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18 votes
0 answers
1k views

What is the strongest nerve lemma?

The most basic nerve lemma can be found as Corollary 4G.3 in Hatcher's Algebraic Topology: If $\mathcal U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of ...
2xThink's user avatar
  • 81
3 votes
0 answers
282 views

Commutator length of the fundamental group of some grope

A popular way to describe a grope as the direct limit $L$ of a nested sequence of compact 2-dimensional polyhedra $L_0 \to L_1 \to L_2 \to \cdots$ obtained as follows. Take $L_0$ as some $S_g$, an ...
Shijie Gu's user avatar
  • 2,083
3 votes
0 answers
547 views

Aut/Inn/Out Automorphism Groups of the unitary group $𝑈(𝑁)$

Given a group $G$, we denote the center Z$(G)$, we like to know the automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences: $$...
wonderich's user avatar
  • 10.5k
2 votes
0 answers
106 views

Minimal symmetry of a fibre bundle

Let $F \to E \to B$ be a topological fibre bundle with fibre $F$ and base $B$. It can be characterized by a map $B \to BAut(F)$. If it can also be characterized as a map $B \to BG$ (or say $G$ is a ...
Student's user avatar
  • 5,230
8 votes
1 answer
387 views

Outer automorphism group of Brieskorn homology sphere?

In this post, it is discussed how a Brieskorn homology sphere $\Sigma(a_1,a_2,a_3)$ with $\displaystyle \frac{1}{a_1}+ \frac{1}{a_2}+ \frac{1}{a_3} < 1$ is an aspherical manifold with a ...
Jeffrey Rolland's user avatar
6 votes
1 answer
426 views

What is known about the discrete group cohomology $H^2(\mathrm{SL}_2(\mathbb C), \mathbb C^\times)$?

The cohomology ring of $\mathrm{SL}_2(\mathbb C)$ as a topological group is straightforward (it's generated by a Chern class), but what is known in the discrete case? I'm particularly interested in $H^...
Calvin McPhail-Snyder's user avatar
7 votes
1 answer
288 views

A finitely presented group whose rational cohomology is not nilpotent

Does there exist a finitely presented (preferably $\text{FP}_{\infty}$) group $\Gamma$ and an element $\alpha \in \text{H}^{\ast>0}(B\Gamma;\mathbf{Q})$ that is not nilpotent? If non-discrete ...
Jens Reinhold's user avatar
16 votes
1 answer
505 views

How many cells needed to build the classifying space $BG$?

Let $G$ be a finitely presented group of cohomological dimension $n$. Apart from the unresolved ambiguity pertaining to the Eilenberg--Ganea conjecture, it is known that we can find an $n$-dimensional ...
Jens Reinhold's user avatar
5 votes
1 answer
384 views

Which groups have undetectable third U(1)-cohomology?

Let $G$ be a finite group. A categorical Schur detector for $G$ is a set $\mathcal{S}$ of proper subgroups $S \subsetneq G$ such that the total restriction map $$ \mathrm{rest}_{\mathcal{S}} : \mathrm{...
Theo Johnson-Freyd's user avatar
6 votes
1 answer
658 views

Generalized Birman exact sequence for surfaces with boundaries

Let $S_g^n$ be a surface of genus g with n boundaries and let $Mod(S_g^n)$ be its mapping class group. We will also denote by $S_{g,m}^n$ a surface of genus g with n boundaries and m punctures. The ...
Philippe Tranchida's user avatar
4 votes
0 answers
164 views

non-abelian tensor products of several groups

R. Brown and J-L. Loday had defined the tensor product of two arbitrary groups acting on each other. Let $G,H$ be groups with actions on each other on the right. each group act on itself by ...
M masa's user avatar
  • 479
0 votes
1 answer
676 views

Second homotopy group of the wedge sum of $S^2$ with the presentation complex of a finitely generated group

I am reading a paper which makes the following claim: let $G$ be a finitely presented group, and let $X$ be the presentation complex of $G$. Let $X' = X \vee S^2$ be the wedge sum of $X$ with the ...
Hussain Kadhem's user avatar
7 votes
3 answers
586 views

Dimension of classifying space of a group

If $N$ is a normal subgroup of a group $G$ such that $G/N= \mathbb{Z}$. Suppose that the classifying space of $G$ is a finite CW-complex of dimension $n$. Does it follow that the classifying space of $...
lab's user avatar
  • 451
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 ...
user avatar
8 votes
2 answers
507 views

Contractible Rips complex from non-hyperbolic group

I heard that the Rips complexes associated to the Cayley graphs of hyperbolic groups are contractible for a sufficiently large radius. Is the converse true? Namely, if a group is non-hyperbolic, then ...
Uzu Lim's user avatar
  • 903
9 votes
0 answers
372 views

Groups with trivial outer automorphism group and prescribed center?

Given an arbitrary abelian group $A$, can we find a group $G$ such that $\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G)=1$, and $Z(G)\simeq A$? Why is this interesting? Given a group $G$, we have ...
Charles Rezk's user avatar
  • 27.2k
5 votes
1 answer
336 views

"Simplicial complex" product of groups?

Let $X=(V,E)$ be a graph, and to each vertex $v \in V$, associate a group $G_v$. The graph product of the groups $G_v$ (as defined e.g. here) is $F/R$; the quotient of the free product of the $G_v$ by ...
Matt's user avatar
  • 208
12 votes
1 answer
625 views

Any group is a quotient of an acyclic group?

As far as I know, for any group $G$ there exists an acyclic group $H$ such that $G$ is a subgroup of $H$. I am wondering about the dual situation. Is any group $A$ a quotient of an acyclic group $B$ ...
ABC's user avatar
  • 530
8 votes
1 answer
513 views

Equivariant (co)homology of flag manifolds, convolution algebra and nil hecke algebra?

For a complex reductive group $G$ and its Borel subgroup $B$, it seems to be well-known that the equivariant homology group $H^G_*(G/B\times G/B)$ forms a nil-Heck algebra $$NH=\Bbbk[y_i,\partial_{j}]...
Cubic Bear's user avatar
14 votes
2 answers
789 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,...
Francesco Polizzi's user avatar
37 votes
1 answer
1k views

If $A$, $B$ are abelian groups such that $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ for all abelian groups $G$, must $A$ and $B$ be isomorphic?

$\DeclareMathOperator\Hom{Hom}$The question is in the title. If the isomorphism $\Hom(A, G) \cong \Hom(B, G)$ is natural in $G$ then this is just the Yoneda Lemma. If $A$ and $B$ are finitely ...
Carlos Esparza's user avatar
1 vote
0 answers
60 views

Restricted wreath product as fundamental group of a space with coinciding Reidemeister and Nielsen numbers

I am studying a group $\mathbb{Z}_n \wr \mathbb{Z}^k$, where $\wr$ denotes the restricted wreath product: $$ \mathbb{Z}_n \wr \mathbb{Z}^k = \bigoplus_{x\in\mathbb{Z}^k}(\mathbb{Z_n})_x\rtimes\mathbb{...
Michael Freimann's user avatar
1 vote
0 answers
113 views

Question on models for $EG$ for a $G$-CW complex

I am having trouble finding information on a definition in P. Hanham's PhD thesis paper. recall that given a discrete group $G$ a $G$-CW-complex $X$ is a CW-complex equipped with a topological $G$ ...
Dominic Petti's user avatar
5 votes
0 answers
140 views

Reference request: Name or use of this group of diffeomorphisms of the disc

Let $k \in \{0,\infty\}$, $G\subseteq \operatorname{Diff}^k(D^n)$ be the set of diffeomorphisms $\phi:D^n\to D^n$ of the closed $n$-disc $D^n$ (with its boundary) satisfying the following: $ \phi(S_r^...
ABIM's user avatar
  • 5,405
13 votes
1 answer
289 views

Powers of the Euler class, torsion free subgroup of Homeo($S^1$)

For any subgroup $G$ of $\text{Homeo}(S^1)$, we have the Euler class $\chi$ in the group cohomology $H^2(G;\mathbb{Z})$. One can think of this class as the pullback of the generator of $H^2(\mathrm{B}\...
Sam Nariman's user avatar
  • 1,003
7 votes
2 answers
494 views

How does the Steenrod algebra act on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb{F}_p)$?

Let $p$ be an odd prime. The $\mathbb F_p$ cohomology of the cyclic group of order $p$ is well-known: $\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$ where $\xi$ has degree 1, $x$ has ...
Theo Johnson-Freyd's user avatar
12 votes
1 answer
522 views

Realizing inner automorphisms on Eilenberg-MacLane spaces

Let $G$ be a discrete group and let $(X,x_0)$ be a based Eilenberg-MacLane space for $G$, so there is a fixed isomorphism $\pi_1(X,x_0) = G$ and the universal cover $\widetilde{X}$ is contractible. ...
Lisa's user avatar
  • 225
15 votes
1 answer
512 views

fundamental groups of complements to countable subsets of the plane

This question is a follow-up of this MSE post and a comment by Henno Brandsma: Question 1. Let $S$ be the set of isomorphism classes of fundamental groups $\pi_1(E^2 - C)$, where $C$ ranges over all ...
Moishe Kohan's user avatar
  • 12.3k
14 votes
2 answers
906 views

Acyclic group and finite CW-complex

Is there a nontrivial example of an acyclic group $G$ such that its corresponding Eilenberg space $K(G,1)$ is homotopy equivalent to a finite CW-complex ?
Paris's user avatar
  • 717
4 votes
1 answer
276 views

Shifting the group homology of a topological group?

Let $G$ be a topological group. It has a classifying space $BG$, which has homology groups $H_{*}BG$. Changing the topology of $G$ affects the space $BG$ and hence its homology groups. For example ...
John Greenwood's user avatar
9 votes
1 answer
386 views

Different definitions of formality for groups

Let $X$ be a space with fundamental group $G$. Recall that the de Rham fundamental group of $X$ is the inverse limit of the Malcev completions of the nilpotent truncations of $G$. This has a Lie ...
Tina's user avatar
  • 383
1 vote
0 answers
132 views

Nilpotency of topological groups

A group $G$ is said to be nilpotent if $G$ has a central series of finite length, that is, a series of normal subgroups $$ \{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G $$ ...
Niall Taggart's user avatar
7 votes
0 answers
333 views

Positive instances of the Eilenberg-Ganea conjecture with families

The original Eilenberg-Ganea conjecture, which remains unsettled, can be formulated as: any (discrete) group $G$ of cohomological dimension $\operatorname{cd}(G)=2$ has geometric dimension $\...
Mark Grant's user avatar
  • 35.9k
6 votes
1 answer
218 views

Free linear group actions on spheres with "strong" angle preservation

Suppose $G$ is a finite group and that $\rho: G\rightarrow O(d)$ is a faithful orthogonal representation, with action on $\mathbb{R}^d$ denoted $\cdot$. Let's say that $\rho$ is "strongly" angle ...
Fred's user avatar
  • 157
12 votes
2 answers
583 views

Do there exist acyclic simple groups of arbitrarily large cardinality?

Recall that a group $G$ is acyclic if its group homology vanishes: $H_\ast(G; \mathbb Z) = 0$. Equivalently, $G$ is acyclic iff the space $BG$ is acyclic, i.e. $\tilde H_\ast(BG;\mathbb Z) = 0$. In ...
Tim Campion's user avatar
  • 63.9k
18 votes
2 answers
1k views

Fundamental group of punctured simply connected subset of $\mathbb{R}^2$

(This question is originally from Math.SE where it was suggested that I ask the question here) Let $S$ be a simply connected subset of $\mathbb{R}^2$ and let $x$ be an interior point of $S$, meaning ...
Thomas Browning's user avatar
17 votes
1 answer
998 views

Where should I search for computations of group cohomology rings of not-too-complicated finite groups?

A computation I'm trying to make uses as input the cohomology rings of not-too-complicated finite groups in low degrees, and I'd like to determine where to search for preexisting computations. ...
Arun Debray's user avatar
  • 6,881
3 votes
1 answer
225 views

Quotient of normalizers is the fixed points of a homogeneous space

Let $G$ be a finite group, with subgroups $A \leqslant H$. Is there an isomorphism of $N_G A$-sets (or just sets) $$ N_G A / N_H A \cong (G/H)^A ?$$ This dropped out of some calculations of Mackey ...
David Barnes's user avatar
4 votes
0 answers
96 views

When are extensions of algebraically good groups algebraically good?

Let $G$ be a discrete group. The pro-algebraic completion of $G$ is a pro-algebraic group $G^{\mathrm{alg}}$ together with a morphism $s:G\to G^{\mathrm{alg}}$ which is initial among all morphisms ...
Patrick Elliott's user avatar
8 votes
0 answers
285 views

Are the braid groups good in the sense of Toën?

In this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good ...
Patrick Elliott's user avatar
6 votes
1 answer
422 views

A finite p-group question: can this happen?

Let all groups here be finite $p$--groups. Given $K<H$, let $r(K,H)$ be the smallest $r$ such that there exists a chain of subgroups $K=L_0 \lhd L_1 \lhd \cdots \lhd L_r = H$, such that each $L_i/...
Nicholas Kuhn's user avatar
17 votes
1 answer
1k views

A finite 2-group containing the dihedral group of order 16?

The dihedral group $D_{16}$ of order 16 has a presentation $D_{16}= \langle a,t \ | \ a^2=t^8=atat=e\rangle$. Question: Does there exist a finite 2-group $G$ containing $D_{16}$ as a subgroup, and ...
Nicholas Kuhn's user avatar
15 votes
1 answer
629 views

Characteristic classes of symmetric group $S_4$

For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...
Bob's user avatar
  • 439
4 votes
1 answer
1k views

First homology group of the general linear group

The abelianization of the general linear group $GL(n,\mathbb{R})$, defined by $$GL(n,\mathbb{R})^{ab} := GL(n,\mathbb{R})/[GL(n,\mathbb{R}), GL(n,\mathbb{R})],$$ is isomorphic to $\mathbb{R}^{\times}$....
user's user avatar
  • 323
7 votes
1 answer
2k views

Automorphism group of the special unitary group $SU(N)$

Let us consider the automorphism group of the special unitary group $G=SU(N)$. We know there is an exact sequence: $$ 0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0. $$ For $G=SU(2)...
annie marie cœur's user avatar
3 votes
2 answers
291 views

How many non-isomorphic extensions with kernel $S^1$ and quotient cyclic of order $p$?

I want to determine how many non-isomorphic extensions (as group they are non-isomorphic) are possible of the form $1 \to \mathbb{S}^1 \to G \to (\mathbb{Z}_p)^k \to 1$, where $G$ is a compact lie ...
mathstudent's user avatar
1 vote
1 answer
312 views

Topological Invariants for Group

Let $\mathbf{Grp}$ be the category of groups and $\mathbf{Top}$ be the category of topological spaces. To each group $(G, \circ_G)$, we can associate a topological space $(G,\tau_G)$ the basis for ...
user avatar
1 vote
0 answers
57 views

$\omega$-nilpotent cover of a recurrent surface

Theorem. Any $\omega$-nilpotent cover of a recurrent Riemannian manifold is Liouville. $\omega$-nilpotent ($\Gamma=\bigcup_{i=1}^{\infty}Z_{i}$, $Z_{i}$ normal in $\Gamma$, where $Z_{n+1}$ maps to ...
Yu Feng's user avatar
  • 391
7 votes
1 answer
1k views

Classifying space of semidirect product of groups

Assume that $G$ and $H$ are two groups and $G\rtimes _\phi H$ is their semidirect product. My question is, how does the classifying space $B(G\rtimes_\phi H)$ of $G\rtimes _\phi H$ relate to $BG$ and $...
Totoro's user avatar
  • 2,535
4 votes
0 answers
136 views

Second homology of finitely presented group with free abelianisation

It is known that for a presented group $G=F/N$ we have $$H_2(G;\mathbb{Z}) \cong \frac{[F,F]\cap N}{[F,N]}.$$ In general, the right side seems to be difficult to calculate. I am in the special ...
FKranhold's user avatar
  • 1,623