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)

3
votes
1answer
19 views

Connected components $0-1$ matrices

Let $M$ be a $0-1$ matrix. Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where ...
1
vote
2answers
31 views

How do I prove this using van-Kampen theorem informally ? (2)

First of all, I feel really sorry to keep asking these questions without a try, but it is because I can't try.. Really if someone wants to see my lecture note I can show it.. Statement of the theorem ...
4
votes
3answers
26 views

How do I informally prove this using Van Kampen theorem?

Let $X$ be the space obtained from two tori $S^1\times S^1$ by identifying a circle $S^1\times\{x_0\}$ in one torus with the corresponding circle $S^1\times\{x_0\}$ in the other. Calculate ...
2
votes
4answers
33 views

How do I prove that a finite covering space of a compact space is compact?

Let $C$ be a finite sheeted covering space of compact space $X$. How do I prove that $C$ is compact? Someone please give me a proof sketch.. Let $p:C\rightarrow X$ be a covering map. Let ...
6
votes
2answers
17 views

Example of closed orientable manifold with a nonzero vector field, but not parallelizable

Can you give a simple example of a closed orientable manifold with an everywhere nonzero section of its tangent bundle, but where the tangent bundle is not trivial? If the manifold is not orientable, ...
1
vote
1answer
25 views

How do I prove this winding number is not zero?

Let $\alpha:[0,1]\rightarrow S^1:t\mapsto e^{2\pi it}$ be a path. Let $f:S^1\rightarrow S^1$ be a continuous map such that $-f(x)=f(-x)$ on $S^1$. How do I show that the winding number $Wnd(f\circ ...
0
votes
1answer
25 views

Non-connectedness in the plane

Let us have open $V \subset \mathbb{R}^2$ and $x\in V$. Now, how can I prove that the quotient space $V\backslash\{x\}$ is not simply connected? Pictorially I understand it as the failure of a loop to ...
0
votes
0answers
13 views

Maximal augmentation of cosimplicial space

I would like to ask the following question: There is a map of cosimplicial spaces $f^*: X^*\to Y^*$ such that for every $n$, the map $f^n:X^n\to Y^n$ is homotopically trivial. Is this true that the ...
2
votes
1answer
33 views

Is every open set in a base space evenly covered?

Let $C$ be a covering space of $B$. Then, does every open set in $B$ evenly covered by a covering map? This must be false but I cannot find a counterexample.. Please help
2
votes
1answer
62 views

Prove that these loops are homotopic [duplicate]

Let $G$ be a topological group with identity element $e$. Let $f,g: (S^1, (1,0)) \to (G,e)$ be loops in $G$ with base point $e$. We define $f * g: (S^1, (1,0)) \to (G,e)$ by $$f * g(s) = f(t) \cdot ...
2
votes
1answer
26 views

Unit close disc to prove a matrix algebra identity?

I need to prove that every $3 \times 3$ matrix with real positive entries has one eigenvector with a positive eigenvalue. Now, how do I prove this using the fact that the set $B=\{x=(x_1,x_2,x_3)\in ...
2
votes
1answer
23 views

Alexander Duality for Relative Homology

Is there a formulation for Alexander Duality for pairs of spaces $(A, B)$ such that $A\subset B\subset S^n$? I can't find a reference for this anywhere, but I think it is as follows, which I arrived ...
2
votes
1answer
46 views

Kernel of induced map between singular chain groups

Let $p : \widetilde X \to X$ be a two-sheeted cover. This induces $p_\sharp : C_n(\widetilde X; \mathbb Z_2) \to C_n(X; \mathbb Z_2)$. I can show that $p_\sharp$ is surjective by noting that every ...
0
votes
1answer
42 views

Fundamental group of composition of function

Let Y be a simply connected space, and let f : X → Y and g : Y → Z be continuous functions. What is π1(g ◦ f )? Prove your answer. I think the fundamental group of the composition would be the ...
0
votes
2answers
72 views

Fundamental group is a homotopy invariant

I am a newbie to topology and am not able to understand how to attack this problem: Any hints would be appreciated Assuming that: $$ f \sim g \Rightarrow \pi_1(f) = \pi_1(g). $$ Prove that the ...
2
votes
2answers
60 views

Map from $n$-sphere to $n$ dimensional torus

Let $n\ge 2$. How can you prove that for every continuous $f:S^n\to T^n$, the induced map on singular homology $f_\star:H_n(S^n)\to H_n(T^n)$ is the zero map? Here, $S^n$ is the $n$ dimensional ...
4
votes
0answers
49 views
+150

Action of $\operatorname{Aut}(G)$ on the Borel construction

I am interested into the (say real) regular representation $\rho$ of $G=(\mathbb{Z}/p)^n$. Considering the universal vector bundle $EG\rightarrow BG$, the Borel construction with the regular ...
2
votes
0answers
32 views

Union of simply connected spaces at a point not simply connected

I came across this example Spanier's Algebraic Topology book (by way of the Munkres Topology book). I kind of have an intuitive idea of why the space isn't simply connected but can't figure out a ...
2
votes
0answers
46 views

covering spaces

A covering space of a Hausdorff space is also Hausdorff. Conversely, a compact Hausdorff finite covering space has a Hausdorff base space. However, in general, a non Hausdorff space may have a ...
2
votes
0answers
34 views

Van Kampen Theorem for a Certain Square

Take a square with all the edges identifies. Choose a point $x$ on the boundary of this square. Take a small neighborhood $B_\epsilon(x)$ of this point. I want to compute $\pi_1(B_\epsilon(x) ...
1
vote
0answers
27 views

generalized cohomology

If I have a generalized cohomology theory $E$, then $E^n(X) = [\sum^{-n}X, E]$. I would like to know what $[\sum^{-n}*, E]$ looks like for $*$ a point. We can assume that $E$ is a nice $CW$ spectra ...
0
votes
2answers
21 views

Wedge Sum Embedding with Inclusions

Let $X$ and $Y$ be two disjoint topological spaces, $x_0\in X$, $y_0\in Y$ and we consider the Wedge Sum (the quotient of the union by the relation $x_0\sim y_0$). I want to proof that $\pi \circ ...
0
votes
0answers
33 views

Showing the sphere is not homeomorphic to a torus (my own question!) (or indeed a circle and a washer) - OR puncturing is not continuous

Motivation imagine a rubber sheet extended over the end of a tube, I am saying: "there is no continuous transformation that can retract this sheet over the side" - it is common place to talk about ...
0
votes
1answer
39 views

Are Knots closed?

In every definition I see, a (classical) knot is an embedding of $S^1$ in $S^3$ or $\mathbb{R}^3$. But my lecturer said that the complement of a knot in $S^3$ is open, hence the knot is closed. But ...
2
votes
1answer
36 views

Understanding the Homotopy Invariance of Fiber Bundle

I'm trying to understand the proof of Theorem 2.1 in "The Topology of Fiber Bundles" found online at http://math.stanford.edu/~ralph/fiber.pdf. What I don't understand is how do we actually define ...
3
votes
1answer
34 views

isomorphism of pointed sets

What is an isomorphism in the category of pointed sets? Is it just an exact sequence $$ 1 \to A \to B \to 1 ?$$ (Note: even though the kernel of the middle map is zero, $A$ might not inject into $B$.) ...
2
votes
1answer
46 views

Embed Torus into Klein Bottle

Is there a continuous map of the torus into the Klein bottle? Can one do this so that it is locally a homeomorphism (or a complete embedding)? My idea is to take the square $[-1,2] \times [-1,1]$ and ...
0
votes
0answers
59 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
votes
1answer
64 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 ...
0
votes
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
votes
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 ...
0
votes
1answer
39 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
votes
2answers
53 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 × ${$1$}$)$, ...
1
vote
1answer
32 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 $ΣX$ ...
0
votes
1answer
45 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
votes
3answers
77 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
vote
1answer
74 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?
4
votes
0answers
67 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
vote
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
vote
3answers
69 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 $$\mathbb{R}P^1=S^1/\{x=-x\}$$ ...
0
votes
1answer
18 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
votes
0answers
30 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
votes
1answer
37 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
votes
0answers
31 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
votes
0answers
110 views
+50

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
57 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 $\langle a,b,c\mid a^2b^2c^2\rangle$?
0
votes
0answers
24 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 ...
1
vote
0answers
25 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
votes
1answer
64 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 ...
0
votes
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 ...