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

learn more… | top users | synonyms (1)

0
votes
2answers
44 views

How to prove the existence of this homomorphism $G_1*G_2\to H$?

Say there is a homomorphism from a group $G_1$ to another group $H$, and there is a homomorphism from $G_2$ to $H$, then there is a homomorphism from $G_1*G_2$ to $H$ induced by the former to ...
2
votes
1answer
44 views

Boundary connected sum of manifolds

I have two related questions about the boundary connected sum of manifolds with boundaries. Let $T=S^1 \times S^1$ be a torus and let $X=T \times [0, 1]$ be the cylinder over the torus. Let $X'$ be a ...
1
vote
0answers
33 views

why does the flipping map $S^1 \wedge S^1 \to S^1 \wedge S^1$ introduce a minus sign in homotopy?

Suppose we have a map $f:S^1 \wedge S^1 \to X$ of pointed spaces ($S^1$ is the circle), let $T:S^1 \wedge S^1$ be the map that flips the factors, (so $T(x,y)=(y,x)$) and let $f'=f \circ T:S^1 \wedge ...
0
votes
2answers
38 views

Liftings in Topology

I am wondering about this: Assume you have a class $[f] \in \pi_1(B,b_0)$ and a covering map $p:E \rightarrow B$. Now, I know that if you take any two paths $g,h \in f$ that are homotopic and they ...
0
votes
1answer
33 views

Showing $K_G (X \times EG) =K(X \times_G EG)$

I know that $K_G (X \times EG) =K(X \times_G EG)$. I've seen the proof that this is the case for the vector space definition of K-theory, but I wanted one that works for the definition where $K_G(X)= ...
2
votes
0answers
25 views

Existence of orientation-reversing automorphisms

Let $M$ be a connected orientable manifold (I don't really care about which category of manifolds we should work in). Since I read that there are nice manidolds without orientation-reversing ...
3
votes
0answers
40 views

Is there fundamental group “with coefficients”

Constructing fundamental group in usual way, one uses $I=[0,1] \in \mathbb R$, so uses real numbers, and gets a group classifying coverings for nice spaces: locally connectible and so on, in fact ...
1
vote
0answers
26 views

Name of this property: all maps from given class of spaces into $X$ are nullhomotopic?

Let $X$ be a topological space and let $\mathcal{C}$ be some class of topological spaces. Is there a standard name for the following property of $X$? For every space $C\in \mathcal{C}$ all maps $C\to ...
5
votes
1answer
85 views

Category theory for knot theory

Where would I start to learn category theory for its use in knot theory? I have a background in physics and Ive read Adams Knot book. I know nothing about category theory. Eventually I want to learn ...
0
votes
1answer
48 views

Simple example of $X$ with torsion in $H^1(X,\mathbb{Z})$?

Question: Is there a simple example of a space $X$ possessing torsion in its first integral cohomology group $H^1(X,\mathbb{Z})$? For reasonable spaces $X$, e.g. CW-complexes, one has ...
1
vote
1answer
59 views

Gluing the ends of a cylinder. Can we get other than a torus?

Let $X=S^1 \times I$ be a cylinder, where $S^1$ is the 1-dimensional circle. If we glue the "bottom" boundary $S^1 \times 0$ and the "top" boundary $S^1\times 1$ by a homeomorphism sending $x\times 0$ ...
1
vote
1answer
47 views

Are the two inverses in the free group same?

Set G is a group, and the set C is contained in G. Set C contains only 2 elements a and b such that they are the inverse of each other.Set F(C) is the free group made by C. Then in F(C), it contains ...
0
votes
2answers
24 views

Nullhomotopic map extended

I have troubles understanding this proof: Let $h:S^1 \rightarrow X$ be a continuous map, then we have that if $h$ is nullhomotopic, $h$ can be extended to a continuous map $k:B^2 \rightarrow X.$ ...
0
votes
2answers
52 views

Proof of the Borsuk-Ulam Theorem

The Borsuk-Ulam Theorem says the following: For any continuous map $g: S^n \rightarrow \mathbb{R}^n$ there exists $x \in S^n$ such that $g(x)=g(-x)$. I'm trying to work through the proof given in ...
2
votes
0answers
82 views

K-theory of $\mathbb{RP}^{\infty}$

what are the $K_0$ and $K_1$ group of $\mathbb{RP}^{\infty}$? Any reference would be good enough.
3
votes
0answers
31 views

About the Chern class of infinite complex Grassmannian

I learned that any characteristic class of rank-$k$ complex vector bundles on paracompact spaces is determined bijectively by a cohomology class in $H^*(Gr_k^\infty(\mathbb C))$, the cohomology ring ...
0
votes
0answers
37 views

What is $H_1(A_\mathbb{C}^{top},\mathbb{Q})$

Let $A$ be an abelian variety defined over a number field. I have seen in a few papers the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ being used. I read up on the singular homology but it ...
0
votes
1answer
32 views

Fibre is open in covering space

I think I don't see the wood for the trees: In my notes I found the remark that if $p:E \rightarrow B$ is a covering map, then for each $b \in B$ we have that $p^{-1}(b)$ in $E$ has the discrete ...
0
votes
0answers
13 views

Vector bundle, nonexistence of Euclidean metric

Milnor-Stasheff "Characteristic classes" problem 2-C says: Any vector bundle over a paracompact base space can be given a Euclidean metric in other words, if $\pi : E \rightarrow B$ is a vector ...
0
votes
0answers
18 views

A remark on representable functors in May's Concise Course

On page 206 of May's Concise Course there is a lemma stating that for well pointed space $X$ and $Y$ and a particular representable functor $A:Spaces \to Groups$ (in particular reduced K-theory) that ...
3
votes
1answer
39 views

Which p-adic groups are simply-connected?

Suppose that we are working over a nonarchimedean local field $F$, for instance $\mathbb{Q}_p$. Which semisimple algebraic groups (or Lie groups) over $F$ are simply-connected? In particular, I am ...
1
vote
1answer
42 views

Introduction to Localization of Topological Spaces

I am trying to learn localization of topological spaces but am not sure where to start. Can anyone recommend some introductory materials? It would be great if it contains detailed motivations, ...
1
vote
1answer
107 views

Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
5
votes
2answers
99 views

Cup product in Morse cohomology

Dualizing the Morse complex, we obtain the Morse cohomology, which is isomorphic to the usual singular cohomology and thus admits a cup product. Does anybody know how this cup product would look like ...
0
votes
1answer
47 views

Is the homotopy class given by the degree?

Let $X$ be a topological space such that $\pi_n(X)=H_n(X)=Z$. A continuous map $f: S^n \rightarrow X$ is an element of $\pi_N(X)=Z$ therefore $[f]_{\mathrm{homotopy}}$ is characterised by an integer ...
2
votes
1answer
33 views

Fundamental group of the quotient of a cylinder by a rotation at either end

The following is a past qual problem: let $X = S^1 \times [0,1]$ be the cylinder, and define an equivalence relation on $X$ by $(z,1) \sim (iz,1)$ and $(w,0) \sim (e^{i\pi /7} w, 0)$. Compute ...
3
votes
0answers
32 views

Homology of non-singular projective algebraic variety

I am unsure whether or not the following claim is true or false and whether or not my proof works or not: Claim: Let $V \subset \mathbb{C}P^n$ be a complex $k$-dimensional, non-singular, projective ...
2
votes
1answer
43 views

An isomorphism between homology and cohomology with $\mathbb{Z}_2$ coefficients

In the proof of a theorem we did in a class (namely: if $M$ is an odd-dimensional, closed manifold, then $\chi(M)=0$), there's the following step: $$H_k(M;\mathbb{Z}_2)\cong H^k(M;\mathbb{Z}_2)$$ ...
1
vote
0answers
35 views

Higher homotopy groups

Theorem 5.1 of this paper describes a map $K_n(R)\to \pi_{n+1}(SK(E(R),1))$, where $S$ denotes the suspension. My question: Do we have a map from $K_n(R)\to \pi_{n+1}(S^2K(E(R),1))$. Any reference is ...
1
vote
1answer
40 views

Covering space description

Given the covering map $f:X\rightarrow Y$ in the picture below, which takes edges to edges preserving labels and orientations, I would like to describe the induced map $f_{\ast}:\pi_1(X,x)\rightarrow ...
5
votes
0answers
41 views

Definition of Hodge structure: is torsion allowed?

I am trying to understand the definition of an integral Hodge structure. Apparently, for $X$ a compact Kahler manifold, $H^n(X,\mathbb R)$, the lattice $H^n(X,\mathbb Z)$ and the Hodge filtration give ...
1
vote
0answers
36 views

Loop space of $S^1$

How concretely can the (based) loop space $\Omega S^1$ of $S^1$ be described? I know it's a space with homotopy groups $\pi_0(\Omega S^1) \simeq \mathbb{Z}$ and $\pi_i(\Omega S^1) \simeq 0$ for ...
4
votes
1answer
65 views

Mapping Class Group

$\newcommand{\MCG}{\mbox{MCG}}$Let $\alpha$, $\beta$ be non-isotopic, non-separating curves on a surface $S$ (meaning that "cutting along " them will not disconnect the surface). How do we show that ...
1
vote
1answer
58 views

The blow up of of the plane and the Moebius band

The (real) blow up of $\mathbb{R}^2$ is defined by $\tilde{\mathbb{R}}^2=\{(p,l)\in\mathbb{R}^2\times\mathbb{R}\mathbb{P}^1|p\in l\}$, with the projection $\pi:\tilde{\mathbb{R}}^2\to\mathbb{R}^2$, ...
0
votes
0answers
19 views

Why is it necessary to define a simplex as the affine subspace or “zero set” of an affine transformation?

Why is it necessary to define a simplex as the affine subspace or "zero set" of an affine transformation? Why can't it just be defined as $\sum{t_iv_i}$ where $\sum{t_i}=1$ and $v_i$ are the vertices ...
3
votes
2answers
68 views

Is there some knot theory behind the Mobius donut?

I was watching this video by Numberphile where a professor cuts a bagel into two interlocking pieces. Is this a torus knot or torus link? I'm trying to interpret in terms of $(p,q)$-torus knots Torus ...
2
votes
1answer
63 views

The cohomology ring of the nerve of a category associated to a vector space

Let $n\ge2$, and let $V$ be an $n$-dimensional vector space over a field $k$. Consider the category $\mathcal{C}$ whose objects are nonzero, proper subspaces of $V$, and whose morphisms are ...
3
votes
1answer
89 views

For a Cantor set $\mathcal{C} \subset S^3$ such that $\pi_1(S^3 \setminus \mathcal{C})=0$, prove $S^3 \setminus \mathcal{C}$ can be split by a sphere.

I'm working from the paper Cantor Sets in $S^3$ with Simply Connected Complements by Richard Skora. On page 184 the second sentence states that any Cantor set $\mathcal{C} \subset S^3$ such that ...
0
votes
0answers
17 views

Gysin sequence and Serre spectral sequence

Given an oriented $S^k$ bundle $E$ over a compact manifold $M$ we get the Gysin Sequence (I am interested in the DeRham cohomology). We can obtain this sequence from the Serre Spectral Sequence if the ...
0
votes
2answers
33 views

CW complex is contractible if union of contractible subcomplexes with contractible intersection

Exercise 0.23 from Algebraic Topology by Hatcher reads: Show that a CW complex is contractible if it is the union of two contractible subcomplexes whose intersection is also contractible. I have ...
4
votes
0answers
134 views

Higher homotopy groups: Basepoint independence.

Let $f,g: [0,1]^n=I^n \rightarrow X$ be contiuous maps s.t. $f(\partial I^n)=g(\partial I^n)=x_1$. If $\gamma:I \rightarrow X$ is a path joining $x_0$ and $x_1$ ...
2
votes
0answers
31 views

Classifying covering spaces of product spaces

Given two covering maps $p\colon \tilde{X} \to X$ and $q\colon \tilde{Y} \to Y$, we can form the covering map $p\times q \colon \tilde{X} \times \tilde{Y} \to X\times Y$. By covering space theory, we ...
1
vote
0answers
32 views

Geometric definition of the stable commutator length

In his book, D.Calegari proves the equivalence of the algebraic and geometric definitions of stable commutator length (Proposition 2.10, p. 15). I actually have some difficulties in understanding the ...
0
votes
1answer
22 views

Fundamental group of disjoint union of two 2-tori identifying them along pairs of points

I'm trying to solve the following problem: let $X$ be the space obtained from the disjoint union of two 2-tori $A,B$ be identifying them along 2 pairs points (resp. three pairs of points). If we ...
1
vote
1answer
39 views

Deck transformations

We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group ...
0
votes
2answers
49 views

Induced map on fundamental groups between surfaces

Let $\Sigma_n$ and $\Sigma_m$ be two closed oriented surfaces of genus $n$ and $m$, with $n \leq m$. We may think about these surfaces as connected sums of tori, so there is an canoical inclusion map ...
2
votes
1answer
52 views

Cell structure of $S^2 \times S^1$

Can anyone please provide the cell structure of $S^2 \times S^1$? I know that there are one cell in each dimension from 0 to 3 but I am not sure about the attaching maps. Thanks in advance.
1
vote
1answer
38 views

cohomology is dual to homology of a spectrum if homology is free

Let $E$ be a multiplicative spectrum (and $X$ a space with $H_n(X; \mathbb{Z})$ free abelian for every $n$). The following excerpt is taken from the notes here claim that item (1) below easily implies ...
0
votes
1answer
35 views

Condition for Orientability of Manifold

Let $M^n$, $n>2$ be a manifold and let $f:D\rightarrow M$ be an embedding of the closed $n-$disk in $M$. Prove or Disprove: $M$ orientable iff $M-f(D)$ is orientable. $M$ is orientable iff all ...
2
votes
1answer
42 views

Quotient of Unit Quaternions by Subgoup (Lie Groups)

Let $G \leq Sp(1)\cong S^3$ (unit quaternions) be a discrete subgroup of order 120 (the Binary Icosahedral Group, not the other one), with presentation $G=<s,t| s^2=t^3=(st)^5>$. $\hspace{2mm}G$ ...