All Questions
14 questions
15
votes
3
answers
926
views
Lower central series quotients in terms of (co)homology
Let $G$ be a group. It is well-known that $H_1(G,\mathbb{Z})=G/[G,G]$. Also (at least up to torsion) $[G,G]/[G,[G,G]]=\Lambda^2H^1(G,\mathbb{Z})/H_2(G,\mathbb{Z})$ as explained, for example, in this ...
- 7,377
14
votes
2
answers
788
views
Restriction of a branched cover to its branch locus
Assume that we have a smooth, compact, complex surface $X$, and a smooth and irreducible divisor $B \subset X$. Let $G$ be a finite group. For every group epimorphism $$\varphi \colon \pi_1(X-B) \to G,...
- 66.3k
12
votes
2
answers
780
views
Where does the term "torsor" come from?
Is there a heuristic reason why principal homogeneous spaces of a group (object) $G$ (in some categories) are called $G$-torsors? Does it have anything to do with the idea of "torsion", ...
- 23.3k
9
votes
0
answers
376
views
Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$
This is related to my previous MathOverflow question Fundamental group of $\mathrm{Sym}^2(C_g)$ minus the diagonal.
Let $C_2$ be a smooth curve of genus $2$ and $X:=\mathrm{Sym}^2(C_2)$ its second ...
- 66.3k
8
votes
1
answer
1k
views
Beyond an intro to topological graph theory...
I'm looking to find out what active areas of research there are in topological graph theory, particularly those that interface strongly with other areas of math (say, group theory, algebraic topology, ...
- 1,180
7
votes
2
answers
376
views
are there finite nonabelian characteristic quotients $G$ of $F_2$ inducing a surjection $Aut(F_2)\twoheadrightarrow Aut(G)$?
Let $F_2$ be the free group of rank 2. Let $K\le F_2$ be a characteristic subgroup, such that $G := F_2/K$ is finite.
Do there exist examples of such nonabelian $G$ such that the induced map
$$Aut(...
- 7,527
6
votes
2
answers
331
views
Epimorphisms from the genus $2$ surface braid group to finite groups
This question is somehow related to my previous MO question Explicit description of a subgroup of the braid group $\mathsf{B}_2(C_2)$; for the reader convenience, let me write down again the relevant ...
- 66.3k
6
votes
2
answers
436
views
Presentation of the fundamental group of a complex variety
Let $X$ be a connected smooth complex algebraic variety and $Z=\bigcup_{i=1}^r Z_i$ be a union of smooth connected hypersurfaces, satisfying that each two intersect transversally. Assume for ...
- 541
5
votes
1
answer
232
views
Finite index subgroups of the mapping class group with geometric meaning
I have got a question that is perhaps not precise in a mathematical sense.
Is there a classification of all coverings of the moduli space of Riemann surfaces which are moduli spaces themselves, that ...
- 471
3
votes
1
answer
432
views
Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups
Let $\Gamma\le SL(2,\mathbb{Z})$ be a finite index subgroup, not necessarily "congruence".
Let $c_4,c_6$ be the number of conjugacy classes of elements of order 4 and 6 respectively, let $c_{-1}$ be ...
- 7,527
3
votes
0
answers
158
views
What is the meaning of local inertia conjugation property?
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030., we have:
Abstract. Let $\widehat{G T}^{1}$ ...
- 119
2
votes
0
answers
119
views
Crossed homomorphism as morphism in the ambient category
Suppose we are given a crossed-homomorphism $\phi:G\to A$ (and an action $\alpha$ of $G$ on $A$)
$\phi(ab)=\phi(a)+\alpha(a)(\phi(b))$. Now, unless the action is trivial, this is not a homomorphism ...
- 199
2
votes
0
answers
167
views
Any abelian group embeds into a Chow group
Let $G$ be an abelian group. Must there exist a perfect field $k$, a smooth projective geometrically connected $k$-scheme $X$ and an integer $i\geq 0$ such that $G$ embeds into the integral Chow group ...
user164740
-1
votes
1
answer
421
views
How does the discrete group act on simplicial set level by level
Suppose that we know a discrete group acts on the geometric realization of a simplicial set. Is there some way to understand how the corresponding action works on the simplicial set?
For example, if ...
- 681