Questions tagged [algebraic-topology]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
11 views

Homology of the complement of a simple closed curve in a surface

Let $S$ be a closed orientable surface of genus $g$. Let $C\subset S$ be a simple closed curve. What is the homology of $S - C$? When I try to apply Mayer-Vietoris to $(S, S-C, T)$, where $T$ is a ...
user avatar
3 votes
1 answer
25 views

What do the elements of the chains of a simplicial complex represent?

I've just started to learn homology and I don't quite understand why we define chains the way we do. For a simplicial complex $S$ we define $C_k$ to be the $k$-chains on $S$ given by an abelian group ...
user avatar
0 votes
1 answer
21 views

Simply connected imply any two paths are freely homotopic

I know that simply connected ($\pi_0(p)=0$ and path-connected) is equivalent to "the space is path-connected and any two paths with the same endpoints are homotopic". What about two paths ...
user avatar
  • 1,479
-2 votes
0 answers
47 views

Motivation behind fundamental group. [duplicate]

I am a graduate student.I am currently studying algebraic topology.Many things in algebraic topology are not clear to me.For example,the study of fundamental groups.I am unable to understand why we ...
user avatar
0 votes
0 answers
17 views

uniqueness of clutching decomposition Hatcher p.$47$ Lemma $208$

In the following accepted questions on mse (first and second) concerning K theory, in particular the uniqueness of the splitting given also in Hatcher p.$47$ Lemma $208$ is addressed by the same ...
user avatar
0 votes
0 answers
14 views

For $U$ open subset of $\mathbb{R}^n$, assume $U=U_1\bigcup ... \bigcup U_m$ where all $U_i$ open and subconvex. Prove $H^j(U)=0$ for $j\geq m$..

A subset is subconvex if it is either convex or empty (so that the intersection of any two subconvex sets is subconvex). Note, $H^*(U)$ denotes the cohomology ring of $U$. The proof for this can be ...
user avatar
  • 1
1 vote
0 answers
45 views

Suspension of a homology 3 sphere

Let $M^3$ be a homology sphere: a connected closed 3-manifold with the same homology groups as $S^3$. Calculate the first fundamental group and homology groups of the suspension $\Sigma M$. Use this ...
user avatar
  • 19
0 votes
1 answer
22 views

link of a vertex in the triangulation of $S^k$ is a triangulation of $S^{k-1}$?

Let $C$ be a triangulation of sphere $S^k$, i.e., $C$ is a geometric simplicial complex and the union of all simplices in $C$ is homeomorphic to $S^k$. For a vertex $v$ of $C$, $$link_C(v):=\{\sigma\...
user avatar
  • 2,069
2 votes
1 answer
46 views

Relative homology of a torus relative to figure eight

Let $X=S^1 \times S^1$ and $A=S^1 \vee S^1$. The question asks to compute $H_n(X,A)$ with coefficients $R$. By the short exact sequences, we have $0\to C_n(A)\to C_n(X)\to C_n(X,A) \to 0$. Since $C_0(...
user avatar
3 votes
1 answer
84 views

Computing singular homology groups of quotient space

I want to compute the homology groups of $X$, the quotient of $S^2 \times S^1$ by the relation $(x,z) \sim (-x,-z)$. I've already computed the homology groups of $S^2 \times S^1$ using Mayer-Vietoris (...
user avatar
-2 votes
0 answers
49 views

generalized Poincare conjecture [duplicate]

How to show that the claim that there exists exactly one differentiable structure on $S^4$ iff smooth four-dimensional Poincaré conjecture is true (homotopy equivalent to S4 implies diffeomorphic to ...
user avatar
0 votes
0 answers
32 views

Induced group action on tangent bundle commutes with structure group?

I am trying to understand how the free and proper action of a discrete group $\Gamma$ on a manifold $X$ by automorphisms changes the structure group of the tangent bundle $\mathcal{T}_X$ of $X$. Let $\...
user avatar
0 votes
1 answer
60 views

Homotopy equivalent to $\mathbb{S}^1$, but not homeomorphic to $\mathbb{R} \times \mathbb{S}^1$

What is an intuitive example of a topological object which is homotopy equivalent to $\mathbb{S}^1$, but not homeomorphic to $\mathbb{R} \times \mathbb{S}^1$?
user avatar
  • 1,155
1 vote
1 answer
79 views

Homotopy equivalence of $BGL_n(\mathbb{R})$ and $BO_n(\mathbb{R})$

I have tried to prove the above thing. My idea was the following: $\iota:O_n(\mathbb{R})\to GL_n(\mathbb{R})$ be the inclusion map which is a group homomorphism. It induces a fibre bundle $B\iota:BO_n(...
user avatar
3 votes
1 answer
47 views

Every continuous $f:\mathbb{R}P^5\rightarrow (S^1\vee S^1)\times T^3$ is homotopic to a constant map.

A practice exam question: Show that every continuous map $f:\mathbb{R}P^5\rightarrow (S^1\vee S^1)\times T^3$ is homotopy equivalent to a constant map. I'm not even sure where to start with this one....
user avatar
1 vote
1 answer
58 views

Why is $X \mapsto hom(\pi_*^{st}X, \mathbb{Q})$ the same as ordinary rational cohomology?

I am trying to understand the notion of Anderson duality from appendix B of this paper https://arxiv.org/abs/math/0211216 by Hopkins and Singer. But I somehow get stuck at the very first steps. I am a ...
user avatar
  • 13
2 votes
0 answers
54 views

Varieties with no vector bundles

What are some examples of algebraic varieties over the complex numbers with no (algebraic) vector bundles other than the trivial ones? The only example I can think of is $ \mathbb{A}^n_{\mathbb{C}} $ ...
user avatar
-3 votes
1 answer
36 views

Simplicial complex [closed]

I started to learn about "simplicial complex" and read about applications but it was very difficult for me to understand these applications, my question is as below what is the importance ...
user avatar
1 vote
0 answers
30 views

homotopy fiber of map induced by the Postnikov tower of $S^2$ is weakly homotopy equivalent to $S^3$

It is well known that for the Postnikov tower for $S^2$, $\mathbb{C}P^{\infty}$ and $P_2S^2$ are weakly homotopy equivalent. Since they are CW complexes, they are actually homotopy equivalent. Now I ...
user avatar
0 votes
0 answers
36 views

When are the two adjunction spaces $X_f$ and $X_g$ homotopy equivalent to each other?

Let $f,g: S^1\to S^1$ be continuous maps, $$X_f=S^1\cup_fD^2=(S^1\cup D^2)/(x\sim f(x)\ | \ x\in S^1)\\ X_g=S^1\cup_gD^2=(S^1\cup D^2)/(x\sim g(x)\ | \ x\in S^1)$$ Show that if $f$ is homotopic to $...
user avatar
  • 11
1 vote
2 answers
54 views

Example of a finite category with no pushouts or pullbacks?

Can anybody prove to me an example of a finite category (in objects and maps) for which there are no pushouts or pullbacks? I'm familiar with some examples that its false for one or the other
user avatar
0 votes
1 answer
25 views

The norm in the n-simplex

İ have this question and my attempt to solve it as the following : Consider an n-simplex $[p_0, p_1,...,p_n]$. If $a,b \in [p_0, > p_1,...,p_n]$, then show that $||a − b|| \leq \sup_i ||a − p_i||$....
user avatar
0 votes
0 answers
18 views

Cellular Approximation for Homeomorphism on a Compact Surface.

In my quest to compute the fundamental group of some object in multiple ways, questions concerning cellular approximations of homeomorphisms came up. The setting is as follows: Suppose $M$ is a ...
user avatar
  • 103
0 votes
0 answers
42 views

Problem on finding Intersection form of compact,orientable $4-$manifolds .

$\mathbf {The \ Problem \ is}:$ Let $M$ be an $\mathbb{F}$-oriented manifold of dimension $2 n$ for a field $\mathbb{F}$. Consider the non-singular bilinear form $H^{n}(M ; \mathbb{F}) \otimes H^{n}(M ...
user avatar
0 votes
0 answers
35 views

Question about the connected sum of two smooth manifolds

I'm a little confused with the following two questions about connected sum: (1) Is the covering space$\widehat{M\# N}$ of the connected sum of two smooth closed manifolds $M, N$ the connected sum $\...
user avatar
1 vote
1 answer
41 views

A continuous map $f:S^2\rightarrow S^2$ such that $f(x)\neq f(-x)$ for all $x$ is surjective

For all $x\in S^2$, let $-x$ denote its antipode. Let $f:S^2\rightarrow S^2$ be a continuous map such that $f(x)\neq f(-x)$ for all $x\in S^2$. Show that $f$ must be surjective. I'm working through ...
user avatar
0 votes
1 answer
42 views

Find the minimum number of colors to color any Map on Torus.

I am finding the minimum number of colors to color any map on torus. I have drawn how complete graph $K_5$ can be embedded on a torus. I know that the chromatic number of this graph is $5$ and we ...
user avatar
1 vote
1 answer
37 views

First homology group of a closed non-orientable 2-manifold vía the cellular homology groups

Let $N_h$ be a closed non-orientable 2-manifold of genus $h\geq 1$. I am trying to compute the first homology groups $H_1(N_h)$. For do so, it is sufficient compute the cellular homology group $H_1^{...
user avatar
  • 493
4 votes
0 answers
39 views
+200

Husemoller: homotopy of linear clutching map (proposition $4.5$, pag. $187$)

Background : I'm currently studying vector bundle through the book of [husemoller,"fibre bundles"] (https://www.maths.ed.ac.uk/~v1ranick/papers/husemoller). The following question concerns a ...
user avatar
1 vote
1 answer
35 views

Show that $S^{1} \vee S^{1}$ is a deformation retract of the space $Y=X \backslash \{x\}$

I have the space $X = [0,1] \times [0,1]$ that's equipped with the equivalence relation pictured below. We let $x_{0} \in X/ \sim$ be the origin $x_{0}=[(0,0)]$. We have two paths defined $\gamma, \...
user avatar
1 vote
1 answer
68 views

Homology homomorphism of the inclusion map of a path component

I'm quite new learning singular homology and from problem 9.9 from Greenberg & Harper's book of algebraic topology I ran into this problem: Let $X$ to be a path component of $X'$ and let $\iota:X\...
user avatar
  • 255
4 votes
2 answers
87 views

A version of Brower's fixed point theorem for contractible sets?

Brouwer's fixed point theorem states that a continuous map $f:B^n\to B^n$ ($B^n\subset\Bbb R^n$ being the $n$-dimensional ball) has a fixed point. It is clear that we can replace $B^n$ with a space $X$...
user avatar
  • 28k
0 votes
0 answers
58 views

The first homology group $H_1(M)$ of a compact manifold $M$ is always $\mathbb Z_i \oplus \mathbb Z_j\oplus...\oplus\mathbb Z_n$ ($i,j,...,n\ge 2 $)?

The dimension of the first homology group $H_1(M)$ is the number of (nonequivalent) loops of the manifold $M$. For all the cases I know, such as $RP^n$, $S^n$, cylinder, n-torus, klein bottle and so ...
user avatar
  • 151
0 votes
0 answers
35 views

If $K(G,1)$ and $K(H,1)$ are Eilenberg-MacLane spaces, show that $K(G\ast H,1)$ is also one.

If $K(G,1)$ and $K(H,1)$ are Eilenberg-MacLane spaces, show that $K(G\ast H,1)$ is also one. Definition. A path-connected space whose fundamental group is isomorphic to a given group $G$ and which has ...
user avatar
2 votes
1 answer
36 views

Computing $\pi_1$ of subset $X\subseteq \mathbb R^2$, $X$ union of 3 simp.conn. subspaces w/ simp.conn. pairwise intersection but empty intersection

Let $X$ be a subset of $\mathbb R^2$, and suppose that $X$ is equal to the union of open and simply connected subspaces $V_1,V_2,V_3$. Moreover, assume that the pairwise intersections $V_i \cap V_j, \...
user avatar
3 votes
2 answers
137 views

The isomorphism $H^1(X;\mathbb Z_2) \rightarrow \operatorname{Hom}(\pi_1(X),\mathbb Z_2)$ and $w_1(E)$

On page $87$ of Hatcher's book Vector Bundles and K-Theory it states that, assuming $X$ is homotopy equivalent to a CW complex ($X$ is connected), there are isomorphisms $$H^1(X;\mathbb Z_2) \...
user avatar
  • 151
-3 votes
0 answers
21 views

is open set in subspace equal to intersection open set in whole space with same subspace? [closed]

Is open set in subspace equal to intersection open set in whole space with same subspace?
user avatar
2 votes
0 answers
63 views

The stack classifying non-oriented triangles is equivalent to a quotient stack $[\widetilde{T}/S_3]$

I come across problems when I try to compute the stack classifying non-oriented triangles. My reference is the book Algebriac Stacks ( K. Behrend, B. Conrad, D. Edidin, B. Fantechi, W. Fulton, L. ...
user avatar
1 vote
1 answer
13 views

Why is $F(N_t \times (a_t , b_t))$ is contained in an evenly covered neighborhood of $F(y_0, t)$ in Theorem 1.7 of Hatcher's Algebraic Topology?

The Proof of Theorem 1.7 of Hatcher's Algebraic Topology says, "Since F is continuous, every point $(y_0, t) \in Y \times I$ has a product neighborhood $N_t \times (a_t, b_t)$ such that $F(N_t \...
user avatar
4 votes
0 answers
21 views

Fractional value of orbifold Euler characteristic: confusing definitions

I'm learning about Euler characteristic of orbifold, and struck by the statement that it can take fractional values. However, I've come across several definitions of orbifold Euler characteristic that ...
user avatar
1 vote
1 answer
59 views

Prove that the homeomorphisms generate action such the projection to quotient is a covering space.

I need to prove that the action in the homeomorphism $\varphi:S^3\to S^3$ defined by $$ \varphi(z_1,z_2) = \Bigl( e^{\frac{2\pi i}{n}}z_1, e^{\frac{2\pi mi}{n}}z_2 \Bigr) $$ is a covering space on the ...
user avatar
  • 83
-1 votes
1 answer
52 views

Compute the fundamental groups of $Z$ and $W$ [closed]

Let $\mathbb{R}^{n}$ and $P^{n}(\mathbb{R})$ denote the Euclidean space and the real projective space endowed with their standard topologies. Let $p$ denote the origin in $\mathbb{R}^{n}$. (b) Let $Z$ ...
user avatar
0 votes
1 answer
37 views

Van Kampen's Theorem: how to find the value of $N$ in $\pi_1 (S^2,x_0) = \frac{e * e }{N}$?

Van Kampen 's Theorem : Let $X= A_1 \cup A_2$,where $A_1 ,A_2 $ and $A_1 \cap A_2 $ are path connected and let $x_0 \in A_1\cap A_2 $ then $\pi_1 (X,x_0) = \frac{\pi_1(A_1) * \pi_1(A_2) }...
user avatar
  • 35
0 votes
0 answers
26 views

Lack of detail in Hatcher's proof of characterisation of the HEP

In page 14 of Algebraic Topology, Hatcher sketches a proof for the fact that a pair $(X, A)$ has the HEP iff $(X\times \{0\})\cup (A\times I)$ is a retract of $X\times I$. However, the argument rests ...
user avatar
  • 595
0 votes
1 answer
35 views

Prove there is no covering spaces

Prove that there are no covering spaces $p: S^{1} \longrightarrow C$ whose base is a convex space and whose total space is $S^{1}$ Covering space def: Let $B$ be a space. A map $p:E\longrightarrow B$ ...
user avatar
1 vote
0 answers
21 views

Example of connectivity of join is strictly greater than the sum of connectivities plus 2?

Given a topological space $X$, $X$ is $k$-connected if any $-1\le \ell \le k$ and continuous map $f:S^{\ell}\to X$ can be extended to $\bar{f}:B^{\ell+1}\to X$, where $S^\ell$ is viewed as the ...
user avatar
  • 2,069
2 votes
1 answer
55 views

Covering the Circle with closed connected sets.

This is a follow up to this previous post . The answer and the question got me to thinking about the case of $S^{1}$. The conditions are again the same . Can $S^{1}$ be covered by finite number of ...
user avatar
  • 563
1 vote
0 answers
40 views

Understanding Leray-Hirsch theorem from Bott and Tu.

What does the statement actually mean? Could anyone please share some ideas on it? Also how does it follow from Künneth formula? Any help in this regard would be warmly appreciated. Thanks for your ...
user avatar
1 vote
1 answer
26 views

Show that with these properties a topology is defined on $\overline{\mathbb{R}}$ where $\overline{\mathbb{R}}$ is a hausdorff-space.

On the set $\overline{\mathbb{R}}:=\mathbb{R}\,\cup\,\{-\infty,\infty\}$ a topology is defined as: A subset $U\subset \overline{\mathbb{R}}$ is open if the following requirements are fulfilled: (i) $\,...
user avatar
1 vote
1 answer
50 views

Sufficient condition for a simply connected subset of $\mathbb{R}^2$ to be contractible.

Simply connected subsets of $\mathbb{R}^2$ are not necessarily contractible, as shown for example by the Warsaw/Polish circle (see the answer to this question). However, the two seem intuitively quite ...
user avatar
  • 267

1
2 3 4 5
381