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

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4
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1answer
227 views

Evaluation and Coevaluation maps of a TQFT

In Lurie's "On the Classification of Topological Field Theories," he states in Proposition 1.1.8 that for an oriented compact manifold $M$ and a TQFT $Z:\mathrm{Cob}(n)\to \mathrm{Vect}_k$, there is a ...
0
votes
1answer
339 views

Reduced suspension and mapping cone.

Reading about Steenrod squares and a result regarding the Hopf invariant the following homeomorphism is used in a proof without being proved $$\Sigma^k(C_f) = C_{\Sigma^kf}.$$ Here $\Sigma$ is the ...
3
votes
2answers
208 views

What restricts the number of cohomologies?

Do different cohomology theories essentially just exist because there are distinguished homology theories associated with them? If yes, is it known if there is always a relation like the Poincaré ...
9
votes
3answers
362 views

Turning cobordism into a cohomology theory

I've recently finished one semester in differential topology (with Milnor's Topology from the Differentiable Viewpoint) and my first semester of algebraic topology. I believe I understand Milnor's ...
0
votes
1answer
145 views

In search of proof that $\widetilde{e^{ix}}:\mathbb{R}\to S^1$ is not epic in $\mathbf{hTop}$

I came across this assertion: There is an epimorphism $X \overset{f}\to Y\;$ in Top such that the homotopy class $X \overset{\tilde{f}}\to Y\;$ of $f$ is not an epimorphism in hTop. Then, by ...
11
votes
2answers
727 views

Nails and strings and paintings

This question is based on the "Picture proof" challenges from Rankk.org... IDEA: You want to hold up a painting using nails on a wall and string. The string is attached to the left and right sides of ...
0
votes
2answers
135 views

integral cohomology groups of an aspherical manifold are isomorphic to the integral cohomology groups of it's fundemental group

(of corresponding dimensions). how can I prove this? I think my main stumbling block is my general ignorance of group cohomology.
4
votes
2answers
221 views

Intuition for induced bundle.

If $X$ is a manifold, $G$ a compact Lie group, $E$ a principal $G$-bundle over $X$ and $V$ be a vector space on which $G$ acts. Then one can form the vector bundle $E \times_{G} V$ over $X$. What is ...
2
votes
1answer
114 views

Opposite Orientation of Boundary in Bordisms

In Lurie's "On the Classification of Topological Field Theories" (and certainly other places) he defines the category $\mathbf{Cob}(n)$ who objects are oriented $(n-1)$ manifolds. Given ...
5
votes
2answers
672 views

Relative homology groups of the torus

I have the following question to problem 2.1.17 in Allen Hatcher's "Algebraic Topology". So far I came up with the following exact sequences (for A and B): $$ \begin{aligned} 0&\rightarrow ...
18
votes
5answers
818 views

Covering spaces - why are they useful?

As someone who trained as a physicist, I have known for ages that $\operatorname{SU}(2)$ is a double cover of $\operatorname{SO}(3)$, so, during an idle day at the office I decided to look up what ...
2
votes
2answers
163 views

Dimension of a subset

For a closed subset $Y$ of a space $X$ we have the following inequlity of topological (covering) dimensions: $$\dim{Y} \leq \dim{X}$$ (assuming at least one of those is finite). I have two questions ...
7
votes
1answer
149 views

Why can all surfaces with boundary be realized in $\mathbb{R}^3$?

I'm having trouble comprehending an informal proof of the fact that all compact surfaces with boundary can be realized in $\mathbb{R}^3$. I'm trying to find a proof of it on the internet, but I can't ...
3
votes
1answer
244 views

Knot with genus $1$ and trivial Alexander polynomial?

I would like to know whether there exists a knot $K$ with genus $g(K)=1$ and trivial Alexander polynomial $\Delta(K) \doteq 1$. A linked question could be: does there exist a Whitehead double with ...
8
votes
3answers
414 views

What is combinatorial homotopy theory?

Edit: After a discussion with t.b. we agreed that this question aims to a different answer from this one, for more information you can read the comment below. Many times I've heard people ...
0
votes
2answers
861 views

What is a good Algebraic topology reference text? [duplicate]

Possible Duplicate: Learning Roadmap for Algebraic Topology The title of the question already says it all but I would like to add that I would really like the book to be about more ...
5
votes
3answers
203 views

Covering map on the unit disk

Let $f: D^2 \rightarrow X$ be a covering map. I am trying to show that $f$ must in fact be a homeomorphism. To do so, I believe it suffices to show that $f$ is injective. Moreover, if only one point ...
8
votes
2answers
207 views

Why is $\pi_* MU$ concentrated in even degrees?

$\pi_* MU$, which is the cobordism ring of manifolds with a complex structure on the stable normal bundle, is a polynomial ring $\mathbb{Z}[x_2, x_4, \dots]$. I'm probably being silly here, but is ...
3
votes
0answers
77 views

Construction of the shift map

Is there a standard way to construct the shift map on an infinite product or coproduct of a direct or inverse system of spectra that induces the standard shift map of abelian groups in homology? Is it ...
4
votes
2answers
163 views

Why is the pullback completely determined by $d f^\ast = f^\ast d$ in de Rham cohomology?

Fix a smooth map $f : \mathbb{R}^m \rightarrow \mathbb{R}^n$. Clearly this induces a pullback $f^\ast : C^\infty(\mathbb{R}^n) \rightarrow C^\infty(\mathbb{R}^m)$. Since $C^\infty(\mathbb{R}^n) = ...
3
votes
0answers
111 views

Homology of subsets of $\mathbb R^n$

Let $E \subset \mathbb R^n$. Must the homology groups $H_k (E)$ be trivial for $k \geq n$? How about just for $k > n$? If not, whats an example? Thanks.
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vote
0answers
78 views

Fundamental groups and covering spaces [duplicate]

Possible Duplicate: Application of Seifert-van Kampen Theorem Let $X$ be as in the following figure. What is the first fundamental group, $\Pi_1 (X)$, of $X$? What are the covering spaces ...
4
votes
2answers
196 views

Algebraic Structure of the Rose with Two Petals

I am trying to determine whether the rose with two petals ($S^1 \vee S^1$ or the figure-eight) has a continuous multiplication with identity element. I know that this is true for the unit circle ...
2
votes
2answers
509 views

Conceptualizing Inclusion Map from Figure Eight to Torus

I'm having some difficulty getting an understanding of this issue: I have an inclusion map $i : S^1 \vee S^1 \hookrightarrow S^1 \times S^1$. So this is an inclusion map from the figure eight to the ...
3
votes
1answer
352 views

A basic question on homology

Let $X$ be Hausdorff. Suppose further that it is triangulable. Let $K$ and $L$ be two simplicial complexes such that their underlying space $|K|=|L|=X$. It is a lot of work to show (see Chapter 2 of ...
7
votes
1answer
407 views

fundamental group intuition

So I understand the mechanics of the fundamental group, but I want to gain a more natural intuition behind it. I imagine the fundamental group $\pi_{1}(X)$ to detect "holes" in a space. For example, ...
2
votes
1answer
75 views

Map into knot complement inducing quasi-isomorphism

I'm looking at the following topology qualifying exam question: Let K be a knot in $S^3$. Construct a map $f: S^2 \vee S^1 \rightarrow (\mathbb{R}^3 - K)$ that induces an isomorphism of integral ...
3
votes
1answer
399 views

Homotopy equivalence with a point need not be a deformation retract

This question arises from problem 8 on pg. 366 of Munkres. Let $X$ be the union of the sets $(1/n) \times I$, $0 \times I$, and $I \times 0$, where $I = [0,1]$, with the topology it inherits from ...
6
votes
2answers
731 views

Application of Seifert-van Kampen Theorem

I am trying to wrap my head around the following problem: I have three objects lined up horizontally, a $2$-sphere, a circle, and another $2$-sphere. It is the wedge sum $S^2 \vee S^1 \vee S^2$. I ...
3
votes
2answers
300 views

The Euler characteristic & a cube with holes? [duplicate]

Possible Duplicate: Is the Euler characteristic $\chi =2$ for the prism with a hole? In attempting to lead a bunch of high school students to an understanding of the Gauss-Bonnet theorem, ...
4
votes
1answer
192 views

Visualizing homologous elements

For the fundamental group it's easy to visualize when two loops are homotopic. I was wondering if there are any ways to look at the equivalent problem for homology? I guess this might be tricky for ...
4
votes
2answers
246 views

Is there a initial “bordism-like” homology theory?

Let $X$ be a space and $x_0\in X$ a base point. The Hurewicz map $$\pi_k(X,x_0)\longrightarrow H_k(X)$$ factors through oriented bordism $$\pi_k(X,x_0)\longrightarrow MO_k(X)\longrightarrow H_k(X).$$ ...
4
votes
0answers
93 views

Why does applying $-\Box_{A//B} A$ to a free coresolution preserve exactness?

Let $A$ be a Hopf algebra over a field $k$, and let $B$ be a normal subHopf algebra of $A$. Suppose we have an $A$-free coresolution of $k$ over the form $F_n=K_n \otimes_k A$. Kochman claims that ...
5
votes
1answer
108 views

Constructing a coresolution

I am working through computing the homotopy of Thom spectra from Kochman's book. Let $A$ be a coalgebra over a field $k$, and let $M$ be a right $A$-comodule. Kochman constructs a coresolution $F$ ...
8
votes
3answers
502 views

Give an explicit embedding from $\mathbb{R}P_2$ to $\mathbb{R}^4$

I have heard that the least dimension $m$ required for $\mathbb{R}P_2$ to be embedded in the Euclidean space is 4, thus I wanted to find an explicit formulae for it. I found two possible strategies, ...
2
votes
3answers
396 views

Euler characteristic of a space minus a point

Let $X$ be a topological space and $*$ be the base point of $X$. How does $\chi(X-*)$ relate to $\chi(X)$ do we have $\chi(X-*)=\chi(X)-\chi(*)=\chi(X)-1$?
4
votes
2answers
123 views

Composed Covers

I have problems solving this seemingly straightforward question. Let $q : X \rightarrow Z$ be a covering space. Let $p : X \rightarrow Y$ be a covering space. Suppose there is a map $r : Y ...
3
votes
1answer
550 views

Null-homotopic Maps from $S^n$ to $S^1$ for $n \gt 1$.

I'm not sure how to answer this one. Is every continuous map $f:S^2 \to S^1$ null-homotopic? If $n > 1$, where $n$ is a natural number, is every continuous map $f:S^n \to S^1$ null-homotopic?
6
votes
1answer
247 views

What is the relation between a ''homotopy fiber bundle'' and a Serre fibration?

Call a continuous map $\pi:E\to B$ between CW complexes a homotopy fiber bundle if for any $x$ in the image of $\pi$, there is an open neighbourhood $U\subset B$ of $\pi(x)$ and homotopy equivalence ...
2
votes
2answers
599 views

Hatcher 2.2 exercise 10

Let $X$ be the quotient space of $S^2$ under the identifications $x\sim −x$ for $x$ in equator $S^1$. I want to compute the fundamental group and homology groups $H_i(X)$. I also want to repeat this ...
3
votes
1answer
214 views

Discrete Subgroup is Regular Covering Space

I'm lacking ideas how to attack the following problem: Given $G$ a topological group, and $H$ a discrete subgroup, I have to show that that $G \rightarrow G/H$ is a regular covering space. Could ...
3
votes
1answer
423 views

a few problems about fundamental groups

I was asked a few "challenge problems". Maybe it's not that hard, but i don't know how to solve them. 1) What's the fundamental group of $R^3 \setminus \{ \{z\text{-axis}\} \cup \{ x^2 + y^2 =1\}\}$? ...
1
vote
0answers
111 views

Retraction and the k-fold projective plane

Let $k > 2$. Is there an embedding of $S^1$ in $\#_k \mathbb{RP}^2$ such that $S^1$ is a retract of $\#_k \mathbb{RP}^2$? I know that this is not correct when $k=1$ (homotopy argument), and this ...
6
votes
1answer
154 views

Does a function space construction always decrease the connectivity of a space?

This question is kind of dual to this question, where I asked if smashing with a space $Y$ always increases (which means ''$\geq$'') the connectivity of a space $X$ and the answer was "yes". The ...
3
votes
1answer
216 views

fundamental group of the union of two subsets

Let $X$ be a topological space and $U,V \subset X$ two open subsets such that $U \cap V$ and $U \cup V$ are both simply connected. How can i show that $U$ and $V$ are simply connected? Thanks in ...
22
votes
4answers
3k views

Learning Roadmap for Algebraic Topology

I am now considering about studying algebraic topology. There are a lot of books about it, and I want to choose the most comprehensive book among them. I have a solid background in Abstract Algebra, ...
1
vote
1answer
509 views

Computing element of fundamental group of Möbius strip

How does one go about computing the element of the fundamental group of a Möbius strip represented by the loop $(\cos 10\pi t, \sin 10\pi t)$.
9
votes
1answer
191 views

Does smashing always increase the connectivity of a space?

Does smashing of a pointed CW complex $X$ with an arbitrary pointed CW complex $Y$ increase the connectivity? The connectivity of a pointed space $X$ is the maximal number $\operatorname{con}(X)$ ...
0
votes
1answer
125 views

Fundamental group of connected compact n-manifolds

Can somebody explain why the fundamental group of a connected compact n-manifold M is finitely generated? I know that this manifold is homotopic to a CW complex (and I guess connected, because M is ...
16
votes
2answers
412 views

Topology on the space of paths

Let $X$ be a topological space, and define a path as a continuous map $\gamma : [a,b] \rightarrow X$. Two paths $\gamma : [a,b] \rightarrow X$ and $\phi : [c,d] \rightarrow X$ are equivalent ($\gamma ...