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

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3
votes
2answers
422 views

Hausdorff condition for CW complexes

Consider the definition of CW complex from wikipedia. It is assumed that the space is Hausdorff. Are there problems if we drop this assumption? What is an example of a space satisfying all the CW ...
3
votes
0answers
187 views

Local contractibility of CW complex

I was trying to understand the notion of CW complex from wikipedia. The very first non-example is: $$\{re^{2\pi i \theta} : 0 \leq r \leq 1, \theta \in \mathbb Q\} \subset \mathbb R^2$$ This is not ...
4
votes
1answer
152 views

Is the pre-image of a cellular map a CW complex?

In general, if we have a map between CW-complexes, $f:X\to Y$, and $f$ is cellular, then is is clear that $f^{-1}(Y)$ (the inverse image, $f$ is not invertible in general) is also a CW-complex? ...
10
votes
2answers
577 views

Homotopy pushouts and induced maps

Suppose we are in a proper closed model category and consider a commutative square $$ \begin{array}{rcl} A&\to& B\\ \downarrow&&\downarrow\\ C&\to&D \end{array} $$ in its ...
11
votes
5answers
2k 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 ...
1
vote
0answers
191 views

homework problem about the projective real space

Sorry for ask this problem, but I am very complicated with this problem :/ . My course it´s of topology, the teacher said that we only need the definition of the quotient topology and of $$ P_R^2 ...
1
vote
2answers
187 views

Why cellular maps induce maps of chain complexes?

If $X$ is a CW-complex and $f:X\rightarrow X$ a cellular map. Then why it induces a map of chain complexes $f_*:C_*(X)\rightarrow C_*(X)$. (Why it commutes with the differential?).
9
votes
1answer
1k views

Fundamental groups of Grassmann and Stiefel manifolds

Could someone provide details on how to compute fundamental groups of real and complex Grassmann and Stiefel manifolds?
0
votes
2answers
172 views

An action of a group on a covering space

We see $S_3$ as the quotient of the free group on two elements and the normal subgroup $R$ generated by $\langle\sigma^3,\tau^2,\sigma\tau\sigma\tau\rangle$ where $\sigma$ and $\tau$ are the ...
2
votes
1answer
532 views

CW complexes exercise from Hatcher

I am having some trouble with this. Any help would be very appreciated. Thanks. Exercise 24. Let $X$ and $Y$ be $CW$ complexes with $0$ cells $x_{0}$ and $y_{0}$. Show that the quotient spaces $X * Y ...
3
votes
1answer
193 views

Self homeomorphisms are homotopic?

Let $X$ be a path connected topological space. Is it always true that any two homeomorphisms $f$ and $g$ from $X$ to itself are homotopic? If not, is there a minimal condition on $X$ which guarantees ...
6
votes
1answer
319 views

Orientability implies separation of space?

If a hypersurface in a manifold separates the ambient space into two disconnected pieces, is the surface necessarily orientable? This seems to be true when one considers the Jordan Brouwer theorem ...
3
votes
2answers
346 views

Dimension of product space

Is there a product space $M \times N$ such that $\dim(M \times N) < \dim(M) + \dim(N)$? Here $\dim$ refers to the dimension of a manifold.
1
vote
1answer
74 views

Inverses in higher homotopies

So when defining a Cohomology theory of a spectrum you define the addition structure via the pinch map. I.E. to define addition on $[X,E_n]$ look at $f,g \in [\sum X, E_{n+1}]$ let $\iota: \sum X \to ...
3
votes
1answer
194 views

Non-vanishing of homology of loop spaces

One of the answers to this MO question implies that loop spaces of $S^n$ for $n>1$ have non-zero homology in arbitrarily high degree. Is there any simple (or, better yet, geometric) way to prove ...
2
votes
1answer
632 views

Cohomology Ring of Projective Space

I would like to see different ways of calculating the cohomology ring of $\mathbb{R}P^n$. I know there are several ways, for example, using Poincare Duality, Gysin sequence, etc... Sketch your ...
2
votes
1answer
372 views

The covering space of a bouquet of 2 circles corresponding to a normal subgroup

Consider $S_3$ with this presentation: $S_3=\left\langle\sigma,\tau:\sigma^2=1, \sigma\tau=\tau^{-1}\sigma\right\rangle$. Let F be the free group with two generators $s,t$ and $R$ the minimal normal ...
2
votes
1answer
491 views

Cohomology of classifying space of torus

I have come heard that the cohomology of the classifying space of a compact torus $T$ is equal to the symmetric algebra over the dual of its Liealgebra $t^*$, where elements of the $t^*$ are of degree ...
6
votes
1answer
545 views

Applications of algebra and/or topology to stochastic (or Markov) processes

Some time back I was reading a PDF about algebra or topology (or algebraic topology, I forget which) and found an extremely enlightening section about an application to stochastic processes. ...
3
votes
0answers
91 views

A theorem by Hopf on surfaces

I am reading a paper on Riemann surfaces and faced a problem about one of the refernces the author gave in the exlaination of one of the results. Here is a summary of what I am reading. Let $X_1$ ...
2
votes
0answers
77 views

The characteristic class of a fibration is a fibre homtopy invariant

Let $p:E \to B$ be a fibration with fibre $F$ (denote the fibration $\xi$). Let us assume that the fibre is $(n-1)$ connected. There is a fundamental class $\iota_F \in H^n(F;\pi_n(F))$. We can define ...
1
vote
1answer
159 views

Differences between finite and infinite CW complex categories

I am currently reading Dyer's Cohomology Theories and he in the very beginning makes the assumption to work in the category of finite CW complexes. This kills of many interesting objects ($K(G,n)$, ...
1
vote
0answers
87 views

Could you help me completing my argument? (on a problem about map of G-modules)

I was trying to answer by myself at this question. This is what I've done: Let $e^\alpha_n$ be an $n$-cell of $X$ (I use the notations in the previous question). We have the attaching map ...
0
votes
1answer
139 views

Relation between a map and its lifting into the covering space

I have the following question: Let $\mathbb{D}$ denote the unit disk. Let $f:X_1 \longrightarrow X_2$ be a continuous mapping between Riemann Surfaces. Let $ \pi_1 : \mathbb{D} \longrightarrow X_1$ , ...
1
vote
1answer
190 views

Distinguishing homeomorphic from equal in identification spaces

This may be more of a philosophical question. The question is thus: Let $X$ be a topological space, and $\mathcal{P_1}, \mathcal{P_2}$, two partitions of $X$. Now consider the identification spaces ...
1
vote
1answer
232 views

Exact meaning of homology [duplicate]

Possible Duplicate: Soft Question - Intuition of the meaning of homology groups I've been studying some homology recently and I know it supposedly counts $n$-dimensional holes. For example, ...
2
votes
1answer
138 views

Embedding a manifold in the disk

I don't understand a sentence made by Hirsch in his Differential Topology at page 175: If $k > n+1$ and $M^n = \partial W^{n+1}$, then an embedding $M^n \hookrightarrow S^{n+k}$ extends to a neat ...
5
votes
1answer
246 views

Groups Associated to Knots

Let $2<n$ and $(2,n)=1$. If $T$ is the standard torus obtained by identifying images of opposite edges of unit square properly, $p\colon \mathbb{R}^2\rightarrow T$ is the quotient map, then ...
2
votes
2answers
98 views

Generators of group $\pi_1(n\mathbb{R}P^2)$

Task: Prove that the group $\pi_1(n\mathbb{R}P^2)$ is generated by elements $a_1,\ldots, a_n,$ obeying unique relation $a^{2}_{1}\cdot\ldots\cdot a^{2}_{n}=1$. I know how to solve it, but I think ...
3
votes
1answer
401 views

Need for computation in pure Mathematics at the highest level?

I'm fourth year undergrad student and I've noticed the skills that I've built up to do computation isn't actually being used. A good example is algebraic topology, I've never really used calculus in ...
0
votes
1answer
134 views

Distinguishing two topological spaces

1) If $X $ is union of two unlinked circles and $Y $ is union of two linked circles, then $\pi_1(\mathbb{R}^3-X)=\mathbb{Z} * \mathbb{Z}$, and $\pi_1(\mathbb{R}^3-Y)=\mathbb{Z} \times \mathbb{Z}$ (See ...
6
votes
2answers
191 views

cohomology ring structure of conf($\mathbb{R}^m$, 3)

I am attempting to compute the (integral) cohomology ring structure of the 3 configuration space of $\mathbb{R}^m$ and have run into a few doubts. Using a result of Fadell and Neuwirth, we have that ...
16
votes
2answers
785 views

Introductory book for homotopical algebra

I'm interested in learning homotopical algebra (by which I mean: model categories, simplicial methods, etc.) However, I've been unable to make heads or tails of any of the "standards" ...
4
votes
1answer
297 views

Associative up to homotopy

I'm reading Adams' book Infinite Loop Spaces. He explains that the product map on a loop space isn't associative, but it is associative up to coherent homotopy. I'm confused about the coherent part of ...
3
votes
2answers
152 views

The fundamental group of $K_{3,3}$ — relationship between its generators and embedding into manifolds

So I've been reading this wonderful PDF textbook on algebraic topology: http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf In particular, I'm very interested in the chapter on graphs. This ...
2
votes
0answers
84 views

Induced map on cohomology for abelian variety

just a simple question: if I have an abelian variety $A$ (choose the base field) and consider the inversion morphism $i: A \rightarrow A$ on it, then this map induces a map on cohomology ...
3
votes
1answer
172 views

Compute the degree of a map on the sphere with a property of symmetry

Suppose a continuous map $f : S^2 \rightarrow S^2$ verifies this property : $$ \forall x \in S^2, f(x) - f(-x) = 2\left(f(x),x\right)x $$ An equivalent expression could be : $$f(x) = X(x) + u(x)x$$ ...
2
votes
0answers
131 views

Complex torus and integral of forms

Start with a complex torus $X$ of dimension $n$ and a basis $e_1,...,e_{2n}$ for its first integral cohomology. Let $f_1,...f_{2n}$ be the dual basis vectors, so they are in $H^{1}(X,\mathbb Z)^*$. ...
3
votes
1answer
289 views

Universal Covering Space

If $X$ is path connected, locally path connected and semi-locally simply connected topological space and $x_0\in X$, consider the set $\chi =\{(X_{\alpha},x_\alpha) \} $ of covering spaces of $X$ with ...
4
votes
3answers
1k views

(Explicitly) Constructing Deformation Retractions

I'm having trouble building the actual deformation retractions, although I understand the concepts behind them, homotopies, etc. For example, when constructing a deformation retraction for ...
5
votes
1answer
187 views

Algorithm for computing the rank of the fundamental group of a graph?

I've been learning a bit about applications of algebraic topology to graph theory and I'm interested in figuring out how to compute the fundamental group $\pi_1(X,x_0)$ of an arbitrary graph ...
5
votes
1answer
610 views

On Frobenius reciprocity theorem

The classical Frobenius reciprocity theorem asserts the following: If $W$ is a representation of $H$, and $U$ a representation of $G$, then $$(\chi_{Ind W},\chi_{U})_{G}=(\chi_{W},\chi_{Res ...
2
votes
1answer
503 views

The homology group of the projective space of dimension $2$

I am reading a book on homological algebra. In order to determine the homology group of the $2$-dimensional projective space, the author identifies the space with the southern hemisphere of $S^2$, ...
2
votes
0answers
763 views

Is $S^\infty$ contractible? Does it have CW structure?

How can I prove that $S^\infty$ is contractible? Is $S^\infty$ a CW complex?
3
votes
1answer
187 views

Supposedly “trivial” implication that injective surfaces are incompressible

My question is about a passage in Algorithmic Topology and Classification of 3-Manifolds by Sergei Matveev. Let $F$ be a surface in some $3$-manifold $M$. $F$ is called incompressible if for every ...
2
votes
1answer
231 views

Associated bundle

Given a principal $G$-Bundle $P\rightarrow X$ and if we let $G$ act on itself by multiplication (denote this action by $\rho$) we obtain an associated bundle $P\times_{\rho} G=(P\times G)/\sim$ where ...
7
votes
1answer
539 views

The Atiyah Hirzebruch Spectral Sequence

I am learning about the AHSS (for complex K-theory) by trying to compute the K-theory of some spaces. I have heard that the AHSS is functorial (maps of spaces induce maps of spectral sequences). Is ...
1
vote
1answer
668 views

$X$ a CW complex is contractible if it's the union of an increasing sequence

I'm doing exercise 11 on page 358 in Hatcher: Show that a CW complex is contractible if it is the union of an increasing sequence of subcomplexes $X_1 \subset X_2 \subset \dots $ such that each ...
4
votes
1answer
348 views

$X$ $n$-connected $\iff $ any continuous map $f:K \rightarrow X$ is null-homotopic

I want to show the following: $X$ $n$-connected $\iff $ any continuous map $f:K \rightarrow X$ where $K$ is a cell complex of dimension $\leq n$ is homotopic to a constant map For this I think I can ...
2
votes
1answer
304 views

Proof: The complement of an annulus embedded in a sphere has two connected components

By the Jordan Curve Theorem we know that the complement of an $S^{n-1}$ embedded into the $S^n$ has exactly two connected components. What if -- instead of a sphere -- we embed an annulus, i.e. ...