Ten papers published by Geometry & Topology Monographs


Geometry and Topology Publications is pleased to announce commencement
of publication of G&T Monograph number 13:


Groups, homotopy and configuration spaces (Tokyo 2005)

Editors: Norio Iwase, Toshitake Kohno, Ran Levi, Dai Tamaki and Jie Wu


This volume is the proceedings of the conference "Groups, Homotopy
and Configuration Spaces" held at the University of Tokyo, July 5-11,
2005, in honor of the 60th birthday of Fred Cohen. The emphasis of the
conference was on cohomology of groups, classical and modern homotopy
theory, geometry and topology of configuration spaces and related
topics. However, the conference was intended to have a broad scope,
with talks on a variety of topics of current interests in topology. The
organizing committee consisted of Norio Iwase, Toshitake Kohno, Ran Levi,
Dai Tamaki and Jie Wu. The conference was supported by the COE program
of the Graduate School of Mathematical Sciences, The University of Tokyo.


The proceedings will consist of twenty-three papers, of which the first
ten have now been published:

(1) Geometry & Topology Monographs 13 (2008) 1-10
String topology of Poincare duality groups
by Hossein Abbaspour, Ralph Cohen and Kate Gruher
URL: http://www.msp.warwick.ac.uk/gtm/2008/13/p001.xhtml
DOI: 10.2140/gtm.2008.13.1

(2) Geometry & Topology Monographs 13 (2008) 11-40
Computation of the homotopy of the spectrum tmf
by Tilman Bauer
URL: http://www.msp.warwick.ac.uk/gtm/2008/13/p002.xhtml
DOI: 10.2140/gtm.2008.13.11

(3) Geometry & Topology Monographs 13 (2008) 41-83
A family of embedding spaces
by Ryan Budney
URL: http://www.msp.warwick.ac.uk/gtm/2008/13/p003.xhtml
DOI: 10.2140/gtm.2008.13.41

(4) Geometry & Topology Monographs 13 (2008) 85-104
Cohomology of Artin groups of type ~A_n, B_n and applications
by Filippo Callegaro, Davide Moroni and Mario Salvetti
URL: http://www.msp.warwick.ac.uk/gtm/2008/13/p004.xhtml
DOI: 10.2140/gtm.2008.13.85

(5) Geometry & Topology Monographs 13 (2008) 105-146
The boundary manifold of a complex line arrangement
by Daniel C Cohen and Alexander I Suciu
URL: http://www.msp.warwick.ac.uk/gtm/2008/13/p005.xhtml
DOI: 10.2140/gtm.2008.13.105

(6) Geometry & Topology Monographs 13 (2008) 147-168
Basis-conjugating automorphisms of a free group
and associated Lie algebras
by F R Cohen, J Pakianathan, V V Vershinin and J Wu
URL: http://www.msp.warwick.ac.uk/gtm/2008/13/p006.xhtml
DOI: 10.2140/gtm.2008.13.147

(7) Geometry & Topology Monographs 13 (2008) 169-193
On braid groups and homotopy groups
by F R Cohen and J Wu
URL: http://www.msp.warwick.ac.uk/gtm/2008/13/p007.xhtml
DOI: 10.2140/gtm.2008.13.169

(8) Geometry & Topology Monographs 13 (2008) 195-201
Odd-primary homotopy exponents of compact simple Lie groups
by Donald M Davis and Stephen D Theriault
URL: http://www.msp.warwick.ac.uk/gtm/2008/13/p008.xhtml
DOI: 10.2140/gtm.2008.13.195

(9) Geometry & Topology Monographs 13 (2008) 203-227
Filtering the fiber of the pinch map
by Brayton Gray
URL: http://www.msp.warwick.ac.uk/gtm/2008/13/p009.xhtml
DOI: 10.2140/gtm.2008.13.203

(10) Geometry & Topology Monographs 13 (2008) 229-259
Homotopy algebra of open-closed strings
by Hiroshige Kajiura and Jim Stasheff
URL: http://www.msp.warwick.ac.uk/gtm/2008/13/p010.xhtml
DOI: 10.2140/gtm.2008.13.229

Abstracts follow

(1) String topology of Poincare duality groups
by Hossein Abbaspour, Ralph Cohen and Kate Gruher

Let G be a Poincare duality group of dimension n. For a given element
g in G, let C_g denote its centralizer subgroup. Let L_G be the graded
abelian group defined by (L_G)_p = oplus_{[g]}H_p+n(C_g) where
the sum is taken over conjugacy classes of elements in G. In this paper
we construct a multiplication on L_G directly in terms of intersection
products on the centralizers. This multiplication makes L_G a graded,
associative, commutative algebra. When G is the fundamental group of
an aspherical, closed oriented n-manifold M, then (L_G)_* = H_*+n(LM),
where LM is the free loop space of M. We show that the product on L_G
corresponds to the string topology loop product on H_*(LM) defined by
Chas and Sullivan.


(2) Computation of the homotopy of the spectrum tmf
by Tilman Bauer

This paper contains a complete computation of the homotopy ring of
the spectrum of topological modular forms constructed by Hopkins and
Miller. The computation is done away from 6, and at the (interesting)
primes 2 and 3 separately, and in each of the latter two cases, a sequence
of algebraic Bockstein spectral sequences is used to compute the E_2
term of the elliptic Adams-Novikov spectral sequence from the elliptic
curve Hopf algebroid. In a further step, all the differentials in the
latter spectral sequence are determined. The result of this computation
is originally due to Hopkins and Mahowald (unpublished).


(3) A family of embedding spaces
by Ryan Budney

Let Emb(S^j,S^n) denote the space of C^infty-smooth embeddings of
the j-sphere in the n-sphere. This paper considers homotopy-theoretic
properties of the family of spaces Emb(S^j,S^n) for n >= j > 0. There is
a homotopy-equivalence of Emb(S^j,S^n) with SO_{n+1} times_{SO_{n-j}}
K_{n,j} where K_{n,j} is the space of embeddings of R^j in R^n which
are standard outside of a ball. The main results of this paper are
that K_{n,j} is (2n-3j-4)-connected, the computation of pi_{2n-3j-3}
(K_{n,j}) together with a geometric interpretation of the generators.
A graphing construction Omega K_{n-1,j-1} --> K_{n,j} is shown to induce
an epimorphism on homotopy groups up to dimension 2n-2j-5. This gives
a new proof of Haefliger's theorem that pi_0 (Emb(S^j,S^n)) is a group
for n-j>2. The proof given is analogous to the proof that the braid
group has inverses. Relationship between the graphing construction
and actions of operads of cubes on embedding spaces are developed.
The paper ends with a brief survey of what is known about the spaces
K_{n,j}, focusing on issues related to iterated loop-space structures.


(4) Cohomology of Artin groups of type ~A_n, B_n and applications
by Filippo Callegaro, Davide Moroni and Mario Salvetti

We consider two natural embeddings between Artin groups: the group
G_{tilde{A}_{n-1}} of type tilde{A}_{n-1} embeds into the group 1G_{B_n}
of type B_n; G_{B_n} in turn embeds into the classical braid group
Br_{n+1}:=G_{A_n} of type A_n. The cohomologies of these groups are
related, by standard results, in a precise way. By using techniques
developed in previous papers, we give precise formulas (sketching the
proofs) for the cohomology of G_{B_n} with coefficients over the module
Q[q^{+-1},t^{+-1}], where the action is (-q)-multiplication for the
standard generators associated to the first n-1 nodes of the Dynkin
diagram, while is (-t)-multiplication for the generator associated to
the last node.

As a corollary we obtain the rational cohomology for G_{tilde{A}_n}
as well as the cohomology of Br_{n+1} with coefficients in the
(n+1)-dimensional representation obtained by Tong, Yang and Ma.

We stress the topological significance, recalling some constructions
of explicit finite CW-complexes for orbit spaces of Artin groups. In
case of groups of infinite type, we indicate the (few) variations to
be done with respect to the finite type case. For affine groups, some
of these orbit spaces are known to be K(pi,1) spaces (in particular,
for type tilde{A}_n).

We point out that the above cohomology of G_{B_n} gives (as a module over
the monodromy operator) the rational cohomology of the fibre (analog to
a Milnor fibre) of the natural fibration of K(G_{B_n},1) onto the 2-torus.


(5) The boundary manifold of a complex line arrangement
by Daniel C Cohen and Alexander I Suciu

We study the topology of the boundary manifold of a line arrangement in
CP^2, with emphasis on the fundamental group G and associated invariants.
We determine the Alexander polynomial Delta(G), and more generally, the
twisted Alexander polynomial associated to the abelianization of G and
an arbitrary complex representation. We give an explicit description
of the unit ball in the Alexander norm, and use it to analyze certain
Bieri-Neumann-Strebel invariants of G. From the Alexander polynomial,
we also obtain a complete description of the first characteristic variety
of G. Comparing this with the corresponding resonance variety of the
cohomology ring of G enables us to characterize those arrangements for
which the boundary manifold is formal.


(6) Basis-conjugating automorphisms of a free group
and associated Lie algebras
by F R Cohen, J Pakianathan, V V Vershinin and J Wu

Let F_n = <x_1,...,x_n> denote the free group with generators
{x_1,...,x_n}. Nielsen and Magnus described generators for the kernel
of the canonical epimorphism from the automorphism group of F_n to the
general linear group over the integers. In particular among them are the
automorphisms chi_{k,i} which conjugate the generator x_k by the generator
x_i leaving the x_j fixed for j not k. A computation of the cohomology
ring as well as the Lie algebra obtained from the descending central
series of the group generated by chi_{k,i} for i<k is given here. Partial
results are obtained for the group generated by all chi_{k,i}.


(7) On braid groups and homotopy groups
by F R Cohen and J Wu

This article is an exposition of certain connections between the braid
groups, classical homotopy groups of the 2-sphere, as well as Lie algebras
attached to the descending central series of pure braid groups arising
as Vassiliev invariants of pure braids. Natural related questions are
posed at the end of this article.


(8) Odd-primary homotopy exponents of compact simple Lie groups
by Donald M Davis and Stephen D Theriault

We note that a recent result of the second author yields upper bounds
for odd-primary homotopy exponents of compact simple Lie groups which
are often quite close to the lower bounds obtained from v_1-periodic
homotopy theory.

(9) Filtering the fiber of the pinch map
by Brayton Gray

A new type of Hopf invariant is described for the fiber of the pinch map:

F --> X union CA --> SA

and this is used to study the boundary map in the fibration sequence of
Cohen, Moore and Neisendorfer:

Omega^2 S^{2n+1} --> Omega F_n --> Omega P^{2n+1} --> Omega S^{2n+1}

The boundary map is shown to be compatible with the Hopf invariant and
a filtration of the spliting is obtained.

(10) Homotopy algebra of open-closed strings
by Hiroshige Kajiura and Jim Stasheff

This paper is a survey of our previous works on open-closed homotopy
algebras, together with geometrical background, especially in terms of
compactifications of configuration spaces (one of Fred's specialities)
of Riemann surfaces, structures on loop spaces, etc. We newly present
Merkulov's geometric A_infty-structure [Internat. Math. Res. Notices
(1999) 153--164] as a special example of an OCHA. We also recall
the relation of open-closed homotopy algebras to various aspects of
deformation theory.

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