User zuriel - MathOverflow

List of commutator identities and equivalences

2012-05-29T06:24:27Z

Let $G$ be a group and let $[a,b]=a^{-1}b^{-1}ab$ be the commutator of $a$ and $b$ in $G$. There are several well-known commutator identities such as

$[x, z y] = [x, y]\cdot [x, z]^y$

and

$[[x, y^{-1}], z]^y\cdot[[y, z^{-1}], x]^z\cdot[[z, x^{-1}], y]^x = 1$.

I wonder if there is a more complete list of commutator identities and commutator equivalences of the form

$\alpha(x_1,\cdots,x_n)\equiv\beta(x_1,\cdots,x_n)\mod{N}$,

where $\alpha$ and $\beta$ are expressions involving commutators and $N$ is some subgroup of $G$, for example the $k$-th term in the lower central series of $G$. Anyone knows such a list? Thank you!

Automorphism group of factor groups

2012-05-21T06:09:30Z

Let $G$ be a group and let $H$ be a factor group of $G$. Is there any result that relates $\operatorname{Aut}(G)$ (the automorphism group of $G$) and $\operatorname{Aut}(H)$?

As a very special case of the question, let $F_2$ be the free group with two generators $x$ and $y$. Let $G_2$ be the factor group of $F_2$ by adding relations such that $[x,[x,y]]=[y,[x,y]]=1$; that is, $G_2$ is the discrete Heisenberg group. Then is there any relation between $\operatorname{Aut}(F_2)$ and $\operatorname{Aut}(G_2)$? Hence or otherwise, how to find out $\operatorname{Aut}(G_2)$?

Profiniteness Condition for Hochschild-Serre Spectral Sequence?

2012-04-25T07:46:32Z

This question may seem elementary to experts but I am quite confused about it:

According to the entry of Lyndon–Hochschild–Serre spectral sequence on wikipedia, for a group extension $1\to N\to G\to Q\to1$, there is a Lyndon–Hochschild–Serre spectral sequence if $G$ is a profinite group and $N$ is a closed normal subgroup of $G$:

http://en.wikipedia.org/wiki/Lyndon-Hochschild-Serre_spectral_sequence

Also in the book Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, the same condition was assumed.

However, according to some other books, for example Brown, Kenneth S. (1972), Cohomology of Groups, and A User's Guide to Spectral Sequences by John McCleary, the profiniteness condition on the group $G$ was NOT assumed.

Why there is such a difference? Do we really need the condition that $G$ is a profinite group and $N$ is a closed normal subgroup of $G$ to construct the Lyndon–Hochschild–Serre spectral sequence?

How to Compute Transgressions in a Serre Spectral Sequence?

2012-04-18T16:23:32Z

For a short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ there is an associated fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$, which can be constructed by realizing the homomorphism $B\rightarrow C$ by a map $K(B,1)\rightarrow K(C,1)$ and the convert it into a fibration. The fiber is $K(A,1)$ (from the associated long exact sequence of homotopy groups).

For a fibration $F\rightarrow X\rightarrow B$, the differential $d_n\colon E_{n,0}^n\to E_{0,n-1}^n$ in the Serre spectral sequence was shown to be equal to the transgression in Hatcher's book on Spectral Sequences (Proposition 1.13). The transgression was defined using (relative) homology groups.

My questions is: From the short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$, is there any method to directly compute the transgression of the associated fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$, at least for the case $n=2$, without constructing $K(G,1)$'s and considering their homologies?

Homology of Covering Spaces

2012-04-18T21:18:23Z

Let $A$ be a subgroup of a group $G$. Then since $A$ is a subgroup of the fundamental group $\pi_1(K(G,1))=G$, there is a covering space $p\colon Y\to K(G,1)$ with $p_*(\pi_1(Y))=A$. So the homology of $Y$ should be completely determined by $A$ and $G$. Suppose that $A$ and $G$ is known, how can one compute $H_*(Y)$, the homology groups of $Y$?

Triply graded spectral sequence?

2012-04-10T02:34:00Z

As we know, most of the spectral sequences are doubly graded. However, this "doubly graded" condition is not a part of the formal definition of spectral sequence. Is there any useful triply (quadruply, quintuply, etc.) graded spectral sequences? If not, is there a hope that some meaningful work can be done with this topic?

Why the underlying function of a monomorphism may not be an injection

2012-04-09T20:48:05Z

In category theory, a monomorphism (also called a monic morphism or a mono) is defined to be a left-cancellative morphism. It seems that this definition generalizes the definition of injections. However, even in a concrete category, a monomorphism may not be an injection. Why could this happen? I know examples of monomorphisms which are not injections but what is the reason behind the existence of such examples? Isn't monomorphism a generalised concept of injection? Similar questions can be asked about epimorphisms and surjections.

A Question on McCleary's book on Spectral Sequences

2012-04-09T20:40:07Z

I am reading John McCleary's A User's Guide to Spectral Sequence and was quite confused about one result: On page 15 of the version I was reading, it says that if $E^{\star,\star}_2$ is the bigraded vector space in Example 1.E, then $P(E^{\star,\star}_2,t)=(1+t^{11})(1+t^4+t^8+t^{12})(1+t^3)$. I am quite confused on how to obtain this result from Example 1.E. It seems to me that $P(E^{\star,\star}_t)$ has a term $t^{11+12+3}=t^{26}$, which by definition of $P(E^{\star,\star}_2,t)$ implies that $\text{dim}_k(\bigoplus _{p+q=26}E^{p,q})=1$. Why is that? I am not sure if I have understood Example 1.E wrongly. Any explanation will be greatly appreciated.

Update: Thank you @Neil Strickland for reminding me. The conditions of Example 1.E are: Suppose $E^{\star,\star}_2$ is given as an algebra by

$E^{\star,\star}_2\cong\mathbb{Q}[x,y,z]/(x^2=y^4=z^2=0)$,

where the bidegree of each generator is given by $\text{bideg}x=(7,1)$, $\text{bideg}y=(3,0)$ and $\text{bideg}z=(0,2)$. Furthermore, suppose $d_2(x)=y^3$ amd $d_3(z)=y$. In this case, the spectral sequence collapses at $E_4$ and, though $x$ and $y$ do not survive to $E_{\infty}$, the product $xy$ does.

Rational Homology of a Covering Space

2012-02-27T02:00:48Z

I have heard that the rational homology of a covering space is easy to compute, compared with the ordinary homology. However, I don't know any details about that. Can anyone help me? Any reference will be greatly appreciated.

Computing the homology groups of spaces in a fibration

2012-02-15T02:50:52Z

Let $F\rightarrow X\rightarrow B$ be a fibration. If we know very well the spaces $F$ and $B$ and wish to compute the homology of $X$. One possible tool is the Serre Spectral Sequence. However, it works under the condition that $\pi_1(B)$ acts trivially on $H_*(F;G)$. If this condition ($\pi_1(B)$ acts trivially on $H_*(F;G)$) does not hold, what other tools can one use to compute the homology of the homology of $X$?

In fact I am interested in the special case that all spaces in the fibration are $K(\pi,1)$ spaces. If any approach works for this particular case it would be wonderful.

Thank you!

Homotopy-Fibre Sequence of Classifying Spaces

2012-01-19T08:44:29Z

Let $G$ be a topological group and $H$ be a normal subgroup of $G$ (I think $H$ is required to be admissible in the sense that the quotient map $G\to G/H$ is a principal $H$-bundle, am I right?). Then there exists a homotopy fibre sequence $BH\to BG\to B(G/H)$, where $BG$ denotes the classifying space of $G$.

My questions is: suppose that we already know the groups $G$ and $H$ and suppose that we know the classifying space of $G/H$ and the classifying space of $H$, to what extent can we decide the classifying space of $G$ from these information? How to find the classifying space of $G$ if we know the classifying space of $G/H$ and the classifying space of $H$?

Any reference will be greatly appreciated.

The Classifying Space of the Discrete Heisenberg Group

2012-01-01T01:48:31Z

What is the Classifying Space of the Discrete Heisenberg Group? Which paper/book contains a detailed proof?

Thank you for your time.

Centerlessness of reduced free group

2011-08-15T08:39:01Z

Let F_n be the free group with n generators, where n is an interger greater than 1. Let RF_n be the reduced free group, which is defined to be the quotient group of F_n obtained from F_n by adding the relations that each generator of F_n commutes with all its conjugates.

Can anyone help to prove or disprove that RF_n is centerless, that is, the center of RF_n is trivial?

Your help will be much appreciated.

Where can one find reference proving that Braid group induces isomorphism between punctured disk and the complement of the braid?

2011-06-14T13:04:23Z

It is a known result that if $B$ is an $n$ braid over a disk, then $B$ naturally induces an isomorphism between the fundamental group of a disk with n points removed and the fundamental group of the space $D\times [0,1]-B$, where $D$ is a disk. My question is, in which book/paper can I find a proof of this result?

Question about some element in a group commutes with its all conjuagates.

2011-06-14T12:56:59Z

Let G be a group and g and h be elements in G. If g commutes with all conjugates of g and h commutes with all conjugates of h, can one conclude that gh commutes with all conjugates of gh?

Thanks!!

Comment by Zuriel

2012-05-21T07:34:02Z

@Mark Sapir, Thank you for your answer! Just to clarify: did you mean that in the free group with generators $x$ and $y$, the normal subgroup generated by $[x,[x,y]]$ and $[y,[x,y]]$ is a fully invariant group? If yes, why is it so?

Comment by Zuriel

2012-04-19T11:05:50Z

@Igor Rivin, for your long answer, may I know which part(s) of the book should I read to find answers to my question?

Comment by Zuriel

2012-04-18T22:16:42Z

Many thanks @Aru Ray! I will find the book and look for related results.

Comment by Zuriel

2012-04-18T21:48:37Z

@Aru Ray, so you mean that the result has nothing to do with $G$?

Comment by Zuriel

2012-04-18T19:38:59Z

@Josh, it seems to be quite a big book. May I know which page/section should I read for this question?

Comment by Zuriel

2012-04-18T19:32:12Z

@Ralph, yes, it is. However, the name "Lyndon-Hochschild-Serre spectral sequence" was not explicitly mentioned in Hatcher's book.

Comment by Zuriel

2012-04-10T04:08:29Z

I am still new to this website and am not sure about many of its functions. Will wait more as you have suggested; thank you!

Comment by Zuriel

2012-04-10T02:47:08Z

@Ben Williams, this change is not mentioned in the book. Does this mean that the case is that this part of the book contains some flaws, not I understood it wrongly?

Comment by Zuriel

2012-04-02T00:51:37Z

@jim stasheff, may I know which book you are refering to by "Borel"? It seems to me that Eilenberg−Moore spectral sequences require the base space B to be simply connected; what if this condition does not hold? Thanks in advance!!

Comment by Zuriel

2012-02-15T03:59:21Z

@Ryan, May I know where I can find any reference on the Serre Spectral sequence for any fibration mentioned in your comment? Thank you!

Comment by Zuriel

2012-01-02T02:57:36Z

@Alain Valette, you have mentioned that $\Gamma$ can be viewed as the semi-direct of $\mathbb{Z}^2$ and $\mathbb{Z}$; did you mean that $n\in\mathbb{Z}$ is mapped to the element $f_n\in Aut(\mathbb{Z}^2)$, such that $f_n(m_1,m_2)=m_1+nm_2$? Then why is $\Gamma$ isomorphic to the semi-direct product? It seems that the central extension $0\rightarrow\mathbb{Z}\rightarrow\Gamma\rightarrow\mathbb{Z}^2\rightarrow0$ does not split; then is there any nature semi-direct structure from this central extension? Thank you!

Comment by Zuriel

2011-08-15T10:15:07Z

@Mark Sapir: as you mentioned in your previous answer, [a, b] is in the center of RF_2 and thus the group has a center. But do we also need to prove [a, b] is not equal to the identity element of the group? I know that it should be true; but how to prove it? Thank you again for your answer.

Comment by Zuriel

2011-08-15T09:21:00Z

Thank you Mark for your answer!! Then can we conclude that the center of RF_n is the intersection of all N_i, as is defined in your answer?

Comment by Zuriel

2011-06-15T10:57:15Z

Thank you for your answer! May I know more details (title, etc.) of those classical papers by Baer and others? I would like to read them.

Comment by Zuriel

2011-06-15T10:54:32Z

Thank you so much for your answer! May I know how $G_1$ and $G_2$ are defined and how do you know that there are exactly 15 elements in each group with the given property?