes
0answers
28 views

Does the word of an attachment map for some cell on a CW complex correspond to the cellular boundary map?

I'm in the middle of reading Hatcher, Chapter 2.2. The definition he gives for the boundary map for cellular homology of a CW complex $X$ is for some $n$-cell, $e^n_\alpha$, $d_n(e^n_\alpha) = ...
0
votes
0answers
23 views

Special Case of Universal Coefficient theorem

Show that if $C_* :$ $ ...\to C_2 \to C_1 \to C_0 \to 0$ be a complex of vector spaces over a field $k$, then $H^n$($Hom_k(C_*,k)) \cong$ $Hom_k(H_n(C_*),k)$.Does this result holds if $k$ is not a ...
0
votes
0answers
37 views

homotopy between constant simplicial sets

Assume that $K,L$ are constant simplicial sets (i.e all the faces maps and the degenerancies maps are equal to the identity and $K_{i}=K_{0}$, $L_{i}=L_{0}$ for $i>0$). Assume that there is a ...
0
votes
0answers
44 views

how to show whether this topological space is metrizable or not?

Let $X$ be a two-element topological space with a discrete topology. Let $J$ be an uncountable indexed set. And let $Z=X^J$ be the Cartesian product endowed with the product topology. Is $Z$ ...
0
votes
0answers
44 views

Simpicial approximation map

Let $\Delta^n$ be the standard $n$-simplex and $f :\delta\Delta^n\to \delta\Delta^n $ ($\delta\Delta^n$ is a boundary of $\Delta^n$ ) be a continuous function such that $f(-x) = -f(x)$. Is there a ...
0
votes
0answers
21 views

Definition of General Position for a Semi-Algebraic Set

I'm looking for a precise definition of what it means for a semi-algebraic set to be in general position. I've found definitions that apply to sets of points, but that doesn't seem to help in the ...
0
votes
0answers
33 views

definition of finite pointed spaces in Lurie Higher Algebra

Let $S$ be the infinity category of spaces. In Higher Algebra 1.4.1.4 Lurie defines $S^{fin}$ as the smallest full subcategory of $S$ which contains the final object $*$ and is stable under finite ...
0
votes
0answers
26 views

proof that $E(K) \leq \frac{1}{2}V(K)(V(K)-1)$

Let $K$ be a connected compact surface that has a triangulation and $V(K)$ be the number of verticals, $E(K)$ be the number of edges. Prove that: $$ E(K) \leq \frac{1}{2}V(K)(V(K)-1) $$ Thank you any ...
0
votes
0answers
37 views

What is the fundamental group of a modular curve $\mathcal{H}/\Gamma$?

Let $\Gamma$ be a finite index subgroup of $PSL_2(\mathbb{Z})$. What is the fundamental group of $\mathcal{H}/\Gamma$? By the Kurosh Subgroup theorem, $$\Gamma \cong F_n * C_2^{*r} * C_3^{*s}$$ ie, ...
0
votes
0answers
25 views

Relation between Thom class and closed Poincaré dual

Let $S,M$ be oriented manifolds without boundary such that $i:S\to M$ embedding and $i(S)$ is closed in $M$. $$ \eta_{S|M}:=\frac{TM_{|S}}{TS} $$ is the normal bundle of $S$ in $M$. I want to find a ...
0
votes
0answers
12 views

How to show for q-cochian, $\delta \sigma = \tau$ is skew- symmetric in cohomology theory?

For sheaf $\mathfrak{F}$, A $q$-cochian $C^q = \{ \sigma_{j_0 \cdots j_k} \}$ is a set of sections $\sigma_{j_0 \cdots j_k} \in \Gamma(U_{j_0} \cap \cdots \cap U_{j_k},\mathfrak{F})$ which are ...
0
votes
0answers
19 views

What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite

When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism ...
0
votes
0answers
23 views

$Z_2$ spaces and $Z_2$-index

Let $(X, \nu )$ be a free $Z_2$-space and let $A,B \subseteq X$ be closed invariant sets (that is $\nu (A) = A$ and $\nu (B)=B$ ) with $X= A \cup B$. Show that $ind_{Z_2}(X) \leq ind_{Z_2}(A) + ...
0
votes
0answers
18 views

Complete regularity for a covering map.

How do I show that if $p:E\to B$ is a covering map, then if $B$ is completely regular then also $E$ is completely regular? Thanks. If I take $x\in E$ and a closed set $x\notin F \subset E$; then I ...
0
votes
0answers
65 views

Covering spaces of $S^1 \vee S^1$: to what subgroups do these ones correspond?

The universal covering space for $S^1 \vee S^1$ is the Cayley graph, $X$, of the free group on two generators, $F\{a,b\}$. The subgroup $F\{b\}$ corresponds to the covering space ...
0
votes
0answers
37 views

Linear maps gives continuous maps between one-point compactification?

If $V\longrightarrow W$ is a linear map, say an orthogonal projection, it gives a map $S^V\longrightarrow S^W$. Is this map between spheres continuous and well-defined?
0
votes
0answers
16 views

An explicit example of a Pullback of equivelent short exact sequences.

I am trying to construct the Pullback of the example found in an answer Example on Ext Functor. Say I have equivalent exact sequences: of the form $ E=0\rightarrow N\rightarrow E_1\rightarrow ...
0
votes
0answers
49 views

Every triangulation on a disk is orientable.

Since a disk doesn't contain a Möbius or any other non orientable surface, it is orientable. I want to prove it rigorously by showing every triangulation on a disk is orientable. For this, I was ...
0
votes
0answers
26 views

Induced mappings of $\pi_1$ and conjugation

Given $f:X\rightarrow Y$ with induced map $f_*:\pi_1(X,x_0) \rightarrow \pi_1(Y,y_0)$, a path $h$ in $X$ from $x_0$ to $x_1$ with inverse $\overline{h}(t) =h(1-t)$, and a $[\phi] \in \pi_1(X,x_0)$, ...
0
votes
0answers
32 views

Curious surface with trivial fundamental group

Today the teacher mentioned a surface that has a trivial fundamental group even though it isn't obvious at first. He called it "the two room house". I thought it was pretty cool so I looked it up ...
0
votes
0answers
19 views

Prove that $J = u([0,1]) \subset X$ isn't a retract of $X$ given that $u$ is homotopic to $v \times v$.

Prove that $J = u([0,1]) \subset X$ isn't a retract of $X$ given that $u$ is homotopic to $v \times v$. $u$ is continuous and closed, $u(t) = u(s) \iff t-s \in \mathbb{Z}$. Also, $u(0) = u(1) = v(0) = ...
0
votes
0answers
56 views

Understanding commutative cochain problem

I have just read about commutative cochain problem(CCP) here and I'm trying understand it. It states that you cannot turn(in nontrivial way) simplicial set $S$ to differential graded commutative ...
0
votes
0answers
15 views

Equivalence of cycles in homology group

Let $X$ be $\mathbb{R}^2/\mathbb{Z}^2$.$\sigma_1:t\mapsto (t,0), \sigma_2:t\mapsto (0,t),\sigma:t\to(t,t)$. We know that $H_1(X) = \langle [\sigma_1],[\sigma_2]\rangle$. I want to prove that ...
0
votes
0answers
46 views

Relation between certain Pontryagin class and Chern class

Let $P$ be a principal $SU(2)$ bundle over base $B$, $P'$ be the principal $SO(3)$ bundle obtained by quotient $\{1,-1\}$. How do we show the first Pontryagin class of $P'\times \mathbb{R}^3$ equals ...
0
votes
0answers
25 views

Equivalence of two definitions of $p$th persistent homology groups

Let $K$ be a simplicial complex and $K_0\subseteq\ldots\subseteq K_n$ a filtration of $K$. I am looking for a formal proof of $$\text{im} f_p^{i,j}(K)\cong \frac{Z_p(K_i)}{B_p(K_j)\cap Z_p(K_i)},$$ ...
0
votes
0answers
25 views

Relative Barycentric Subdivision question

I have been stuck at this question for days, partly because I don't know how to visualize Barycentric Subdivision of high dimensional simplexes. Question: Let $L$ be a simplicial complex and let $B$ ...
0
votes
0answers
36 views

Question on a covering mapping.

I have this question: Let $Y$ have the discrete topology. show that if $p:X\times Y \rightarrow X$ is a projection on the first coordinate then $p$ is a covering map. My answer, I am looking at: ...
0
votes
0answers
15 views

Equivariant exponential Law of function spaces

Let $G = G_1 \rtimes (G_2 \rtimes G_3)$ and $G_3$ be not normal in $G$. Let us denote $map_G(X,Y)$ be the set of all $G$-equivariant maps from $X$ to $Y$. I am searching for a candidate $G$-space $Z$ ...
0
votes
0answers
42 views

Why is $\mathbb{R}^2/G$ homeomorphic to the Klein bottle?

Let $G$ be the group of transformation generated by $a,b:\mathbb{R}^2\to \mathbb{R}^2$ where $a(x,y)=(x+1,y-1)$ and $b(x,y)=(x,y+1)$. We note than $bab=a$ and that $G$ acts properly discontinuously ...