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

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2
votes
1answer
48 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
1answer
45 views

For which $n$ can one infer from these assumptions that there is a $y \in S^n$ such that $f(y) = y$?

In this problem $S^n$ means $\{x \in \mathbb{R}^{n + 1} \ | \ \|x\| = 1\}$ and $f\colon S^n \rightarrow S^n$ is a continuous map that satisfies $f(x) = f(-x)$ for every $x \in S^n$. For which $n$ can ...
1
vote
1answer
138 views

Tangent bundle of P^n and Euler exact sequence

I'm thinking about the Euler exact sequence for complex projective space, and I'm a bit confused. In the topological category, one has $$ T\mathbb{P}^n \oplus \mathbb{C} \cong L^{n+1} $$ where $L$ ...
2
votes
1answer
97 views

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 ...
3
votes
0answers
43 views

Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
4
votes
0answers
25 views

Show that $M^n-N^{n-1}$ has exactly two components with $N^{n-1}$ as the topological boundary of each.

This is the problem 6.8.1. from "Topology and Geometry" by Glen E. Bredon. The problem is, If $M^n$ is a connected, orientable, and compact $n$-manifold with $H_1(M^n;\mathbb{Z}) = 0$ and if ...
1
vote
3answers
43 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 ...
3
votes
1answer
45 views

What to do when this theorem can't be applied: How to calculate $H_1$?

Consider the following theorem (Lee's book on topological manifolds, page 369): (Homology Effect of Attaching a Cell) Let $X$ be any topological space and let $Y$ be obtained from $X$ by attaching a ...
4
votes
1answer
59 views

Degree of a map $S^0 \to S^0$

By definition the degree of a map $f: S^n \to S^n$ is $\alpha \in \mathbb Z$ such that $f_\ast(z) = \alpha z$ for $f_{\ast}:H_n(S^n) \to H_n(S^n)$. What is the definition of the degree of $f: S^0 \to ...
1
vote
1answer
43 views

Degree of this attaching map — or how to define this attaching map?

Consider the cell complex consisting of two zero cells $e_0^1, e_0^2$ connected by two 1 cells $e_1^1,e_1^2$ with one 2 cell $e_2$ in the middle (Picture: Imagine $S^1$ with one $0$-cell at the north ...
2
votes
1answer
43 views

How do modern categories of spectra manage to avoid “cells now, maps later”?

In Adams' definition, a map $f: X \to Y$ of CW spectra consists of a cofinal subcomplex $X'\subseteq X$ and maps $f_n: X'_n \to Y_n$ that commute with the structure maps. This definition is reproduced ...
0
votes
0answers
24 views

Cellular homology $d_n$ definition by example

I'm trying to understand cellular homology. Consider the following diagram taken from these lecture notes: I'm trying to understand what the maps $d_n$ are. As I understand $H_n(X^n, X^{n-1})$ is ...
2
votes
2answers
317 views

K-fold covering

I'd like some help with this homework: Let $p: E\to B$ be a covering map; let $B$ connected. Show that if $p^{-1}(b_0)$ has $k$ elements for some $b_0 \in B$, then $p^{-1}(b)$ has $k$ elements for ...
2
votes
1answer
46 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
votes
1answer
44 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 ...
3
votes
0answers
284 views

Homology groups of a 2-sphere with q cross caps

Fraleigh(7ed) Section43 Exercise9 The below is the full solution. But I can't understand the red underlined parts. 1) Why the piece that contains the i-th crosscap have boundary $z_i-2a_i$? I ...
4
votes
2answers
45 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 ...
1
vote
2answers
36 views

2-dimesional cell complexes with fundamental group isomorphic to the following.

I have been asked to give examples of 2-dimensional cell complexes whose fundamental group isomorphic to the following $$ \Bbb Z_4 * \Bbb Z_5$$and $$\Bbb Z_4\times \Bbb Z5$$ I know in the first ...
11
votes
4answers
1k views

Wedge sum of circles and Hawaiian earring

The (countably infinite) wedge sum of circles is quotient of disjoint countable union of circles $\amalg S_i$, with points $x_i\in S_i$ identified to a single point, while the Hawaiian earring H is ...
0
votes
2answers
34 views

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

Want to form the mapping cone of a map $f: X\rightarrow Y$ in the homotopy category. I am hoping that some one can give easy examples to show that mapping cone $Y \cup_f CX$ does not dependent on f ...
2
votes
0answers
66 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 $X$, ...
2
votes
1answer
34 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
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
49 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
votes
1answer
26 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
358 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 ...
3
votes
0answers
39 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
49 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
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
vote
0answers
34 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 ...
6
votes
1answer
69 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
38 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
votes
0answers
38 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 ...
6
votes
2answers
39 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
votes
1answer
44 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
35 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
50 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
34 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
159 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 ...
28
votes
0answers
732 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
48 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 ...
2
votes
1answer
32 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
votes
0answers
55 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
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 ...
4
votes
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 ...