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| A Survey of the Homotopy Properties of Inclusion of Certain Types of Configuration Spaces into the Cartesian Product |
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Citation： |
Daciberg Lima GONCALVES,John GUASCHI.A Survey of the Homotopy Properties of Inclusion of Certain Types of Configuration Spaces into the Cartesian Product[J].Chinese Annals of Mathematics B,2017,38(6):1223~1246 |
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Net amount： 1595 |
Authors： |
Daciberg Lima GONCALVES; John GUASCHI |
Foundation： |
The work in this paper was partially supported by the
CNRS/FAPESP programme n\textsuperscript{o}~226555 (France)
and n\textsuperscript{o}~2014/50131-7 (Brazil). The first author would like to thank FAPESP~--~Funda\c{c}\~ao de
Amparo a Pesquisa do Estado de S\~ao Paulo, Projeto Tem\'atico Topologia Alg\'ebrica, Geom\'etrica 2012/24454-8 (Brazil)
for partial support, and the Institute for Mathematical Sciences, National University of Singapore, for partial support
and an invitation to attend the Combinatorial and Toric Homotopy Conference
(Singapore, 23\textsuperscript{rd}--29\textsuperscript{th} August
2015) in honour of Frederick Cohen. |
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| Abstract： |
Let $X$ be a topological space. In this survey the authors consider
several types of configuration spaces, namely, the classical (usual)
configuration spaces $F_n(X)$ and $D_n(X)$, the orbit configuration
spaces $F_n^G(X)$ and $F_n^G(X)/\sn$ with respect to a free action
of a group $G$ on $X$, and the graph configuration spaces
$F_n^{\Gamma}(X)$ and $F_n^{\Gamma}(X)/H$, where $\Gamma$ is a graph
and $H$ is a suitable subgroup of the symmetric group $\sn$. The
ordered configuration spaces $F_n(X)$, $F_n^G(X)$, $F_n^{\Gamma}(X)$
are all subsets of the $n$-fold Cartesian product
$\prod\limits_1^n\, X$ of $X$ with itself, and satisfy
$F_n^G(X)\subset F_n(X) \subset F_n^{\Gamma}(X)\subset
\prod\limits_1^n\, X$. If $A$ denotes one of these configuration
spaces, the authors analyse the difference between $A$ and
$\prod\limits_1^n\, X$ from a topological and homotopical point of
view. The principal results known in the literature concern the
usual configuration spaces. The authors are particularly interested
in the homomorphism on the level of the homotopy groups of the
spaces induced by the inclusion $\map{\iota}{A}[\prod\limits_1^n\,
X]$, the homotopy type of the homotopy fibre $I_{\iota}$ of the map
$\iota$ via certain constructions on various spaces that depend on
$X$, and the long exact sequence in homotopy of the fibration
involving $I_{\iota}$ and arising from the inclusion $\iota$. In
this respect, if $X$ is either a surface without boundary, in
particular if $X$ is the $2$-sphere or the real projective plane, or
a space whose universal covering is contractible, or an orbit space
$\St[k]/G$ of the $k$-dimensional sphere by a free action of a Lie
group $G$, the authors present recent results obtained by themselves
for the first case, and in collaboration with Golasi\'nski for the
second and third cases. The authors also briefly indicate some older
results relative to the homotopy of these spaces that are related to
the problems of interest. In order to motivate various questions,
for the remaining types of configuration spaces, a few of their
basic properties are described and proved. A list of open questions
and problems is given at the end of the paper. |
Keywords： |
Configuration space, Equivariant configuration space, Fibration,& Homotopy fibre, $K(\pi, 1)$ space, Braid groups |
Classification： |
55P15, 55R80, 20F36, 55Q40, 55R05 |
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