Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.
1
vote
0answers
12 views
Relation amongst Chern classes for line bundles on $S^2\times S^2$
I am considering complex line bundles $L$ over $S^2\times S^2$, and I want one such that $2c_2(L)<c_1(L)^2$. Well $c_1$ gives an isomorphism between complex line bundles and $H^2(S^2\times ...
0
votes
0answers
19 views
Multiplication by an integer on a symmetric spectrum
The map $n : X \rightarrow X$ is used in the definition of the mod-n spectrum $X/n$. But what is this map? How does it look like?
I know how this works for the sphere spectrum (just use the map $z ...
7
votes
2answers
103 views
Ideas for a present to my topology teacher
Tomorrow is the my final lecture in my favorite course, algebraic topology. I want to give a present to my prof. as a keepsake, something along the lines of this
only something I can make due ...
0
votes
1answer
28 views
Defining the winding number for a general curve
In the Complex Analysis text by Ahlfors, he says that we can define the winding number $n(\gamma,a)$ for any continuous, closed curve $\gamma$ which doesn't pass through the point $a$ ...
0
votes
0answers
24 views
On wedge of H-spaces
I've read that the wedge of two cyclic maps, $f\vee g$, does not need to be cyclic. Well, I understood the counter-example (see below) except by the fact that $S^1\vee S^1$ is not an H-space.
Where ...
1
vote
0answers
20 views
Framed cobordism classes and homotopy spheres
This is another question that has been bothering me for days:
What information about a manifold does one gain from knowing whether or not its framed cobordism class contains a homotopy sphere?
My ...
1
vote
0answers
33 views
Homology groups of $\mathbb{R}^3$ relative to a disjoint union of two copies of $S^1$
Problem #5: Let $C_1$, $C_2$ be two copies of $S^1$ disjointly embedded in $\mathbb{R}^3$. Compute $H_i(\mathbb{R}^3,C_1\cup C_2)$ for all $i\in\mathbb{N}$.
If I am understanding this correctly, ...
0
votes
1answer
26 views
Consequence of injectivity of projections from covering spaces
We have the theorem which says that the induced homomorphism $p_* : \pi_1(\tilde X,\tilde x_0)\rightarrow \pi_1(X,x_0)$ is injective (hence a monomorphism). Here $\tilde X$ is a covering space of $X$.
...
9
votes
1answer
112 views
The simplest nontrivial (unstable) integral cohomology operation
By an integral cohomology operation I mean a natural transformation $H^i(X, \mathbb{Z}) \times H^j(X, \mathbb{Z}) \times ... \to H^k(X, \mathbb{Z})$, where we restrict $X$ to some nice category of ...
2
votes
0answers
43 views
$\operatorname{Hom}_{\mathbb{Z}_2}(S^n,\mathbb{R}) \simeq S^n \times_{\mathbb{Z}_2} \mathbb{R}$
Let $\operatorname{Hom}_{\mathbb{Z}_2}(S^n,\mathbb{R})$ the set of $\mathbb{Z}_2$-maps from $S^n$ to $\mathbb{R}$ and $S^n \times_{\mathbb{Z}_2} \mathbb{R}$ the fiber product of $S^n$ and ...
0
votes
0answers
64 views
Proof of that SO(3) is not simply connected.
I want to prove that $\pi_1(SO(3))\cong \mathbb{Z}/2\mathbb{Z}$. I have already proved that there exists a surjection $\mathbb{Z}/2\mathbb{Z}\rightarrow \pi_1(SO(3))$.So I want to show that ...
5
votes
1answer
70 views
+50
Relation between quadratic refinement and quadratic form
The question in the title has now been bothering me for days. I first came across the term quadratic refinement when I read about the Kervaire invariant when reading Kervaire's 1960 paper. The ...
2
votes
1answer
47 views
Fundamental Group of an Identification Diagram
I'm looking for help with the following question, specifically the Fundamental Group part. I won't complain if the Homology Groups are calculated as well, but I think I should be okay on this part: ...
4
votes
1answer
44 views
Why are the integers appearing in lens spaces coprime?
I have a past paper question for a first course in algebraic topology, which asks one to calculate the first three homology and homotopy groups for the space $L_n$, defined as follows:
Let ...
4
votes
2answers
72 views
Hatcher 3.3 Exercise 31
The following is a question from Hatcher's "Algebraic Topology":
Let $M$ is a compact $R$-orientable n manifold, then the boundary map $\partial : H_n(M,\partial M;R) \to H_{n-1} (\partial M)$
...
8
votes
1answer
85 views
Non-orientable 3-manifold has infinite fundamental group
I'm doing past papers for a first course in algebraic topology.
The question is:
Let $M$ be a 3-dimensional, closed, connected, non-orientable manifold. Show that $M$ has infinite fundamental ...
2
votes
1answer
26 views
Importance of triangulation
Kervaire's seminal 1960 paper A manifold which does not admit any Differentiable Structure starts "An example of a triangulable closed manifold $M_0$ of dimension 10 will be constructed."
What is the ...
2
votes
1answer
54 views
Confusion regarding various definitions in defining singular homology
In defining singular homology,
A singular $n$-simplex is a continuous mapping $\sigma_n$ from the
standard $n$-simplex $\Delta^n$ to a topological space $X$. Notationally,
one writes ...
0
votes
0answers
49 views
How does one show that definition of Betti number and its “informal definition” are equal?
When formally defining Betti number, we often use homology group - but I am not sure how we can use that definition to prove the informal definition of Betti number - that talks about "unconnected and ...
3
votes
1answer
47 views
trefoil knot and meridian/longitudinal cycles
I hope this is a simple question...
For the trefoil knot 3_1, whose knot group is given by a presentation of the fundamental group, $\pi_1(M) = \langle a,b: aba = bab \rangle$, where the meridian and ...
2
votes
1answer
45 views
Brouwer degree and homotopy invariance
Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ and let $f \in C^1(\overline{\Omega})$. Following Topological degree theory and applications, we define the degree of $f$ with respect to some ...
0
votes
3answers
84 views
Function to uniquely map a set of rectangles in space to a number?
I am trying to build a new way of indexing spatial data.
Is there a function that takes as parameter a number of rectangles in euclidean space, and outputs an unique number?Can such a function be ...
3
votes
1answer
35 views
A comparison between the fundamental groupoid and the fundamental group
Are there two path connected topological spaces $X,Y$ such that the fundamental groupoid of $X$ is not isomorphic to the fundamental groupoid of $Y$ but the fundamental group of $X$ is isomorphic to ...
1
vote
0answers
29 views
What is the closed orientable surface of genus 2?
I just have a very simple question. Could someone please explain to me what the closed orientable surface of genus 2 is? Thank you so very much
6
votes
2answers
81 views
Are there any simply connected parallelizable 4-manifolds?
On pp. 166 of Scorpan's "The Wild World of 4--manifolds", he gives an example of a parallelizable 4-manifold ($S^1\times S^3$) and then asserts: "there are no simply connected examples". It's ...
1
vote
0answers
14 views
Critical group and morse Lemma
I have this theorem with a part of it's prove
I have two questions:
1) what is the spectral decomposition of A ?
2)How to see that $B_{\varepsilon}\cap f_0 = \lbrace x\in H,||x||\leq \varepsilon ...
1
vote
0answers
30 views
Fiber product and $G$-invariant maps
Let $S^n \times_{\mathbb{Z}_2} \mathbb{R}$ be the fiber product of unitary sphere $S^n$ and $\mathbb{R}$ over $\mathbb{Z}_2$, where $\mathbb{Z}_2$ acts on $S^n$ by antipodal relation and on ...
3
votes
1answer
48 views
On the Hopf invariant
It was an important problem of topology to determine for which dimensions the Hopf invariant was one. There are several clear expositions giving the definition of the Hopf invariant including the ...
1
vote
0answers
19 views
Morse Theory and critical groups
Please i have a question ,
What is the relation between Morse theory and critical point theory ?
I studied the Morse inequalities and critical groups, but i can not not find or at least i do not ...
1
vote
1answer
69 views
Is there a compact contractible manifold?
Does there exist a compact connected manifold (without boundary), that has a trivial homotopy type?
3
votes
1answer
48 views
Surgery on manifold
In this article on surgery on manifolds it is explained that from an $n$-manifold $M$ an $n$-manifold $M'$ can be constructed by cutting out $S^p \times D^q$ and gluing in $D^{p+1}\times S^{q-1}$.
...
1
vote
0answers
34 views
Definition of cohomology with compact support
We can define a cohomology on open manifold:
Define a simplicial cochain group
$$ \Delta^i_c(X;G)$$ consisting of cochains that are compactly supported in the sense that they take nonzero values on ...
4
votes
2answers
74 views
Contractible spaces has trivial fundamental group.
I have to prove the following: Show that if $X$ is contractible (the def. I have is that $I:X\rightarrow X$ the identity function is homotopic to the constant function $p$ for some $p\in X$), then its ...
1
vote
1answer
40 views
Is the product of covering maps a covering map?
I have a question about covering maps. If $\phi_1: X_1 \rightarrow Y_1$ is a covering map, and $\phi_2: X_2 \rightarrow Y_2$ is a covering map, then is it true that $\phi_1 \times \phi_2: X_1 \times ...
3
votes
1answer
33 views
Universal covering space of connected open subset of $\mathbb R^n$
Is the universal covering of an open connected subset $U$ of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?
4
votes
2answers
105 views
'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)
In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms ...
4
votes
2answers
62 views
Two defintions of simply connected
I'm showing that the first definiton here implies the second (the other implication is obvious). My thoughts: Let $p,q$ be two paths in the space $X$. Then since $X$ is path connected there are two ...
2
votes
0answers
43 views
Quotients of infinite dimensional sphere
Recall $$S^\infty = \cup S^n,\ {\bf RP}^\infty = \cup {\bf RP}^n,\
{\bf CP}^\infty = \cup {\bf CP}^n$$
Hence $S^\infty / {\bf Z}_2 ={\bf RP}^\infty$. And I think that the following is possible : ...
1
vote
0answers
33 views
Geometric interpretation of $R$-orientations
For an $n$-dimensional manifold $M$ and a ring $R$ one can define $R$-orientability, e.g. by choosing local orientations, i.e. generators of $H_n(M, M - \{x\}; R)$ for every point $x$ in the manifold, ...
3
votes
1answer
46 views
Visualizing the group operation in higher homotopy groups
I'm having trouble picturing the homotopy group operation of concatenation between two pointed spaces.
For $n$-spheres, we have for $f,g: S^n \to X$
$$(f * g)(s_1,\ldots, s_n) = \begin{cases}
...
1
vote
0answers
35 views
Different point-set level definitions of spectra
I've been trying to understand the Adams spectral sequence and one of the more accessible sources is the (unfinished) book on spectral sequences by Hatcher. The usual (only?) construction of the ...
23
votes
3answers
379 views
How did we know to invent homological algebra?
Update: Qiaochu Yuan points out in the comments that the title of the question is misleading, as homological algebra did not begin with long exact sequences as I'd thought.
(Original question ...
2
votes
1answer
35 views
What would be an example of a nontrivial, i.e. multi-sheeted, covering of the torus?
What would be an example of a nontrivial, i.e. multi-sheeted, covering of the torus? I would greatly appreciate any help that you could give me.
2
votes
0answers
31 views
Product of two Kähler manifolds
Let $M$ and $N$ be Kähler manifolds. I know that the cartesian product $M\times N$ is a kahler manifold. I was wondering which form has the Kähler form on $M\times N$. Here's what I thought:
Let ...
3
votes
2answers
65 views
covering space of a particular CW complex
I am trying to find all connected covers of the following space $X$ (up to isomorphisms)
$X$ has one $0$-cell, two $1$-cells labeled $a$ and $b$, and three $2$-cells attached via $a^2$, $b^2$ and ...
1
vote
0answers
34 views
Intersection between cycles and dominant maps
i must say in advance that i'm not very familiar with algebraic cycles and intersection theory, so i hope my question is not too trivial.
Let $X$ and $Y$ be two K3 surfaces. Let ...
5
votes
2answers
95 views
Motivation of stable homotopy theory
A stable homotopy category can be obtained by modifying the category of pointed CW-complexes: objects are pointed CW-complexes, and for two CW-complexes $X$ and $Y$, we take
$$\lbrace X,Y \rbrace = ...
1
vote
0answers
41 views
Vector field and a solution of an ODE
I have this:
And my questions are :
1)what is :"The local theory of differential equations in a Banach space"
2)Why it's implies that each solution of (6) is equal to $\eta$
Please
Thank you .
...
6
votes
0answers
100 views
When is a fibration a fiber bundle?
In this question I am using Wiki's definitions for fibration and fiber bundle.
I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
5
votes
1answer
100 views
Explanation of example 3F.7 in Hatcher
The section I am refering to is the following example on page 314 of Hatcher's Algebraic Topology:
I'm a bit confused by his statement about relations and can't quite see what he is trying to say. ...
