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
0answers
44 views

Reduced homology and colimits

I would like to prove that colimits commute with reduced homology, i.e that if $ X = \operatorname{colim}\limits_{n \in \Bbb{N}} X_n$, then $$ \displaystyle\tilde H_k(X) = ...
0
votes
0answers
14 views

Does the singular cohomology theory agree with Alexander-Spanier's for compact metric spaces?

From http://en.wikipedia.org/wiki/Alexander%E2%80%93Spanier_cohomology, we know that the Alexander–Spanier cohomology groups coincide with Cech's for compact metric spaces, and coincide with singular ...
0
votes
1answer
21 views

a generalization of punctured cylinder

Let $S^1\times \mathbb{R}$ be the infinite cylinder. Pucture it, we have $S^1\times \mathbb{R}-*$. Then $(S^1\times \mathbb{R}-*)\simeq Skeleton^1(S^1\times \mathbb{R})\simeq S^1\vee S^1$. How ...
0
votes
1answer
46 views

Punctured $3$-Space Fundamental Group Calculation [closed]

How do I calculate the fundamental group of $\mathbb R^3\setminus \{(0,0,0)\}$, at base point $(1,0,0)$?
-1
votes
1answer
52 views

Mayer-Vietoris sequence [closed]

How do I compute the homology of the space obtained by taking three copies of $D^n$ and identifying their boundaries with each other?
0
votes
0answers
15 views

Mapping cylinder of punctured plane reflection

Suppose I define the mapping cylinder of a reflection about the $x$-axis for a punctured plane missing $(x, y)$ and $(x, -y)$. Obviously this quotient map would be homeomorphic to a one hole ...
0
votes
0answers
14 views

Explanation of CW Complexes

We recently studied about CW complexes in algebraic topology class, and I find it hard to understand how can I think of one. For example, can you please tell me how to find the CW complex of a torus? ...
0
votes
1answer
57 views

Which of these are homotopy equivalent? $S^1, \mathbb{R}, \{*\}$

Which of these spaces are homotopy equivalent: $S^1, \mathbb{R}, \{*\}$? I found a homotopy equivalence between $\mathbb{R}$ and the one point space $\{*\}$, so they are homotopy equivalent. The ...
1
vote
2answers
43 views

Proving that a continuous map is homotopic to the constant map

How can I prove that a continuous map $f : \mathbb{R}P^2 \to S^1$ is homotopic to the constant map? I know that in the projective space every point is a line but I do not get why the above has to be ...
0
votes
0answers
17 views

cohomology ring of punctured spaces [closed]

What is the cohomology ring (algebra) of $$ H^*(S^{1}\vee S^1;\mathbb{Z})? $$ $$ H^*(\vee_k S^n;\mathbb{Z})? $$ The punctured torus $$ H^*(\prod_n S^1-*;\mathbb{Z})? $$ and $$ H^*(\prod_n ...
1
vote
1answer
62 views

Quotient space of $S^n$ and the projective plane

The quotient space on $S^n \times I$ obtained from equating $(x, 0) \sim (-x, 1)$ seems like it might have the same fundamental group as the projective plane, but I'm not entirely sure how to prove ...
2
votes
2answers
67 views

The singular homology and cohomology of manifolds vanishes in high dimensions

Let $M$ be an $n$-manifold. It seems that there are two results that the $p$-th singular homology and cohomology of $M$ are zero if $p>n$. But I can not find them in my books of algebraic ...
8
votes
2answers
197 views

What's the difference between cohomology theories of varieties and topological spaces

There is defined several cohomology theories for algebraic varieties, but in the situation is very different for topological spaces (up to homotopy) for which there is only one cohomology theory for ...
1
vote
1answer
42 views

cohomology ring of a quotient space

what is the cohomology ring $$ H^*((S^3\times S^3\setminus \{e\})/(a,b)\sim (ab,b^{-1});\mathbb{Z}_2)? $$ Here the unit $e$ and the product $ab$ is of the Lie group $S^3=Sp(1)$.
1
vote
1answer
28 views

cohomology ring of some spaces [closed]

What is the cohomology ring $$ H^*(\mathbb{R}^2\times S^1\setminus (0,1);\mathbb{Z})? $$ $$ H^*(\mathbb{R}^2\times S^3\setminus (0,e);\mathbb{Z})? $$ Here $0=(0,0)\in \mathbb{R}^2$, $1=\exp(i0)\in ...
1
vote
1answer
69 views

Reference request: Zero set of global section

Things are in the complex algebraic setting. Assume that a vector bundle $V$ of rank $n$ over a $\mathbb{P}^n$ has a global section $\sigma$. Is it true that the zero set of $\sigma$ is a ...
1
vote
1answer
48 views

Mapping torus with homotopic homeomorphisms

Suppose I define the mapping torus $M_f$ in the usual way by identifying $(x, 0)$ and $(f(x), 1)$. If I have a homeomorphism $f: X \rightarrow X$ and another homeomorphism $f': X \rightarrow X$ that ...
5
votes
2answers
70 views

Why are contact structures studied from a cohomological, rather than homological, perspective?

As far as I know, contact structures are studied from a "cohomology/dual" perspective, meaning mostly from the perspective of contact forms and their respective kernels, instead of from a ...
0
votes
0answers
26 views

Quotient of homology groups

I know that for good pairs $(X,A)$ there is an isomorphism on the reduced homology: $$ \bar{H}_*(X/A) \approx H_*(X,A) $$ I am wondering if taking the quotient of homologies ever makes sense and ...
0
votes
1answer
14 views

If the reduced homology of K is nonzero, k is evasive.

Where can I find a proof for this theorem of Kahn, Saks and Sturtevant? K is a simplicial complex. Theorem $\textbf{10.1}.$ If $\tilde H_*(K)\neq 0$, where $\tilde H_*(K)$ denotes the reduced ...
0
votes
0answers
19 views

CW approximation

I was reading several proofs of the CW approximation theorem. If $X$ is a space then the idea is to make $n$-equivalences $f_n:K_n \to X$ where $K_n$ is a $n$-dimensional CW-complex. This goes by ...
2
votes
1answer
38 views

Direct Sum of Homology Groups and Connected Sums - Something's gone wrong.

I know that for any surfaces, $K, L$ forming $K$#$L$ (where # denotes the connected sum) is done by deleting a disk from each and gluing together the boundary. I also know a few facts about the ...
1
vote
0answers
43 views
+50

Convention of a continued fraction presentation of a lens space

I want to clarify the following two conventions on a surgery description of a lens space. Let $p$ and $q$ are relatively prime integers. Express $$ ...
2
votes
0answers
31 views

Definition of CW complex

In Hatcher, a CW complex is defined by inductively attaching cells, where we begin with $X^0$, a discrete space and then attach $1$-cells etc. We then get spaces $X^0,X^1,\cdots$ where ...
1
vote
1answer
20 views

Lifting properties of Serre fibrations

Suppose that $p:X\rightarrow B$ is a Serre fibration. I want to prove that $p$ has the right lifting property with respect to all maps of the form: $$S^{n-1}\times ...
1
vote
0answers
42 views

De Rham Cohomology of the complement of an ellipse

I'll call $A$ the complement of an ellipse in $\mathbb{R}^2$. I know that $H^0(A)=\mathbb{R}^2$ and $H^k(A)=0$ if $k \ge 3$. What about $k=1,2$? I know that $A$ has two connected components, so ...
1
vote
1answer
37 views

Degree of an antipodal map

Let $f:S^n\to S^n$ a continuous map, $n>0$; we consider the induced homomorphism $f_* : H_n(S^n)\to H_n(S^n)$, and, recalling $H_n(S^n)\simeq\mathbb Z$, define $deg(f)\doteq f_*(1)$. I'm asked to ...
1
vote
0answers
28 views

cohomology of labelled configuration space & relation with braid space [on hold]

Let: $M$ be a manifold (if we want, we can let $M=S^2$ , $S^1\times \mathbb{R}$, etc.); $(X,*)$ be a pointed topological space. $F(M,k)$ be the ordered configuration space of $k$-tuples on $M$; ...
3
votes
1answer
41 views

Prove that there is no entire function such that for every $w$ in the complex plane $f^{-1}(w)$ consists of exactly 2 points.

So I came across the following problem. Prove that there is no entire function such that for every $w$ in the complex plane $f^{-1}(w)$ consists of exactly 2 points. So here is what I was thinking. ...
1
vote
1answer
48 views

Adjoining an $(n+1)$-cell is an $n$-equivalence

Suppose $X$ is a topological space and $x_0 \in X$. Let $$ X' = X \cup e^{n+1} $$ be obtained from adding a $(n+1)$-cell (so $X'$ is the pushout of the map $\partial e^{n+1} \to e^{n+1}$ and the ...
0
votes
1answer
104 views

Fundamental group of complement of circle with finite number of lines through origin

In $\mathbb{R}^3$, consider the space missing a circle and a finite number of distinct lines passing through the center of the circle. In the case of one line, one can show that it deform retracts to ...
3
votes
0answers
58 views

Calculating the first homology group

Suppose all vertices on a polygon are identified and the polygon is $abcb^{-1}a^{-1}c$. Is it enough to simply switch to additive notation, get $2c$ and realize that $H_1(X) = \mathbb{Z}_2 * ...
2
votes
0answers
22 views

Short exact sequence for topological join: split needed

I am desperately trying to solve the following problem: Let $X$ and $Y$ be topological spaces and $X * Y$ their join. Prove that there is a short exact sequence $$ 0 \to \tilde{H}_k(X * Y) \to ...
4
votes
0answers
93 views

A space that deformation retracts into the cylinder and Möbius band doesn't embed in $\Bbb R^3$.

Consider the Möbius band, and take the middle circle in it (so that it deformation retracts onto it). Glue the upper boundary of a cylinder through it. This gives a space that deformation retracts ...
2
votes
1answer
52 views

Why is Hawaiian earring not semilocally simply connected?

Let $H$ denote the Hawaiian earring: We defined a space $X$ to be semilocally simply connected if every point in $X$ has a nbhd. $U$ for which the homomorphism from the fundamental group of $U$ to ...
0
votes
0answers
21 views

Fibration with CW-complex as basespace admits retraction

Suppose $f:E\rightarrow B$ has the right lifting property with respect to all CW-pairs $(X,A)$. Then $f$ is a Serre fibration and also a weak-homotopy equivalence. But want i want to prove is the ...
4
votes
2answers
74 views

Homotopy of maps $(D^n, S^{n-1}) \longrightarrow (X,A)$ relative $S^{n-1}$

Suppose that for a map $f: (D^n, S^{n-1}) \to (X,A)$ (where $(X,A)$ is an arbitrary pair of spaces) there exists a homotopy $H: D^n \times I \to X$ with $H(\_,0)=f$, $H(s,t) \in A$ for $s \in S^{n-1}$ ...
1
vote
0answers
30 views

Retraction and intersection

Let $X$ be a topological space, and consider two open subsets $U$, $V$ of $X$ such that there exist two continuous maps $r_{U}: X\longrightarrow U$, $r_{V}:X\longrightarrow V$ which are homotopically ...
2
votes
0answers
26 views

cohomology of orbit space by a free group action

Let $G$ be a group. Let a principal $G$-bundle $G\to E\to B$. Then we have a fiber sequence $G\to E\to B\to BG$. Let $k$ be a field. Suppose $H^*(BG;k)$ and $H^*(E,k)$ are known. How to get ...
1
vote
0answers
32 views

cohomology of orbit space

Let $p$ be an odd prime. Let $T^p=S^1\times\cdots \times S^1$ be the $p$-dimensional torus. Then $$H^*(T^p;\mathbb{Z}_p)=\otimes_pH^*(S^1;\mathbb{Z}_p)=\otimes_p\Lambda_{\mathbb{Z}_p}[a].$$ Here ...
3
votes
0answers
38 views

Fundamental groups of the projective plane

I'm practicing for my topology final. Munkres' Topology, Theorem 74.4 states that: Let $X$ denote the $m$-fold projective plane, that is the space obtained by taking the connected sum of the ...
3
votes
1answer
31 views

Practice Problem Fundamental Group of 7-figured polygon

The question is from Munkres: Consider the space $X$ obtained from a seven-sided polygonal region by means of the labelling scheme $abaaab^{-1}a^{-1}$. Show that the fundamental group of $X$ is the ...
2
votes
0answers
43 views

Punctured plane

What does one point compactification of singly, doubly, triply punctured plane $\mathbb{R}^2$ look like? What would their fundamental groups look like? I'm trying to visualize but can't seem to draw ...
1
vote
2answers
58 views

Join of topological spaces; Mayer-Vietoris

let $X$ and $Y$ be topological spaces, $X\star Y:=\frac{X\times Y\times [0,1]}{\sim}$, where $\sim$ is genereted by $(x,y_1,1)\sim (x,y_2,1)$ and $(x_1,y,0)\sim (x_2,y,0)$ for $x, x_1,x_2\in X$, $y, ...
1
vote
0answers
31 views

Method for defining a number of connected components of real algebraic surface

The question is simple: given the concrete polynomial $f(x,y,z)$ ($x$,$\,$ $y$ and $z$ are real numbers), is there any method for answering this question for a surface $f(x,y,z) = 0$? I'm interested ...
1
vote
1answer
36 views

Contractible CW-complex

Let Z be a CW complex so that for all $n \in \mathbb{N}$ every continuous $f:S^n\rightarrow Z$ is homotopic to a constant map, where $S^n:=\{x \in \mathbb{R}^{n+1}$ | |x|=1}. Then there is a ...
2
votes
2answers
87 views

Computing $\pi_4(S^3)$ using Serre spectral sequence

I'm following Davis & Kirk's computation that $\pi_4(S^3)=\mathbb Z/2$ using the Serre spectral sequence but I'm having problems at the very end. We consider a homotopy fibration $X\to S^3 \to ...
1
vote
1answer
47 views

definition of a $\Delta$ - complex

I've been given this image from hatchers algebraic topology as an example of a $\Delta$ complex but with the explicit definition as follows A $\Delta$-complex structure on a space X is a collection ...
0
votes
1answer
24 views

Group of deck transformations cyclic

Given a pointed topological space $(X,x_0)$, let $p\colon (\tilde{X}, \tilde{x}_0)\to (X,x_0)$ be a covering of that space. Write $p^{-1}(x_0)= \{\tilde{x}_0, \tilde{x}_1,\ldots,\tilde{x}_n\}$. I'd ...
1
vote
1answer
41 views

Fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$. ...