| Title:
|
Free loop spaces and cyclohedra (English) |
| Author:
|
Markl, Martin |
| Language:
|
English |
| Journal:
|
Proceedings of the 22nd Winter School "Geometry and Physics" |
| Volume:
|
|
| Issue:
|
2002 |
| Year:
|
|
| Pages:
|
151-157 |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an $A_\infty$-operad. The classical model for such an operad consists of Stasheff's associahedra. The present paper describes a similar recognition principle for free loop spaces. Let ${\cal P}$ be an operad, $M$ a ${\cal P}$-module and $U$ a ${\cal P}$-algebra. An $M$-trace over $U$ consists of a space $V$ and a module homomorphism $T:M\to\text{End}_{U,V}$ over the operad homomorphism ${\cal P}\to\text{End}_U$ given by the algebra structure on $U$. Let ${\cal C}_1$ be the little 1-cubes operad.\par The author shows that the free loop space $\wedge X$ is a trace over the ${\cal C}_1$-space $\Omega X$. This trace is related to the cyclohedra in a way similar to the relation of ${\cal C}_1$ to the associahedra. Given a ${\cal P}$-module $M$ and a ${\cal P}$-algebra $U$ one can define the free $M$-trace over $U$ like one can construct free ${\cal P}$-al! (English) |
| MSC:
|
18D50 |
| MSC:
|
55P48 |
| idZBL:
|
Zbl 1032.55006 |
| idMR:
|
MR1982442 |
| . |
| Date available:
|
2009-07-13T21:49:30Z |
| Last updated:
|
2012-09-18 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/701714 |
| . |
