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

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17 views

Question about $\pi_0(X)$

I'm studying the fundamental group of a topological space and I've studied a proof checking that $\pi_1(X)$ is a group. I'm thinking if the space of path components $\pi_0(X)$ is a group or not. ...
2
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1answer
34 views

Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
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0answers
27 views

Definition of boundary in a topological invariant way

I'm reading through Aguilar & Prieto lecture notes "Fiber bundles" (available online by googling it, ...
4
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2answers
52 views

Null-homotopic covering space map

I'm stuck with the following question, which looks quite innocent. I'd like to show that if a covering space map $f:\tilde{X}\to X$ between cell complexes is null-homotopic, then the covering space ...
3
votes
1answer
35 views

Homotopy classes of maps from the projective plane to $S^1 \times S^3$

I have a past qual question here: characterize the space $[(\mathbb{RP}^2,x),(S^1 \times S^3,y)]$ of homotopy classes of maps from $(\mathbb{RP}^2,x)$ to $(S^1 \times S^3,y)$, where here $x \in ...
2
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2answers
25 views

intersection number of twocompact oriented manifolds

I have an oriented manifold M of n dimension and 2 oriented submanifolds, one of dimension k and the other of dimension n-k , I have to understand which is the intersection number of those manifolds. ...
0
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1answer
58 views

Homology of 3-sphere minus an embedding of $S^1 \times \mathbb{D}^2$

I'm having trouble with the following past qual question: Let $\phi \colon S^1 \times \mathbb{D}^2 \hookrightarrow S^3$ be an embedding, where $\mathbb{D}^2$ is the open unit disk in $\mathbb{R}^2$. ...
3
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1answer
35 views

Homotopy groups relating to toric varieties

It is known that the toric variety $X_\Sigma$ of a simplicial fan $\Sigma$ can be constructed as a quotient $$X_\Sigma = \bigl(\mathbb C^N \setminus V(B)\bigr)/G.$$ Here $N$ is the number of rays, ...
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0answers
33 views

Link complement simply-connected if codimension $\geq 3$

In Rolfsen, page 50 says that "an easy general position argument shows that a PL link $L^k$ in $S^n$ has simply-connected complement if $n - k > 3$," where $L^k$ is a $k$-dimensional link in $S^n$. ...
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0answers
60 views

Why Must the Degree of this Map be 0? [on hold]

Let $f:S^3 \rightarrow S^1\times S^1\times S^1$ be a continuous map. Show that it's degree must be $0$. (Just a hint would be good)
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2answers
67 views

Cohomology Ring of Klein Bottle over $\mathbb{Z}_2$

I am trying to show that the cohomology ring of the Klein bottle with $\mathbb{Z}_2$ coefficients is $H^*(K,\mathbb{Z}_2) \cong \mathbb{Z}_2[x,y]/(x^3,y^2, x^2y)$. What I know: ...
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1answer
21 views

Group action and set define via their quotient topology open/closed equivalence relations

If we have a topological space $X$ and a subset $A \subset X$, we can define $X \backslash A$. My question is: Is it true that this equivalence relation is closed iff $A$ is closed as a subset of $X$ ...
0
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1answer
29 views

Fundamental polygon

So, I have seen fundamental polygons quite a few times now and I was always wondering what they are actually good for. Let's take the sphere. It's fundamental polygon can be seen here image. Does ...
3
votes
2answers
70 views

How do we get a simplicial homology functor?

The $n$-th simplicial homology group $H_n(A)$ of an abstract simplicial complex $A$ depends on the choice of an orientation for $A$ (but for different orientations, the homology groups are ...
2
votes
2answers
76 views

Stiefel-Whitney classes of 3-manifolds are trivial

Is there a simple way how to show that Stiefel-Whitney classes of a compact closed 3-manifold $M$ are zero? This is exercise 11-D in Milnors Characteristic classes. The available tools in the ...
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0answers
21 views

$C_g \simeq SX$ and $C_h \simeq SY$ [on hold]

Hi need some help with this problem: Let $f : X \to Y$ . Then we can form the cofiber sequence $X \to Y \to C_f \to C_g \to C_h$ where $g: Y \to C_f$, $h: C_f \to C_g$, and $i: C_g \to C_h$. Show ...
2
votes
2answers
65 views

The meaning of $\omega$ = df?

I'm reading Algebraic Topology by William Fulton and I am a little lost about 1-forms, which I assume to be the number of differentials according to the number of dimensions (correct me if I'm wrong. ...
0
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2answers
55 views

Closed 4-manifolds have CW-complex?

It looks like for compact 4-manifolds this question is open: When is a compact topological 4-manifold a CW complex? How about if we just consider closed 4-manifolds, does that have an answer/make the ...
1
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0answers
41 views

Covering Space of $\mathbb{C}-\{a,b\}$ via Multivalued Function

Consider the multivalued complex function $f(z)= \sqrt{z-a}+\sqrt{z-b}$, where $a\neq b$, defined in the domain $U=\mathbb{C}-\{a,b\}$. The graph $W$ of $f$ defines a regular covering space $W ...
4
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0answers
34 views

Compute the induced map on $\mathbb{CP}^n$

Let $d>0$ and $f:\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$ be given by $f(z_0,...,z_n)=(z_0^d,...,z_n^d)$. Let $F:\mathbb{CP}^n \rightarrow \mathbb{CP}^n$ be the induced map by $f$. Compute ...
2
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1answer
31 views

The cone minus its apex deformation retracts onto its basis

Let $X$ be a topological space and $$C(X)=X\times [0,1]/X\times \{0\}$$ be the cone on $X$. Call $P$ the apex of the cone. I want to show that $C(X)-P$ deformation retracts onto $X\times \{1\}$. My ...
2
votes
2answers
70 views

Isomorphism of modules arising from algebraic topology

While studying for a course in algebraic topology, the following question popped out: Let $S,R$ be two commutative rings with unit, $A,B$ two $S$-modules, and assume that $R$ is also an ...
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0answers
22 views

AI / Machine Learning related to high/modern/front mathematics [on hold]

I major math and cs. and i'm interested in ai/machine learning/data mining. so i want to know what math subjects are used in frontier of these technology. especially, high mathematical tool, like ...
1
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0answers
40 views

Relative cohomology of a vector space module non-zero vectors

I am trying to explicitly compute the relative cohomology groups $H^m(\mathbb R^n,\mathbb R^n_0;\mathbb Z)$, where $\mathbb R^n_0$ is all the non-zero vectors in $\mathbb R^n$. I think that the answer ...
1
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0answers
41 views

Why is $(X\times EG)/G\to X/G$ a fibration if $G$ acts freely on $X$?

Suppose that $G$ acts freely on $X$, and let $EG$ be a contractible space on which $G$ acts freely. According to many references, the projection $(X\times EG)/G\to X/G$ is a fibration. However, I ...
3
votes
1answer
42 views

Extending cellular maps between aspherical complexes

In a paper I read, the author seemed to use a property similar to: Let $X, Y$ be two aspherical CW-complexes and $f : X^{(2)} \to Y^{(2)}$ be a cellular map between their 2-skeletons. Then $f$ ...
3
votes
2answers
72 views

vector bundles on $X \times S^2$ built via clutching functions

Let $X$ be a compact Hausdorff space and $E \to X$ a (complex) vector bundle over $X$. We can build a vector bundle on $X \times S^2$ by clutching functions, i.e an automorphism $f:E \times S^1 \to E ...
2
votes
4answers
285 views

Should I read about Manifolds or Algebraic Topology?

I really enjoy doing maths and it fills quite a lot of my spare time. I'm starting my first year in the university on october and I probably won't have that much time for independent reading once ...
0
votes
1answer
35 views

What are the deck transformations of this threefold cover of the figure 8?

Hatcher lists some examples of covers of a figure 8 (page 58). One of them corresponds with the group with two generators $a$ and $b$ and the relations $a^2, b^2, aba^-1, bab^-1$. I thought ...
2
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0answers
44 views

Role of the Thom space in the Pontryagin-Thom construction

I am trying to understand the Pontryagin-Thom theorem; especially how the Thom space comes into play. Just to bring everyone on the same page: I am specifically talking about the construction of an ...
3
votes
0answers
44 views

Surgery and Euler Characteristic

I am trying to find out how a $(p,n-p)$ surgery affects the Euler Characteristic of an orientable, $n-$ dimensional, compact manifold. Call the initial manifold $M$ and the post-op manifold $M'$. This ...
11
votes
3answers
155 views

Fundamental group of the product of 3-tori minus the diagonal

I have a past qual question here: let $T^3 = S^1 \times S^1 \times S^1$ be the 3-torus, and let $\Delta = \{ (x,x) \in T^3 \times T^3 \colon x \in T^3 \}$ be the diagonal subspace. Compute $\pi_1(T^3 ...
0
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2answers
37 views

Why is $H_k(X^n) = H_k(X)$, $k < n$, where $X^n$ is the $n$-skeleton of the CW-complex $X$?

Why is $H_k(X^n) = H_k(X)$, $k < n$, where $X^n$ is the $n$-skeleton of the CW-complex $X$? I am probably overlooking something trivial. I tried using the fact that $H_k(X^n,\emptyset)\cong ...
3
votes
1answer
38 views

Question about the proof of the universal coefficient theorem

When deriving the universal coefficient theorem, in class we proceeded as follows: We have the SES: $$0\to ...
1
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0answers
36 views

Homology group of the join

Prove that $$\tilde{H_n}(X*Y) \cong H_{n-1}(X \times Y/X \vee Y)$$ Firstly, I set $$A=X \times Y \times [0,1) / (x,y_0,0) \sim (x,y_1,0) $$ $$B=X \times Y \times (0,1] / (x_0,y,1) \sim (x_1,y,1) $$ ...
3
votes
1answer
50 views

Eilenberg-Maclane space $K(G\rtimes H, 1)$ for a semi-direct product.

We know that $K(G\times H, 1)=K(G,1)\times K(H,1)$. Do we know something like this for a semi-direct product, where $K(G,1)$ denotes the Eilenberg-Maclane space.
2
votes
2answers
50 views

Find a CW complex with prescribed homology groups

A past qual question asks to construct a CW-complex $X$ with $H_0(X) = \mathbb{Z}$, $H_5(X) = \mathbb{Z} \oplus \mathbb{Z}_6$, and $H_n(X) = 0$ for $n\not= 0, 5$. One can build a CW-complex $Y$ by ...
1
vote
1answer
34 views

Show this Map $f:M\rightarrow S^n$ is a Diffeomorphism

Let $M$ be a compact, connected smooth manifold of dimension $n>1$, and $f:M\rightarrow S^n$ a smooth map such that the differential $df_x:T_xM\rightarrow T_{f(x)}S^{n}$ has full rank $= n$ for all ...
2
votes
1answer
29 views

Homology groups of a simplicial complex

I have a question from a qualifying exam: let $X$ be the simplicial complex that consists of the 3-simplices $(v_1,v_2,v_3,v_4)$, $(v_3,v_4,v_5,v_6)$, $(v_1,v_2,v_5,v_6)$, where the $v_i$'s are all ...
2
votes
0answers
48 views

Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1]. Is there a known tight upper bound in the number of polytopes in ...
0
votes
1answer
24 views

Connected components of the complement of a closed geodesic on a hyperbolic surface.

Let $M$ be homeomorphic to a 2-sphere with a finite number $\geq 3$ of points removed. This implies that $M$ can be equipped with a complete, finite area hyperbolic metric. I imagine $M$ as an ideal ...
1
vote
1answer
30 views

Number of connected components of the complement of a closed curve.

Let $\gamma:[0,1] \rightarrow \mathbb{R}^2$ be a continuous, closed curve (i.e. $\gamma(0) = \gamma(1)$). My question is about the number of connected components of the complement $\mathbb{R}^2 ...
0
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1answer
56 views

A detail in the proof of Jordan's theorem

The (usual) proof with homology first inductively shows that complements of embedded disks are acyclic. In doing so, Mayer-Vietoris is applied, and this assumes that their complements in the sphere ...
3
votes
1answer
67 views

Algebraic question (In Hatcher's book, exercise: 1.1.16-c:)

In Hatcher's book, exercise: 1.1.16-c: Show that there are no retractions $r :X \rightarrow A$ in the following cases: (c) $X = S_1 × D_2$ and A the circle shown in the figure. Page 39 By ...
3
votes
1answer
28 views

Two definitions of framed manifolds/cobordism

One notion of framed cobordism is that $\Omega_{fr}^n(X)$ is a certain equivalence class of $n$-manifolds $M^n$ in $X$ with a trivialization of the normal bundle $N M^n\subseteq TX$. For compact ...
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2answers
63 views

How to call two subsets that can be deformed into each other?

Given a topological space $X$, is there a canonical name for the equivalence relation generated by the following relation on the subsets of $X$? $A \sim B :\Leftrightarrow \exists \text{ continuous } ...
6
votes
2answers
107 views

Is this Space Homotopy Equivalent to $S^2$

Let $X$ be the space $S^1$ with two $2$-cells attached via maps of relatively prime degrees. This space is simply connected and has the homology of $S^2$, but is it homotopy equivalent to $S^2$?
2
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0answers
33 views

Cellular structure on a manifold [duplicate]

Is it always possible to put a cell structure on a manifold? in other words is it possible to decompose a manifold as a CW complex? I know that by Morse theory we always have a handle decomposition of ...
5
votes
1answer
78 views

Tensor product of real line bundles is trivial as a map $\mathbb{R}P^\infty\to\mathbb{R}P^\infty\times\mathbb{R}P^\infty\to\mathbb{R}P^\infty$

The tensor product of a real line bundle with itself is trivial, as is easily seen by looking at the transition functions or checking the Stiefel-Whitney class. Real line bundles are classified by the ...
2
votes
1answer
56 views

Proof of an algebraic topological lemma

I have been given the following result without proof, so I would like to show it is true: Let $I=[0,1]$, then: $$H^\bullet(I,\partial I;R)\cong H^\bullet(I/\partial I,*;R)\cong ...