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

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70 views

Why does this imply that two homotopic maps $h,k:S^1→ S^1$ must have the same degree?

I want to show that if two maps $h,k:S^1→ S^1$ are homotopic, then they have the same degree. We define the degree of a continuous map $h: S^1 \rightarrow S^1$ as follows: Let $b_0$ be the point ...
6
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1answer
67 views

Is the Whitney embedding theorem tight for all $n$?

Whitney's embedding theorem states that any smooth $n$-manifold $M$ can be smoothly embedded into $\Bbb R^{2n}$. In dimension 1 this is tight: the circle cannot be embedded into $\Bbb R^1$. It is a ...
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0answers
31 views

Homeomorphism of CW complex

In exercise 2.2.13 in Allen Hatcher's Algebraic Topology, we consider (I quote directly) the 2-complex $X$ "obtained from $S^1$ with its usual cell structure by attaching two 2-cells by maps of degree ...
2
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2answers
37 views

Proving that the orientation bundle of a non orientable manifold is isomorphic to every other oriented 2-coverings of such manifold

I've got some problem proving this statement, recalling that for me, an orientable manifold is a manifold which admits an atlas such that the transition functions have always local degree $1$ (we are ...
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1answer
45 views

How can I show $G_0$ and $G_1$ are conjugate subgroups?

Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the induced homomorphism from the fundamental group of $E$ to the fundamental group of $B$. Let $e_o$ and $e_1$ be points in E ...
4
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2answers
64 views

How do I show that $S^1$ is the suspension of $S^0$?

How do I show that $S^1$ is the suspension of $S^0$? I have all the definitions here, I'm just bad at applying them. The suspension of a topological space $X$ is the quotient $CX / (X × ${$1$}$)$, ...
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1answer
41 views

Is the suspension space contractible?

Let $X$ be a topological space. The suspension of $X$, denoted $ΣX$, is the quotient $CX / (X × ${$1$}$)$, where $CX$ is the cone on $X$, the quotient space $(X × [0, 1]) / (X × ${$0$}$)$. Is $ΣX$ ...
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1answer
49 views

How to prove the Cone is contractible?

Let $\sim$ be an equivalence relation on a topological space $X$ with a subset $A ⊂ X$ such that the equivalence classes are: $A$ itself and Singletons $\{x\}$ such that $x ∉ A$. Then define $X/A$ ...
2
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3answers
82 views

What space is this Homeomorphic to?

Let $X$ be a topological space and let $A ⊂ X$. Let $\sim$ be an equivalence relation on $X$ such that the equivalence classes are: $A$ itself and Singletons $\{x\}$ such that $x ∉ A$. Then ...
1
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1answer
77 views

Looking for a homeomorphism $\mathbb{C}P^1 \cong S^2$

I want to show $\mathbb{C}P^1 \cong S^2$ by explicit construction. Everything I tried so far did not work out unfortunately :( Any hints?
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0answers
76 views

Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups

I am interested in the following question. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective embeddings $f:K_1\rightarrow K_2, g:K_2\rightarrow ...
1
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2answers
20 views

Possibility of the cellular decomposition of a manifold

I am working to find if it's possible to find a cellular decomposition of $S^2\times S^1$ as following: $e^0\cup e^1\cup e_1^2\cup e_2^2\cup e^3$. I cannot find such a decomposition. And I try to ...
1
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3answers
71 views

Is $S^1$ homeomorphic to $\mathbb{R}P^1$?

I am supposed to construct a homeomorphism of $S^1$ and $\mathbb{R}P^1$ but I am not toally sure that this is even possible. I think I have learned at some point that $$\mathbb{R}P^1=S^1/\{x=-x\}$$ ...
1
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1answer
23 views

Contradictory Orientations of Faces in Simplicial Complexes

From what I've read, an orientation of a simplex is an equivalence class of total orders on its vertices, where equivalence means up to even permutation. An orientation on an $n$-simplex induces an ...
0
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0answers
31 views

Difference between simplex and simplicial complex

First I know the definition of simplex intuitively as follows, Simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. And the defintion of simplicial ...
2
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1answer
38 views

Homology as Boundary of “Submanifold”

In the plane, imagine a horizonal figure eight, $\infty$. Let $\alpha$ be the curve which is convex from the leftmost point of the figure to its "middle", and concave from the middle until the ...
2
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0answers
34 views

Homology of $S^n - S^k\vee S^\ell$

Does anyone know a good trick to computing homology groups of the sphere minus the wedge of two spheres of possibly different dimension $S^n \setminus S^k\vee S^\ell$ ? Any particular $k$ and $\ell$ ...
7
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0answers
146 views

Why the Steinberg idempotent is idempotent?

Consider the group $GL_n(\mathbb{F}_p)$. We have the following subgroups : -$\Sigma_n$ the symmetric group (permutation matrices) -$B_n$ the Borel subgroup (upper triangular matrices) -$U_n$ the ...
4
votes
1answer
60 views

on Cayley diagrams

is the picture the Cayley Graph of the group $\langle a,b,c\mid a^2, b^2,c^2\rangle$ ? What would it be for $\langle a,b,c\mid a^2b^2c^2\rangle$?
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0answers
25 views

Are the face posets of CW-complexes Eulerian?

Suppose we had a CW-complex $X$ with decomposition $X_{i}$ Is its face poset, consisting of cells and covers generated by the attachment of cells to one another, an Eulerian poset? What would be the ...
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0answers
26 views

Homology and Homotopy in the Plane II

This question arose from Homology and Homotopy in the Plane, where it was one of several questions asked (but not answered). I'm posting it separately so I could accept one of the answers there. Is ...
0
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1answer
65 views

what is a path that cover all of $S^n$?

Here is the meaning of "cover" which I can't understand: Prove that if $n\ge 2$, then $S^n$ is simply connected. hint: Use Exercise 2.5 to show that every loop in S" is homotopic to a loop that does ...
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3answers
50 views

Geometric Homotopy as Chain Homotopy

In Can we think of a chain homotopy as a homotopy, I learned that chain homotopy can be defined in an analogous fashion to homotopy, i.e from the product with an interval object etc. What about the ...
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0answers
13 views

Fundamental group and Fuchsian group

We know that the fundamental group of a compact surface of genus larger or equal to 2 is a Fuchsian group, i.e. a discrete subgroup of the automorphism group of the hyperbolic plane PSL(2,R). And any ...
3
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1answer
80 views

Non homeomorphism

I want to show that the sphere $S^2$ and the torus $T^2$ are not homeomorphic, using the notion of intersection modulo $2$. I have to show that any two loops on the sphere $S^2$ have an even number of ...
1
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1answer
29 views

Covering space action on an orientable manifold $M$ implies $M/G$ orientable (Hatcher)

I'm trying to solve the following problem from Hatcher (3.3.4) Given a covering space action of a group $G$ on an orientable manifold $M$ by orientation preserving homeomorphisms, show that $M/G$ is ...
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0answers
23 views

Understanding the “least normal subgroup” in Seifert van Kampen

The Seifert van Kampen theorem implies that if $V$ is simply connected then there is an isomorphism $k: \pi_1(U, x_0) / N \rightarrow \pi(X, x_0)$ where $N$ is the least normal subgroup of $\pi_1(U, ...
1
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1answer
35 views

Proving a fundamental group is NOT abelian

I was wondering if the following approach would be possible in proving the fundamental group of $X$ was not abelian. If one can show there exists a homomorphism: $\pi_1(X, x_0) \rightarrow ...
3
votes
1answer
49 views

Manifold with special cohomology group

I am trying to find if there is a orientable compact manifold $M$ of dimension 10 with the 5th cohomology group of De Rham $H^5_{DR}(M)\cong \mathbb R$. But, I can find such an example or prove that ...
3
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2answers
68 views

Show $H_2(M, \mathbb{Z}) = \mathbb{Z^r}$ if $M$ is orientable, $\mathbb{Z^{r-1}} \oplus \mathbb{Z_2}$ if nonorientable

I'm trying to solve this problem from Hatcher 3.3.24. Let $M$ be a closed connected 3-manifold, and write $H_1(M, \mathbb{Z})$ as $\mathbb{Z^r} \oplus T$ where $T$ is torsion. Show that $H_2(M, ...
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1answer
85 views

Let ($\mathbb{R}^n$,*) be a division algebra over $\mathbb{R}$, $n>1$, $\Rightarrow n$ is even

I want to prove: Let ($\mathbb{R}^n$,*) be a division algebra over $\mathbb{R}$, $n>1$, $\Rightarrow n$ is even. I'm stuck. My thoughts are: I want to use the hairy ball theorem and I want to ...
1
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2answers
111 views

Fundamental group a collection of six 2-spheres touching pairwise in a cyclic order, and each touching a common plane.

Find the fundamental group of the space comprising a collection of six 2-spheres touching pairwise in a cyclic order, and each touching a common plane. Touching means having one point in common. I ...
1
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0answers
15 views

Alexander Method for non-orientable Surface

From Farb and Margalit,Primer on Mapping Class Group, p.62, we know For example for torus with four puncture we can choose following claret red curves. Curves are choosen such that their complement ...
2
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1answer
59 views

Converse to the Eilenberg-Steenrod theorem?

For the purposes of this question, a homology theory is a covariant functor from the homotopy category of finite pointed CW complexes to graded abelian groups, and a collection of connecting ...
1
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1answer
25 views

Fundemental group of $D^2\setminus\{x\}$

Let $D^2=\{x\in\mathbb{R}^2:||x||\le1\}$, $x\neq a\in D^2$. Find $\pi_1(X\setminus\{x\},a)$ if: a. $x\in\partial D^2$ b.$x\in \text{int} D^2$ about the first one I think the fundamental ...
1
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3answers
60 views

Complete metric space, not simply-connected

I've been going over the algebraic topology part of Munkres and this question has stumped me. If we have a complete metric space that is not compact, must it be simply-connected (path-connected plus ...
2
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0answers
30 views

A space having exactly three coverings up to equivalence

Q: Give an example of a topological space having exactly 3 coverings up to equivalence (including a covering by the space itself). Proof: There is a theorem that says that given a topological ...
2
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1answer
43 views

Counterexamples to Brouwer’s fixed point theorem

Brouwer’s fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least ones fixed point. If we replace the convex condition with let's say ...
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1answer
31 views

Show that for a degree 1 map $f: M \rightarrow N$ the induced map $f_*: H_1(M) \rightarrow H_1(N)$ is a surjection

I'm trying to solve the following problem: Show that for a degree 1 map $f: M \rightarrow N$ of connected, closed, orientable manifolds, the induced map $f_*: \pi_1(M) \rightarrow \pi_1(N)$ is ...
1
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2answers
141 views

Does the projective Fermat curve have singular points?

I want to check if the projective Fermat curve, $$X^n+Y^n+Z^n=0, n \geq 1$$ has singular points. Could you give me a hint how we could do this? Do we have to check maybe if the curve is ...
4
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1answer
71 views
+50

Reference Request: Thom Spectrum of a virtual vector bundle

Given a vector bundle $\xi$, one can construct a Thom spectrum $\mathbb{T}h(-\xi)$ associated to its additive inverse. I unsuccessfully browsed through literature to find a reference for an explicit ...
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4answers
59 views

Is a continuous bijection function from a hausdorff space to a compact space is a homeomorphism?

We know a continuous bijection from a compact space to a Hausdorff space is always a homeomorphism. But I am wondering what happened if we switch the domain and codomain. Is a continuous bijection ...
0
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1answer
92 views

Is the complex projective plane a compact manifold with or without boundary (closed manifold)?

my question is the one in the title. (My motivation is to understand in which way Freedman's classification of compact simply-connected 4-manifolds implies the Poincare conjecture for 4-manifolds, as ...
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0answers
19 views

Finish the proof of Borsuk-Ulam theorem (Hatcher)

Hatcher at page 229 proposes to prove the Borsuk-Ulam theorem using the fact that any continuous map $f \colon\mathbb R P^n \to \mathbb RP^m$, $n > m$, induces the trivial map in cohomology with ...
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0answers
22 views

$X$ is a topological space of infinite cardinality which is homeomorphic to $X × X$. [duplicate]

$X$ is a topological space of infinite cardinality which is homeomorphic to $X × X$.Then which is true A. X is not connected B. X is not compact C. X is not homemorphic to a subset of R D. none of the ...
0
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2answers
49 views

How to show the covering space of an orientable manifold is orientable

I'm trying to prove this using purely topological arguments, no differential geometry as I haven't been exposed to it. I've been playing around with definitions a bit and here's what I have so far. ...
2
votes
1answer
59 views

Original Papers on Singular Homology/Cohomology.

I am currently reading Singular Homology Theory and Cohomology on my own mainly from Hatcher's "Algebriac Topology" and "Topology and Geometry" by Bredon. Quite often it happens that it takes a lot of ...
4
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1answer
49 views

Manifolds with vanishing Stiefel-Whitney classes but are not stably parallelizable

It is known that if a manifold is stably parallelizable, then it's Stiefel-Whitney classes must vanish. Is the converse true? Note that we know that the converse cannot hold if stably parallelizable ...
2
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3answers
101 views

$M \times N$ orientable if and only if $M, N$ orientable

For two manifolds $M$ and $N$ I'm trying to prove that $M \times N$ is orientable if and only if $M$ and $N$ are orientable. My attempt so far: $\impliedby)$ Assume $M, N$ are orientable. Then ...
2
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2answers
71 views

Is $\text{Hom}(A\oplus B, G) = \text{Hom}(A, G)\oplus \text{Hom}(B, G)$ true?

I'm reading Hatcher's Algebraic Topology, and in the proof of the Universal Coefficients Theorem (Page 192), it says for abelian groups $A$ and $B$, and an arbitrary group $G$, we have ...