0
votes
1answer
22 views

How does one compute a group modulo a torsion group?

Let's say I have some group $G$ and a subgroup $H$ such that $H$ is a torsion group (i.e. $\forall h \in H$, $h$ has finite order. How do I compute the factor group $\frac{G}{H}$? What effect does the ...
-1
votes
0answers
27 views

An Advice Concerning Master's Programme [on hold]

Which of these programmes is a better choice, if one wants to pursue a degree in pure mathematics? (In Geometry, Topology and Algebra, in particular, algebraic geometry) 1) ...
3
votes
1answer
27 views

Constructing a surjection from fundamental group of a mapping cone to Hawaiian Earring to $\prod_\infty \mathbb{Z} / \oplus_\infty \mathbb{Z}$

If X is the subspace of $\mathbb{R}$ consisting of 1, 1/2, ... together with its limit point 0, C is the mapping cone of the quotient map $SX \rightarrow \sum X = $ (the Hawaiian Earring) which ...
1
vote
1answer
57 views

Is an configuration space contractible?

Let $\mathbb{R}^\infty$ be the union $\cup \mathbb{R}^n$(with inclusions). Let $F(\mathbb{R}^\infty,k)$ denote the $k$-th configuration space of $\mathbb{R}^\infty$, i.e., the subspace of ...
0
votes
1answer
16 views

homeomorphism classes of compact surfaces with addition operation is a monoid

This is essentially pg 6 of serge lang's algebra's discussion about an interesting example. Homeomorphism classes of compact surfaces with the addition operation defined as following. Say M and $M'$ ...
4
votes
1answer
44 views

map between classifying spaces induced by group homomorphism

Let $\Sigma_n$ be the permutation group of order $n$. Then the regular representation of $\Sigma_n$ gives an injective homomorphism $f:\Sigma_n\to O(n)$. Why $f$ induces a map between their ...
1
vote
0answers
58 views

How to prove a direct sum?

$X=X_1\cup X_2$, $X_i, i=1,2$ are closed and disjoint. $i_{2_*}: H_k(X_2)\rightarrow H_k(X),$ induced by the inclusion $i_2: X_2\rightarrow X$ and $j_{1_*}: H_k(X)\rightarrow H_k(X,X_1)$ induced by ...
0
votes
1answer
66 views

Direct Sum on Homology

I have a big problem and i don't know how to solve it i have no idea So, let $i_2: X_2\rightarrow X$ an inclusion and $j_1: X\rightarrow (X,X_1)$ we have that $i_{2_*}: H_k(X_2)\rightarrow H_k(X)$ is ...
0
votes
1answer
27 views

smash product of Eilenberg-Maclane spaces

Let $G$ be an abelian group and $K_n=K(G,n)$ be the Eilenberg-Maclane space. How to obtain $K_m\wedge K_n$ is $(m+n-1)$-connected? (Hatcher's book page 404)
2
votes
3answers
52 views

homotopy groups of wedge sum

Let $X_\alpha$ be connected CW-complexes. Then from Hatcher's book, $$\pi_{n}(\prod_{\alpha} X_{\alpha})=\prod_{\alpha}\pi_{n}(X_{\alpha}).$$ Is it true in general $$\pi_{n}(\bigvee_{\alpha} ...
1
vote
3answers
117 views

Some practical questions on cohomology and the ring $\mathbf{Z}[x]/(x^2)$

So I know that the cohomology ring of $S^n$ is $\mathbf{Z}[x]/(x^2)$ with "$x$ in degree $n$"; if we ignore the grading then this ring fails to distinguish the spheres. What is actually meant by "in ...
7
votes
1answer
141 views

Does the rank of homology and cohomology groups always coincide?

Let $(C_i)_{i \in \mathbb{Z}}$ be a chain complex of free abelian groups. Does the rank of the homology and cohomology groups of $(C_i)_{i \in \mathbb{Z}}$ always coincide, i.e. is ...
2
votes
1answer
65 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 ...
1
vote
0answers
38 views

Characterising subgroup

Let $\omega $ be a path in $\hat{X}$ with $\omega(0), \omega(1) \in p^{-1}(x_0)$, where $p$ is a covering map $p:\hat{X} \rightarrow X$. Let $\alpha=[p \circ \omega] \in \pi_1(X,x_0)$. Then we have ...
2
votes
1answer
100 views

Cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ as a continuous group or a Lie group?

Is there any example of Borel cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ for any $G$ such that $H^n(G, \mathbb{R}/\mathbb{Z})$ is a continuous group? Such as a Lie group? Most of the examples ...
-1
votes
1answer
51 views

Powers of Orbifold Fundamental Groups

I have reduced a problem to $\pi(Y)^n/G^n$ where Y is a manifold and G is a group acting on the manifold. Can I "factor out," the $n$? i.e. $(\pi(Y)/G)^n$. Note that $\pi(X)$ is the fundamental group ...
2
votes
2answers
65 views

A comment on a proof of equivalent knots

The following theorem and its proof is from A First Course in Algebraic Topology By Czes Kosniowski pp. 219-220 ...
6
votes
1answer
98 views

Airy differential equation and Galois group

Consider the Airy equation $y^{(2)}=ry$ where $r \in \Bbb{C}(z)$ but not constant. How do you show that $G^0=G$, where $G$ is the galois group of the picard vessiot extension of solutions over ...
0
votes
1answer
55 views

Soft Question: Geometry of Rings

So.. I was thinking about something last night and literally have no idea where to start. Can the ring axioms be connected to topology? If so, how? Basically, the motivation for my question has to ...
0
votes
1answer
36 views

invariants of a Lie algebra

What does it mean by "constructing invariants" in algebraic topology or algebra in general? How to define a "invariant" in algebra? What does it mean by the "invariant of a Lie algebra"?
6
votes
1answer
63 views

Baumslag–Solitar $B(1,2)$ is not hyperbolic

I have a question which asks me to show that the Baumslag–Solitar $B(1,2)$ is not hyperbolic by considering its Cayley graph and showing that triangles can be arbitrarily fat. The Cayley graph can be ...
6
votes
1answer
73 views

Describe all the quotient groups of $\mathbb{Z} \oplus \mathbb{Z}$

I'm thinking about a problem in algebraic topology of how to determine all the $k$-fold covers of $\mathbb{T}^2$, where $k$ is a certain integer. In thinking about the problem, I came upon the ...
0
votes
1answer
41 views

Question on Hamming distance

Let V_n be n-dimentional vector space over GF(q). E is k-dimentional vector subspace which is a linear q-ary (n,m,d) code and also consider the radius e = [(d-1)/2]. Assume that E is not a perfect ...
1
vote
1answer
30 views

A question about subexponential growth about group

Recall a group $\Gamma$ is said to have subexponential growth if lim sup$|E^{n}|^{1/n}=1$ for every finite subset $E\subset \Gamma. (E^{n}=\{g_{1}g_{2}...g_{n}: g_{i}\in E\}.)$ My question is: Can we ...
1
vote
0answers
37 views

Real Spectrum of a Commutative Ring

Could anyone explain briefly what is a real spectrum of a commutative ring? If it possible please give me some easy example. Many thanks in advance. $\text{Spec}_{r}(A)$
0
votes
1answer
62 views

Relative singular chains basis

If $(X,A)$ is a pair, then $S_k(X,A):=S_k(X)/S_k(A)$ is free on the singular simplicies of $X$ with image not contained in $A$. Why is this so? I tried to give a proof by checking the mapping property ...
1
vote
0answers
27 views

What is the topological relation between $ C^\infty (E, S^1) $ and $ E $

What is the topological relation between $ C^\infty (E, S^1) $ and $ E $? Here, $ C^\infty (E, S^1) $ is the algebra of all continuous functions from $ S^1 $ to $ E $. $ E $ is a four dimensional ...
2
votes
1answer
53 views

$\mathrm{Homeo}(S^1)$ and the Mapping Class Group

Is there a full description of $\mathrm{Homeo}(S^1)$ (i.e. the group of self-homeomorphisms of the circle)? By full description I mean a presentation/list of subgroups ect. Basically anything ...
1
vote
0answers
26 views

Reference: Topology on Ind-object

In some articles I've recently seen authors mention that pro-finite groups or pro-finite algebras possess a topology, but they do not explicitly describe it. I was wondering how is the topology ...
4
votes
0answers
137 views

Difficulty of exercises in several books

Since I am teaching myself, I have no idea of the difficulties of exercises in many books. I want to know whether I am not understanding the content of these books if I cannot easily solve the ...
4
votes
1answer
78 views

Inverse question: Recognizing when a phenomenon behaves like an algebra

I might be phrasing this question incorrectly, but my students asked me about it and I did not have a good answer. I am in statistics and I look at all sorts of social data, network data, climate ...
2
votes
3answers
109 views

finding a topological group with specific conditions

I have a question, it sounds difficult. The question is the following: Let $X$ be a topological group such that the binary operation defined on it is $*$. For any two points $a$ and $b$ in $X$ ...
0
votes
2answers
72 views

Question on relative homology

i have this: where $|\tau|$ is the support of the chain $\tau$, i don't understand the first part why $[\sigma]=0$ in $H_{m-1}(\phi^{c+\varepsilon},\emptyset)$ ??? Please, thank you.
2
votes
1answer
66 views

On a coalgebra structure of simplicial homology

Are there any results on the homology group of an abstract simplicial complex with coefficients in a field $k$ being a $k$-coalgebra? Are there any assumptions and restrictions on the topological or ...
2
votes
1answer
64 views

Basic constructions for graded algebras.

I'm reading about the Weil algebra of a Lie group and it involves some constructions I'm not very familiar with, for instance the "free graded-commutative graded algebra on $a_1...a_n$ with degrees ...
1
vote
1answer
91 views

relation between the group O(3) and SU(2)

Base on relations between groups $O(3)$, $SO(3)$ and $SU(2)$: a) $O(3)=SO(3)\otimes \{1,-1\}$ b) $SO(3)\simeq SU(2)/Z_2$ Can I say $\{1,-1\}$, i.e. $Z_2$, also the center of the group $O(3)$? If ...
3
votes
5answers
98 views

Free product of the trivial group with another group

I'm new to the idea of a free product.. Basically I was wondering if G is an arbitrary group and 1 is the trivial group then is $1\star G \cong G$. If not.. what whould it look like?
0
votes
1answer
74 views

Why call them cycles and boundaries?

I have a small question. Why we have this designation: $n$-cycles for $Z_n$ and $n$-boundaries for $B_n$ ? Why they are called cycles and boundaries ? ...
3
votes
0answers
76 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
1
vote
3answers
121 views

Symmetric monoidal products that preserve limits and colimits

Are there common examples of a symmetric monoidal product $\otimes$ that preserves both limits and colimits in each variable? This question is worded incorrectly, I now realize: (A) I originally ...
3
votes
1answer
84 views

Surjective inclusions in Van Kampen's Theorem

Let $X = U \cup V$ be an open cover. Assume $U,V$ and $U \cap V$ path-connected. We have the inclusions $u : U \cap V \to U$ and $v : U \cap V \to V$ with induced maps $u_* : \pi_1(U \cap V) \to \pi_1 ...
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
3answers
92 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 ...
4
votes
2answers
167 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 ...
2
votes
1answer
80 views

filtered modules (LNAT, Davis & Kirk)

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 240, there is written: Q1: Is convergence of the filtration assumed in the first underline? Otherwise $\forall p: F_p=A$ is a ...
5
votes
1answer
114 views

Showing path connected matrices of a group $G$ is a normal subgroup

Let $G$ be a subgroup of $GL_n(\Bbb{R})$. Define $$H = \biggl\{ A \in G \ \biggl| \ \exists \ \varphi:[0,1] \to G \ \text{continuous such that} \ \varphi(0)=A , \ \varphi(1)=I\biggr\}$$ Show that $H$ ...
4
votes
2answers
184 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 ...
1
vote
1answer
80 views

filtration on the (co)homology of a space from the filtration of a space

Fix $n\!\in\!\mathbb{N}$. Let $X$ be a topological space and $X_0\subseteq X_1\subseteq X_2\subseteq \ldots$ subspaces of $X$. Let $\iota_k:X_k\rightarrow X$ be the inclusion. Let ...
-1
votes
1answer
62 views

Isomorphism $\left(\mathbb{C}^{n}\setminus\{0\}\right)/\mathbb{Z}$ with $S^{1} \times S^{2n-1}$

I have to prove that there is this isomorphism: $$\frac{\mathbb{C}^{n}{\setminus\{0\}}}{ \mathbb{Z}} \simeq S^{1} \times S^{2n-1},$$ where there is this equivalence relation in the left side: $(w_1, ...
3
votes
2answers
80 views

free groups and bouquet of circles

For any free group $F$ generated by the set $S$, one can construct a graph specifically a bouquet of circles $X$ s.t $\pi_1(X)=F$. My question is: Does this mean free groups are isomorphic to a free ...