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

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5
votes
1answer
116 views

If $i\colon A\to X$ is a cofibration then $1\times i\colon B\times A\to B\times X$ is a cofibration for any space $B$. Is that true?

In Algebraic Topology (Hatcher, pg 14) I find: A pair $\left(X,A\right)$ has the homotopy extension property if and only if $X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ is a retract of ...
2
votes
1answer
91 views

Classifying space for finite-dimensional torus

Note that $K({\bf Z},1)=S^1$ but $BS^1 = {\bf CP}^\infty$. For finite groups $H$, $G$, $$K(G\times H,1) = K(G,1)\times K(H,1)$$ Does it works for classifying spaces of continuous groups ? As far ...
4
votes
2answers
325 views

Local homeomorphisms which are not covering map?

I am trying to find examples of maps between topological space which are local homeomorphism but not covering maps. Especially, how twisted has to be such a counterexample : can it be a local ...
1
vote
0answers
31 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 $f:S^n \rightarrow S^n$ is a continuous map that satisfies $f(x) = f(-x)$ for every $x \in S^n$. For which $n$ can one ...
0
votes
1answer
83 views

Regarding 3-fold connected coverings of the $S^1 \vee \mathbb{R} P^2$

As in the question, I need to determine all of the 3-fold connected coverings of the wedge of the unit circle and the real projective plane. Here's what I think: I know that the fundamental group ...
0
votes
0answers
72 views

Clarification needed - the fundamental group of the circle

I am reading the proof of $\mathbb{Z}\approx\pi_1(S^1)$ from Hatcher and didn't understand the last paragraph in the picture (the homomorphism part): Isn't $\tau_m\tilde{\omega}_n:I\to \Bbb{R}$ ...
2
votes
0answers
54 views

Toric Varieties as quotient of Lie groups

let $X_\Sigma$ be the toric variety of a smooth and complete fan $\Sigma$. Then $X_\Sigma$ has no toric factors and can nicely be represented as the quotient $$X_\Sigma \simeq \left( \mathbb{C}^r - ...
2
votes
0answers
60 views

Hurewicz isomorphism in equivariant stable homotopy

Let $G$ be a finite group and let $X$ be a $G$-CW-complex. Denote by $\pi_{\ast}^G(X)$ the $G$-equivariant stable homotopy groups of $G$ and by $\mathrm{H}_{\ast}^G(X,A(-))$ the Bredon homology of $G$ ...
2
votes
1answer
71 views

Path space vs loop space

If we consider the path fibration over a topological space $X$ we have $$ \Omega(X;p,p) \hookrightarrow P(X) \to X .$$ Where I denote with $\Omega(X;p,p)$ the set of paths $\omega:[0,1] \to X$ such ...
0
votes
1answer
112 views

Can't understand proof of excision property

I was reading the proof here. I understood almost all the parts after Exercise 4, such as defining $bs_n ^X$ inductively, a chain homotopy $(R^X )$, and that we get a small chain $(bs_n ^X)^k $ by ...
2
votes
0answers
94 views

Torsion of fundamental group is abelian

On Riemannian manifold with Ricci curvature bounded below (For example, flat torus. It has ${\bf Z}^2$ as fundamental group, which is nilpotent (=almost abelian). In fact Ricci curvature bouned ...
4
votes
0answers
195 views

Homology of Compact Manifolds

Perhaps this is obvious and I am overlooking something, but why are the homology groups of compact manifolds finitely generated?
1
vote
0answers
40 views

Definition of HEP correct?

In Algebraic Topology (Allen Hatcher, pg 14) I read a definition of HEP: ...Suppose one is given a map $f_{0}:X\rightarrow Y$, and on a subspace $A\subset X$ one is also given a homotopy ...
0
votes
1answer
71 views

Prove that there exists a long exact sequence…

Let $f, g : X \to Y$ be two continuous maps. Consider the space $Z$ which is obtained from the disjoint union of $Y$ with $(X \times [0, 1])$ by identifying $(x, 0) \sim f (x), (x, 1) \sim g(x),$ for ...
3
votes
0answers
77 views

Applications of a theorem of Cartier and Gabriel

In a representation theory course I took we stated and proved the following Theorem due to Cartier and Gabriel: Theorem: Suppose $H$ is a cocommutative Hopf algebra over a field $k$ such that $ ...
2
votes
1answer
39 views

Pointed space mapping clarification and isomorphisms between different ordered Homotopy groups.

If I am given a pointed pair of spaces $(X,A,x_{0})$ and define $P(X;x_{0},A) \subset X^{I}$ as the subspace given by the paths $\alpha$ in $X$ such that $\alpha(0) = x_{0} \text{ and } \alpha(1) \in ...
4
votes
2answers
87 views

Derivation and meaning of a long exact sequence of a Homotopy Groups for pairs of spaces

I read that there are long exact sequences of homotopy groups for each pair of pointed spaces $(X,A,x_{0})$. Now I know that for an exact sequence that, as the example below denotes $f \text{ and } ...
0
votes
2answers
68 views

Surgery on $S^m$

On page 4 of the book "ALGEBRAIC AND GEOMETRIC SURGERY" by Andrew Ranicki, after the definition of surgery has written: Example View the $m$-sphere $S^m$ as $$S^m=\partial (D^{n+1} \times ...
3
votes
1answer
151 views

Topics of Group Theory Required to Understand Betti Numbers

I am studying Group Theory. I made sure I have a problem at hand, as part of the motivation for my study. I have chosen the problem as being able to understand as well as compute Betti Numbers where ...
3
votes
1answer
75 views

Prove that any subgroup of $F_5$ of index 3 is isomorphic to $F_{13}$

Let $F_n$ denote the free group on $n$ elements. Prove that any subgroup of $F_5$ of index 3 is isomorphic to $F_{13}$. I noted that the wedge product of 13 copies of $S^1$ is a 3 fold covering ...
3
votes
1answer
67 views

singular (co)homology over various fields of same characteristic

Is the following true: if $K$ and $F$ are fields with the same characteristic and $X$ is a topological space, then for any $n$ there holds $$\dim_K H_n(X;K) = \dim_F H_n(X;F)\text{ and }\dim_K ...
3
votes
1answer
322 views

Fundamental group of the complement of the solid torus

Let $T$ be a solid torus, how to calculate the fundamental group $\pi_1(\mathbb R^3- T)$? intuitively, I think it's a free group with one generator. So if it is so, how to obtain it, and what the ...
1
vote
1answer
93 views

The fundamental group of the unit disc with one point removed from its boundary

If $y\in \partial (\mathbb D^2)$, then how to find $\pi_1(\mathbb D^2-\{y\})$? I know that if $y$ was an interior point then the answer will be $\mathbb Z$. But why both cases would be similar ?
3
votes
1answer
72 views

Yoneda's lemma and $K$-theory.

The K-theoretic form of Bott periodicity is that $\tilde{K}(X)\cong \tilde{K}(\Sigma^2 X)$, i.e., $\langle X, BU\times \mathbb{Z} \rangle \cong \langle \Sigma^2 X, BU\times \mathbb{Z} \rangle$. The ...
5
votes
1answer
100 views

$K$-theoretical interpretation of Bott periodicity.

We consider complex Bott periodicity $\pi_{i-1}(U) \simeq \pi_{i+1}(U)$. Why we can say that the $K$-theoretical interpretation of this assertion is $\tilde{K}(X) \cong \tilde{K}(\Sigma^2(X))$? Where ...
3
votes
3answers
86 views

Definition of the fundamental group

Why are the elements of the fundamental group of a space equivalence classes? Why isn't the group defined to be the set of all possible loops at a base point with the product operation of paths? What ...
2
votes
1answer
232 views

Loop space suspension/adjunction

Let be $X,Y$ Hausdorff spaces. I denote with $\langle \cdot, \cdot \rangle$ the basepoint preserving homotopy classes of maps, with $\Sigma$ the reduced suspesion and with $\Omega$ the loop space. How ...
0
votes
1answer
72 views

Why is this homotopy an isotopy?

I am trying to follow the proof of Willem's quantitative deformation lemma and I get everything except the justification for the (iv) property which states: $\eta_t(u)$ is a homeomorphism for ...
5
votes
1answer
64 views

A step in computing the cohomology ring of $\mathbb{C}P^n$

On page 250 of Hatcher's Algebraic Topology, he uses a certain corollary to compute the cohomology ring of $\mathbb{C}P^n$. The relevant section is below for convenience: I understand the proof ...
5
votes
1answer
183 views

The loop space of the classifying space is the group: $\Omega(BG) \simeq G$

Why does delooping the classifying space of a topological group G return a space homotopy equivalent to G. In symbols, why $\Omega(BG) \cong G$, where $G$ is a topological group and $BG$ its ...
2
votes
1answer
56 views

Graphic interpretation of path fibration.

Let $S^2$ the unit sphere. We can consider the associated path fibration $$ \Omega(S^2) \rightarrow P(S^2) \rightarrow S^2 .$$ I have to explain path fibration so I think that it is useful to make a ...
3
votes
3answers
91 views

Is $\langle abab^{-1}\rangle$ a normal subgroup of $\langle a,b|\varnothing\rangle$?

Let $G=\langle a,b | \varnothing\rangle$ and let $H\leq G$ s.t $H=\langle abab^{-1}\rangle$. Is $H\triangleleft G$? I'm asking this question in order to understand the fundamental group of the Klein ...
3
votes
0answers
74 views

Is a complex polynomial a regular covering? What is its group of deck tranformations?

We know that a complex polynomial $P$ of degree $n$ is an $n$-sheeted covering from $$\{\mathbb{C} - P^{-1}\{\text{critical values of }P\}\} \to \{\mathbb{C} - \{\text{critical values of }P\}\}. $$ ...
2
votes
0answers
67 views

Loop space and $K$-theory

How can I proove without using Yoneda's lemma that $$ \Omega^2(BU \times \mathbb{Z}) \cong BU \times \mathbb{Z} ?$$ In particular how can I define a cellular map $$ f: \Omega^2(BU \times \mathbb{Z}) ...
2
votes
1answer
81 views

How to Define Product Orientations for Topological Manifolds

When working with smooth manifolds, $M^m$ and $N^n$, it is straightforward to see how orientations at points $p\in M$ and $q\in N$ (i.e. ordered bases for the tangent spaces) give rise to an ...
2
votes
1answer
87 views

Is a surjective map a quotient map when its kernel is given by the action of a finite group

Let $X, Y$ be topological spaces and let $f\colon X \to Y$ be a surjective and continuous map. Let $G$ be a group acting continuously on $X$: for every $g \in G$ the map induced from the group action ...
3
votes
2answers
72 views

$\mathbb{R}P^2$ and its lines

I have been solving some past exam questions and I came across the following question. Let $r$ and $s$ two distinct lines in $\mathbb{R}P^2$, and let $X$ the space obtained contracting $r \cup s$ to a ...
6
votes
2answers
326 views

Uniqueness of Preferred Framing of a Solid Torus in $S^3$

One way to state my question tersely is: For a homeomorphism $f : S^1 \times \mathbb{D}^2 \rightarrow S^1 \times \mathbb{D}^2$, does $f|_{S^1 \times S^1}$ determine the isotopy class of $f$? This is ...
5
votes
0answers
73 views

Leray spectral sequence for complexes

Let $f:X\rightarrow S$ be a morphism of schemes. Let $0\rightarrow C_1 \rightarrow C_2 \rightarrow C_3 \rightarrow 0$ be an exact sequence of Abelian sheaves on $X$. Is there a general procedure to ...
0
votes
1answer
62 views

Cannot understand while reading simplicial=singular homology

I was reading http://www.math.toronto.edu/mgualt/MAT1300/Week%2010-12%20Term%202.pdf , and I can't understand the last paragraph of pg 29, and the first paragraph of pg 30. It says that "To compute ...
5
votes
2answers
179 views

Does the hairy ball theorem follow from Borsuk-Ulam?

The proofs I have seen for the hairy ball theorem all use either degree of a map defined in e.g. by homology or direct computations using stereographic projections in order to use homotopy arguments ...
3
votes
1answer
54 views

$J(X)$ and exotic spheres.

I read that we can realize exotic sphere as the coker of $J$-homomorphism. SO we can consider the exotic sphere $S^7$ realized using an identificantion of $B^4 \times S^3$ (where I donote the four ...
8
votes
1answer
161 views

covering space of $2$-genus surface

I'm trying to build $2:1$ covering space for $2$- genus surface by $3$-genus surface. I can see that if I take a cut of $3$-genus surface in the middle (along the mid hole) I get $2$ surfaces each one ...
2
votes
0answers
83 views

Fundamental group of the quotient of $\partial I$ with a simply connected space

I was wondering under what conditions will the quotient of I at its endpoints $\partial I$ with a simply connected space will give $\pi_1 = \mathbb{Z}$. To be clear, the end points do not need to ...
-3
votes
1answer
104 views

Unit Interval is Simply Connected

Given the definition of simply connected space to be a topological space $X$ whose every connected covering is homeomorphic to $X$, i want to show that $[0,1]$ is simply connected.
3
votes
1answer
150 views

Homology of symplectic manifolds

Could you show me some example of compact symplectic 4-manifold $M$ with the torsion in $H_{2}(M;\mathbb{Z})$
5
votes
3answers
233 views

“Nice” application of the fundamental group

I'm looking for an example of a topological result which is easy to prove using the fundamental group, but hard or impossible elementarily. First I thought about something like ...
4
votes
2answers
159 views

definition of convergence of a spectral sequence

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 241, there is written: What do $E^\infty$ and $\lim_{r\to\infty}E^r_{p,q}$ mean? If a spectral sequence is not first-quadrant and ...
3
votes
1answer
139 views

“Cut-off” of the Adams exact couple in A. Hatcher's “Spectral Sequences in Algebraic Topology”

I have been reading Chapter 2. of A. Hatcher's "Spectral Sequences in Algebraic Topology", which is freely available at the author's website. I have trouble understanding the Adams exact couple, ...
3
votes
0answers
72 views

Help calculating the second homology group of $\mathbb{R}P^2 \times S^1$

I need help calculating the second homology group of $\mathbb{R}P^2 \times S^1$. I found all the other homology groups using the Mayer-Vietoris sequence. Any suggestions? I can't use Kunneth.