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

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0
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0answers
7 views

Covering map is proper $\iff$ it is finite-sheeted

Prove that a Covering map is proper if and only if it is finite-sheeted. First suppose the covering map $q:E\to X$ is proper, i.e. the preimage of any compact subset of $X$ is again compact. Let ...
0
votes
1answer
11 views

What does it mean when people say the co fiber $C_f$ of $f: X\rightarrow Y$ does not dependent on f functorially?

I am hoping that some one can give easy examples to show that co fiber $C_f$ of $f: X\rightarrow Y$ does not dependent on f functorially and comments on when one should be careful about this. Thanks! ...
0
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0answers
10 views

Degree of this attaching map

Consider the cell complex consisting of two zero cells connected by two 1 cells with one 2 cell in the middle (picture closed disk with two dots on boundary representing the zero cells). Let $f_1: ...
1
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2answers
19 views

Difference between cellular and simplicial homology

Can someone tell me if there is any difference between cellular and simplicial homology? It seems to me that when I calculate for example the homology groups of the torus it makes no conceptual ...
0
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0answers
20 views

Is this map homotopically trivial?

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 induced map $a(f):a(X^*)\to a(Y^*)$ is also ...
2
votes
0answers
30 views

Dual of path in a space.

Is there a notion dual to the notion of a path in a topological space? Given that a path in a space $X$ is a continuous function from the interval $[0, 1]$ to X, what would the dual of this notion be, ...
0
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1answer
17 views

Steps involved when showing this induced map on homology is welldefined

I am showing that $H_0(X,R)=R$ when $X$ is a path-connected topological space. Let the zero boundary map be $\partial_0 : C_0(X) \to R$, $c \mapsto 0$. Define a map $\varphi : C_0(X) \to R$ by ...
6
votes
1answer
349 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathbf{Top}$, a $k$-closed subset $Y\subset X$ is a set such that $u^{-1}(Y)$ is closed in ...
2
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0answers
20 views

Minimal resolutions of singularities in higher dimensions and their cobordism class.

For a given singular algebraic variety over $\mathbb{C}$, how many minimal resolutions can exist? More specifically, consider a projective quadratic cone $Q$ in $\mathbb{C}P^{4}$ (with homogeneous ...
2
votes
1answer
48 views

Homology of universal cover of $S^1 \vee S^1 \vee S^2$ is not the same as homology of $\Bbb R^2$

I want to show, that although the homology groups of $X :=S^1 \vee S^1 \vee S^2$ and the torus $T^2$ are isomorphic, the homology groups of their universal covers are not. Let $U_X$ be the universal ...
1
vote
2answers
108 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
26 views

$E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering

Let $q:E\to X$ be a covering map. Then $E$ is compact if and only if X is compact and $q$ is a finite-sheeted covering. My question is regarding the $"\implies"$ direction: If $E$ is compact, then ...
4
votes
2answers
35 views

Two-sheeted covering of the Klein bottle by the torus

Prove that there is a two-sheeted covering of the Klein bottle by the torus. OK, so we take the the polygonal representation of the torus and draw a line in the middle as follows: Then there are ...
6
votes
1answer
64 views

Proving the Cone is Contractible: Is my Proof correct?

Is my answer/proof correct? Please help me make my proof more rigorous and accurate. I need everything to be absolutely clear and rigorous. Thank you. Question: Let $X$ be a topological space. The ...
2
votes
1answer
37 views

If $X$ is Hausdorff, then so is $E$

Let $q:E \to X$ be a covering map. If $X$ is Hausdorff, then so is $E$. OK, suppose $X$ is Hausdorff and let $x,y \in E$ with $x\neq y$. Let $V$ denote the evenly covered neighbourhood for $q(x)$, ...
0
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0answers
34 views

A question regarding singular homology (The geometrical interpretation of cycles in singular homology)

From Hatcher's Algebraic topology : Cycles in singular homology are defined algebraically, but they can be given a somewhat more geometric interpretation in terms of maps from finite Δ-complexes. To ...
1
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0answers
30 views

Is there a characterization of contractible hypersurfaces in $\mathbb{C}^2$.

Let $V$ be an irreducible, algebraic hypersurface in $\mathbb{C}^2$ which is contractible as a topological space. I would like to know the algebraic characterization of such objects. For example, ...
0
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0answers
31 views

How do I show $G_0$ and $G_1$ are conjugate subgroups? Please improve my answer.

Is my solution below correct? Please read through it and tell me if it seems complete or to make sense. Question: Let $E$ be path-connected. Let $p : E → B$ be a covering map and $p_∗$ be the ...
2
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0answers
52 views

A more detailed, rigorous proof that a suspension space is not necessarily contractible

Is my answer/proof correct? Please help me make my proof more rigorous and detailed. I need everything to be absolutely clear. Question: Let $X$ be a topological space. The suspension of $X$, ...
6
votes
2answers
30 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
104 views

What is an “essential loop”?

I'm a bit confused. Is an essential loop in a topological space $X$ just a loop $\alpha$, which is not-contractible (i.e. $[\alpha] \neq 0$ in the fundamental group of $X$), or is there something more ...
3
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1answer
22 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
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2answers
33 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 ...
2
votes
4answers
38 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 ...
4
votes
3answers
33 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 ...
9
votes
0answers
156 views

Show $\mathbb{CP^2/CP^1}$ is not a retract of $\mathbb{CP^4/CP^1}$.

So, I have shown that the natural projection $\pi\colon \mathbb{CP^n}\rightarrow \mathbb{CP^n/CP^k}$ induces a monomorphism $\pi^*\colon H^*(\mathbb{CP^n/CP^k},\mathbb Z)\rightarrow ...
26
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0answers
706 views
+100

Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the ...
2
votes
1answer
47 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 ...
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 ...
2
votes
1answer
28 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 ...
1
vote
1answer
29 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 ...
5
votes
1answer
122 views

Proof for the existence of a second fixed point in Poincaré's last geometric theorem

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem: "An area-preserving transformation of an annulus that moves the boundary circles in opposite ...
4
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0answers
50 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
1answer
35 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
4
votes
0answers
74 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 ...
4
votes
1answer
64 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 ...
2
votes
1answer
68 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
0answers
81 views
+50

Is the inverse limit of simplicial maps between finite directed graphs also a graph?

I think I have an intuitive understanding of why the following statement might be true, but I am not sure how to go about proving it. It's also possible my intuitive understanding is wrong and the ...
11
votes
2answers
919 views

What does the Hodge conjecture mean?

I read from the Internet that according to the Hodge conjecture, a certain harmonic differential form in a projective, non-singular algebraic variety is a rational linear combination of the cohomology ...
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
0answers
75 views

Describe two non-homeomorphic connected irregular covering spaces of $S^1\vee S^1$

I am having trouble with the following qual problem. If someone could help me get started it would be great. Describe two non-homeomorphic connected irregular covering spaces of $S^1\vee S^1$, each ...
0
votes
2answers
73 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 ...
4
votes
1answer
81 views

What does having a basepoint buy us in algebraic topology?

This may be a vague quesion. I am confused between the basepointed case and non-basepointed case in algebraic topology. Is there any convenience in base pointed case? For example, it leads to the ...
6
votes
3answers
251 views

Computing this fundamental group

What is the fundamental group of $$X = \left\{\left(\sqrt{x^2+y^2}-2\right)^2 + z^2 = 1\right\}\cup \left\{(x,y,0)\;\; :\;\; x^2 + y^2 \leq 9\right\}\subset\mathbb R^3\,?$$ I would say that it is ...
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 ...
10
votes
0answers
124 views
+50

Definition of bordism - gluing manifolds with structure

In general, when you have a "cobordism category" (as defined in Stong's Notes on Cobordism Theory) you define two objects $M,N$ to be bordant when there exist $U,V$ such that $M \amalg \partial U ...
0
votes
1answer
43 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 ...
2
votes
1answer
38 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 ...
90
votes
2answers
3k views

Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
1
vote
1answer
26 views

Restricted join operation on the simplicial complex?

Let $A$ and $B$ be two simplicial complexes (or CW-complexes) containing a common subcomplex C. Assume that C is contractible in both A and B. Is it true that the space obtained gluing A and B over ...