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

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2answers
20 views

Manifolds with a finite but not trivial fundamental group

I came across this nice result: Theorem: If $M$ is a connected smooth manifold with finite fundamental group, then its first de Rham cohomology is trivial: $$H^1_{dR}(M)=0.$$ However, I don't ...
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0answers
13 views

Spaces of Labeled Complexes (Munkres)

The following is taken from Munkres' Algebraic Topology book. I tried to determine which spaces (e.g. Mobius Strip, Klein bottle, etc) these complexes are, but to no avail. I computed the Euler ...
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1answer
48 views

Is every simply connected open subset of $\Bbb R^n$ contractible?

Question: Is every simply connected open subset of $\Bbb R^n$ contractible? I know the result is true for $\Bbb R^2$ because by the Riemann Mapping Theorem every simply-connected proper open ...
3
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0answers
17 views

$\mathscr{O}(G/H, G/K) \cong (G/K)^H?$

What I am about to ask is related to the question presented here. 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 ...
2
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0answers
30 views

Why does there exist a deck transformation mapping here?

See Kevin Dong's answer here. Let $\widetilde{\gamma}$ be a path in $X_N$ based at $x_0^N \in X_N$ with $p(\widetilde{\gamma}) = \gamma$ a nontrivial path in $X$. Since the cover is normal, there ...
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0answers
10 views

Given a triangulation (labeled complex), how do we determine the space?

Given a triangulation, how do we tell which space it is representing? My first idea would be to calculate the Euler Characteristic, but that would still leave some ambiguity, e.g. both the Projective ...
1
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1answer
25 views

Intersection preserves homotopy equivalence

Let $Z$ be a topological space with subspaces $X$, $Y$, $X'$ and $Y'$. Suppose that $X$ is homotopy equivalent to $X'$ and $Y$ is homotopy equivalent to $Y'$ do we have that $X\cap Y$ is homotopy ...
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1answer
30 views

Is this a valid triangulation of a space? (Algebraic Topology)

On the surface, it looks like it would be a Mobius Strip due to the "twist". However, there are some inconsistencies like b is adjacent to d on the left, but not on the right of the figure. ...
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3answers
43 views

Requirement “closed under finite intersection” in Van-Kampen-Theorem

Given a topological space $X$, a point $x_0 \in X$ and an open cover $\mathcal{U}$ of $X$ of path-connected subsets containing $x_0$ which is closed under finite intersections, we have by Van-Kampen ...
1
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1answer
41 views

Counter example to existence of Mayer-Vietoris sequence

Every open cover $X = U \cup V$ gives an exact sequence (called mayer vietoris sequence) $$ \ldots \to H_n(U \cap V) \to H_n(U) \oplus H_n(V) \to H_n(X) \to H_{n-1}(U \cap V) \to \ldots $$ Do $U$ and ...
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0answers
38 views

Nth Homotopy Group Isomorphic to [T^n, X]

Following Spanier's book on algebraic topology chapter 1, section 6 about suspensions, I'm wondering about the following questions: 1) We know that $S^n$ is an H co-group for all $n\geq1$ because ...
7
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1answer
155 views

Topological idea of orientability of manifold

While reading Poincare Duality a new idea of orientability of manifold came in my mind.I dont know wheather this idea is new or not, or even true or false. My idea is following... A $n$-dim manifold ...
11
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0answers
71 views

Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

$\Bbb{CP}^{2n+1} \# \Bbb{CP}^{2n+1}$ supports a complex structure: $\Bbb{CP}^{2n+1}$ has an orientation-reversing diffeomorphism (complex conjugation!), so this is diffeomorphic to the blowup of ...
2
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0answers
20 views

Euler Integral of a self-overlapping tube with a cusp singularity

I am studying in depth the following paper on Euler calculus applied to target enumeration: https://www.math.upenn.edu/~ghrist/preprints/eulerenumerationpart1.pdf Within this paper there is an ...
3
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0answers
56 views

Computational Topology Codes

I am working on a project with a PI that 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 ...
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0answers
48 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
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0answers
24 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
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2answers
65 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
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1answer
66 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 ...
16
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1answer
225 views
+200

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
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2answers
68 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 ...
7
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1answer
100 views
+50

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 $X$. ...
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0answers
46 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
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1answer
47 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
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0answers
21 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
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1answer
42 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
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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
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1answer
50 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
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3answers
913 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
39 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
208 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
48 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
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1answer
32 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
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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
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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
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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
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2answers
65 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
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1answer
57 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
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2answers
72 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 ...
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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 ...
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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 $n$. ...
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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
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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 ...
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0answers
34 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?
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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 ...
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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 ...
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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 $ ...