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)

2
votes
0answers
30 views

Computational Topology Codes

I am working on a project with a that PI thinks could be solved with computational topology tools. For this project, we will be looking at the persistent homology of objects in 3D images. I tried ...
0
votes
0answers
38 views

Homotopic family of curves

I stumbled over the following question. Imagine we have a two homotopic curves on the sphere $\mathbb{S}^1$ namely $\gamma_1,\gamma_2$. Then we can write them as $\gamma_{i}(t) = e^{i \alpha_i (t)}$ ...
2
votes
0answers
19 views

The homology groups of an infinite product of spaces

Suppose $I$ is some index set and let $\{X_i \}_{i\in I}$ be a collection of topological spaces (as nice as you like them to be). What is known about the (say) singular homology of $X := \prod_{i\in ...
3
votes
2answers
60 views

Definition of Cech-De Rham complex. Can't understand its definition!!!!

Let $M$ be a manifold and let $\mathcal{U}:=\{U_\alpha\}_{\alpha\in I}$ be an open covering, $I$ be a totally ordered set. For every $p$ and for every $\alpha_0<\dots<\alpha_p$, $$ ...
1
vote
1answer
47 views

Action of $S^1$ on homotopy groups of an $S^1$-space

I am interested in the following question : Let $S^1$ be the $1$-sphere, seen as a topological group by being the unit sphere in the complexe plane $\mathbb{C}$. Let $X$ be a (good, ... etc) pointed ...
10
votes
0answers
98 views

Prove Euler characteristic is a homotopy invariant without using homology theory

I was flipping through May's Concise Course in Algebraic Topology and found the following question on page 82. Think about proving from what we have done so far that $\chi(X)$ depends only on ...
2
votes
2answers
61 views

Sum of Betti numbers of a degree $d$ hypersurface of $\mathbb{P}^n_{\mathbb{C}}$

Let $X \subseteq \mathbb{P}^n_{\mathbb{C}}$ be the zero set of a homogeneous polynomial of degree $d$. Assume that $X$ is smooth. Is there a way to find the sum of the Betti numbers of $X$ (i.e. the ...
5
votes
0answers
47 views

Is the space of $G$-maps $G/H \to X$ naturally homeomorphic to $X^H$?

Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h \in H\} \subset X;$$$X^H$ is the $H$-fixed point subspace of $X$. ...
4
votes
0answers
43 views

Are these “Stiefel-Whitney numbers” invariants of the (smooth)topology of the total space?

Let $E$ be a smooth real vector bundle over a smooth compact $n$-dimensional manifold $M$. It is relatively easy to see that the Stiefel-Whitney classes $w_i(E)$ are not diffeomorphism invariants of ...
2
votes
1answer
39 views

Question about maps to $K(G,1)$

I have an unfortunately basic confusion about two results in Hatcher's Algebraic Topology. Let $G$ be an abelian group. Theorem 4.57 specialized to $n=1$ says that there is a bijection $$\langle X, ...
3
votes
0answers
24 views

branched cover over a symplectic surface

Consider a branched cover $X$ of $(B^4,\omega_{st})$ over a symplectic surface $F$. Is it true that there is a naturally induced symplectic structure on $X$? Of course, there is a natural symplectic ...
2
votes
0answers
20 views

Properties of spectra

I've been trying to understand spectra better, and I was wondering what would be a good source to learn from. My issue with the classical treatments (e.g., Adams, Switzer) is that people keep telling ...
0
votes
1answer
40 views

Normality of a covering space.

I am having some trouble trying to do problem 1.3.24 in Hatcher's Algebraic Topology: Given a covering space action of a group G on a path-connected, locally path-connected space X, then each ...
1
vote
0answers
25 views

Reference request about Thm which use Transversality to compute Homotopy Groups [duplicate]

I'm following the following notes, and my attention was caught by Theorem $1.1.4$. I am unable to find any reference of the proof. Could you suggest me some books in which there is a proof of this ...
1
vote
1answer
48 views

The cone of a topological space is contractible (why is the homotopy well defined?)

If $X$ is a topological space, define ${\rm Cil}(X) = X \times I$ the cylinder over $X$, and the cone over $X$, $\operatorname{Con}(X) = \operatorname{Cil}(X)/{\sim}$ the quotient by saying that ...
51
votes
3answers
885 views

Topological spaces admitting an averaging function

Let $M$ be a topological space. Define an averaging function as a continuous map $f:M \times M \to M$ which satisfies $f(a,b) = f(b,a)$ for all $a,b \in M$ and $f(a,a) = a$ for all $a \in M$. These ...
4
votes
1answer
38 views

Ways to link the unknot to a pole

Is there a way to show that the following ways of linking an unknot to an infinite horizontal pole are inequivalent? Perhaps the Wirtinger presentation would work, but I am not sure because of the ...
6
votes
3answers
196 views

How can one compare these two 4-manifolds

We would like to compare the following two real 4 dimensional manifolds: 1)$M$=The tangent bundle of $S^{2}$ 2)$N$= The total space of the canonical line bundle over $\mathbb{C}P^{1}\simeq S^{2}$ ...
2
votes
1answer
45 views

Todd genus as homomorphism from complex cobordism

It's well-known that the Todd genus/arithmetic genus $\chi(\mathcal{O}_X)$ (or probably preferably $\int_X \text{td}(T_X)$ so as to define it in purely terms of the complex structure) is a genus in ...
1
vote
1answer
29 views

What is the Wrong in this Triangulation of the Torus

On pg 133 of Roman's Introduction to Algebraic Topology it is stated that one requires at least 14 triangles in any triangulation of the torus. Admittedly, I do not have a very good understanding of ...
0
votes
0answers
53 views

Trying to prove that a continuous function $\,f: \mathbb{D}^2 \rightarrow \mathbb{R}^2$ has a zero using the fundamental group?

Prove: For a continuous $\;f: \mathbb{D}^2 \rightarrow \mathbb{R}^2,\;$ $\,f\left(\omega,0\right) \notin f\left(S^1\right),\,$ given a non constant function $\left(f_{\mid S^1}\right)_{\ast} : ...
1
vote
1answer
30 views

Trouble With a Triangulation of the Torus

On pg. 133 of Rotman's Introduction to Algebraic Topology, we have a figure which claims to be a triangulation of the torus. Now a triangulation of a topological space is defined as Definition. ...
1
vote
1answer
24 views

Homology groups of $D^2\times S^1$, full torus

I know what are the homology groups of a torus $T=S^1\times S^1$, in sense that $$\tilde{H}_1(T)=\mathbb{Z}^2,H_2(T)=\mathbb{Z}$$ but I wonder what happens if we fill it. What are the homology groups ...
0
votes
1answer
14 views

Interior of simplex that is a proper face not open?

If a simplex $\sigma$ is a proper face of another simplex $\tau$, why is its interior not open in $\tau$? I can't seem to understand as, let's take $\sigma=ab$, a line segment that is a proper face ...
1
vote
2answers
64 views

Prove: If $f: X \subset \mathbb{R}^n \rightarrow Y$ has a continuous extension to all $\mathbb{R}^n$ then $f_\ast$ is trivial.

Prove: If $f: X \subset \mathbb{R}^n \rightarrow Y$ is continuous and has a continuous extension to all $\mathbb{R}^n$ then $f_\ast$ is trivial. I'm not sure how the fact that there exists an ...
1
vote
1answer
56 views

Spectrum of the Ring of Continuous Functions on a Space

I was wondering when exactly we can recover the topological space, $X$, from its ring of continuous functions into $\mathbb C$ (or some sort of sufficient topological group). For any topological ...
1
vote
2answers
69 views

Attaching maps for the CW-decomposition of the 3-torus

I want to calculate the homology of the $3$-torus via cellular homology. I figured out a CW-decomposition of the $3$-torus: $1$ $0$-cell, $3$ $1$-cell, $3$ $2$-cell, $1$ $3$-cell. So the chain complex ...
0
votes
1answer
39 views

Is my proof that the product of covering spaces is a covering space correct?

Let $p_1:\tilde X_1 \rightarrow X_1$ and $p_2:\tilde X_2 \rightarrow X_2$ be two covering spaces. Prove: $p = p_1 \times p_2:\tilde X_1 \times \tilde X_2 \rightarrow X_1 \times X_2$ is a covering ...
1
vote
1answer
49 views

If $r:X\to A$ is a Retraction, Then $H_n(X)\cong H_n(A)\oplus H_n(X,A)$

$\DeclareMathOperator{\im}{Im}$ Let $A$ be a subspace of a topological space $X$ such that there is a retraction $r:X\to A$ of $X$ onto $A$. Then $H_n(X)=H_n(A)\oplus H_n(X, A)$ for all $n$. ...
1
vote
2answers
31 views

Question of maps in Mayer-Vietoris sequence

We obtain MV-seq. from short exact sequence $$ 0\to C_n(A\cap B) \to C_n(A)\oplus C_n(B)\to C_n(A+B)\to 0 $$ So map i wonder that map $H_n(A\cap B)\to H_n(A)\oplus H_n(B)$ maps $[a]$ to $([a],[-a])$. ...
2
votes
1answer
66 views

The sphere $S^2$ is not contractible

I heard that in topology the sphere $S^2$ cannot be continuously deformed to a point, i.e. $S^2$ is not contractible. Sorry for my ignorance, but I really don't get it. Can't we just push all the ...
2
votes
2answers
76 views

In algebraic topology, for a function $f$ what does $f _\ast$ mean?

In algebraic topology, for a function $f$ what does $f_{ \ast}$ mean? I'm solving some exercises and this is something that's appearing, often relating to homotopic functions, and I'm not sure what ...
2
votes
0answers
33 views

Why is the singular $p$-simplex called singular?

I am learning homology. I cannot see in which sense the singular $p$-simplex is singular. It seems quite regular. Any idea?
0
votes
0answers
22 views

What does a homotopic lift mean when talking about homotopic functions? [closed]

I misunderstood the problem I was working on, this should be closed.
1
vote
1answer
42 views

Find homology $S^n-f(X)$ where f is injective

Let $f\colon X\to S^n$ be an injective function. Find the homology groups of $S^n-f(X)$ where: a. $X=S^k\sqcup S^r$ b. $X=S^k\vee S^r$ The question above gives hint to look in both ...
0
votes
3answers
91 views

Definition of compactness unnecessarily verbose?

The definition of a compact set is given as a set, $X$, for which all open covers have a finite subcover. This seems unnecessarily verbose to me. Wouldn't it be sufficient to simply say that $X$ has ...
2
votes
1answer
48 views

branched cover over slice disc

I know some examples in 4 dimensions of rational homology balls (meaning that a manifold such that its rational homology groups are as a ball $B^4$) which are branched covers over a slice disc. Is the ...
0
votes
0answers
14 views

$G$-CW complex structure of a certain space

Let $G$ denotes the dihedral group $\langle x,y : x^2 = y^6 =1 ,xy=y^{-1}x \rangle $ and $H = \langle xy \rangle .$ Then , what is the $G$-CW complex structure of $G/H \ast G/H \ast \cdots \ast G/H $ ...
0
votes
1answer
33 views

To prove that the projection map $S^2 \to \mathbb RP^2 $ is a covering map via group action

I am reading Algebraic Topology and I got some problem in covering map. Please help me. Thnx in advance. I want to show that the projection map $S^2 \to \mathbb RP^2 (\text{ real projective plane })$ ...
1
vote
1answer
42 views

Is it (not) possible for two vector fields on the Klein bottle to be a basis?

Lefschetz fixed point theorem (for example) implies that a compact manifold with a nowhere vanishing vector field must have Euler characteristic zero. Is there a way to draw stronger algebraic ...
4
votes
1answer
39 views

Hirzebruch's $L$-polynomial and $\mathbb{C}P^n$

Hirzebruch's $L$-polynomial is the formal power series \begin{equation} L(x) = \frac{x}{\tanh x} = 1 + \frac{x^2}{3} + \cdots \end{equation} This defines a multiplicative sequence and a genus $L(M)$ ...
2
votes
0answers
32 views

$\Delta$-Complexes Are Hausdorff

I am using the definition of a $\Delta$-complex as given in Hatcher's book here on pg 103. Now on pg. 104, just before the section on Simplicial Homology, Hatcher remarks that if $X$ has a ...
3
votes
2answers
47 views

CW-decomposition of quotient space

Let $X$ be the space that results form $D^3$ by identifying points on the boundary $S^2$ that are mapped to one another by a $180°$-rotation about some fixed axis. I want to calculate the cellular ...
0
votes
0answers
29 views

Trouble With the Definition of $\Delta$-Complex in Hatcher's Book

On Pg. 103 of Hatcher's Algebraic Topology the author has defined what a $\Delta$-complex is: This is what I have gathered from what the author writes: A $\Delta$-complex is a collection of ...
6
votes
3answers
184 views

What is the sheafification of the presheaf of the one point compactification?

Okay, so I had this idea for a presheaf that is quite peculiar. Instead of being based on algebraic category (i.e. abelian groups), it is based on a topological one, the category of compact ...
2
votes
1answer
31 views

Irregular (branched) cover

I need to know the definition of an irregular (branched) cover. I heard this somewhere but I am not able to find any definition on the internet.
1
vote
2answers
32 views

Isomorphism of chain complexes

In my notes it says $C^{sing}_n(\sqcup_{i\in I} X_i;R) \cong {\bigoplus}_{i \in I} C^{sing}_n(X_i;R)$, where $C^{sing}_n$ denotes the n-th singular chain complex and $R$ is a ring, $S_n(X)$ is the set ...
2
votes
1answer
60 views

Does trivial fundamental group imply contractible?

Let $X$ be a path-connected topological space with a trivial fundamental group: $$\pi_1(X,x_0)=\{e\}.$$ Does $X$ have to be homotopic to a point? I know that the converse is true: a ...
1
vote
0answers
39 views

Closed unit ball is a retract of $R^2$

I was asked whether a closed unit ball is a retract of the euclidean space $R^2$. I think the answer is yes and the retraction might be defined as follows: for all the points in $R^2$ join them with ...
3
votes
2answers
52 views

Help finding the fundamental group of $S^2 \cup \{xyz=0\}$

let $X=S^2 \cup \{xyz=0\}\subset\mathbb{R}^3$ be the union of the unit sphere with the 3 coordinate planes. I'd like to find the fundamental group of $X$. These are my ideas: I think the first thing ...