# Configuration spaces are not homotopy invariant

Дата публикации: 2018-05-27 15:22

Еще видео на тему «Spaces of homotopy self equivalences a survey»

It is generally difficult to show that two spaces are homotopy equivalent. Some of the better spaces are CW complexes, where the Whitehead theorem holds.

## Homotopy theory and classifying spaces

The proof uses no notions of homotopy groups, but the result was found by starting with the well known fact that a homotopy equivalence $f: Y \to Z$ of spaces induces an isomorphism of homotopy groups, and then generalising replacing $(S^n,x)$ by a pair $(X,A)$ with the Homotopy Extension Property. The theorem first appeared in the 6968 edition of the book now titled "Topology and Groupoids" . Its origin is also in special cases due to . Whitehead. There is also a dual "cogluing theorem", for fibrations and pullbacks, rather than cofibrations and pushouts. The result has also been set up in various model categories, but an advantage of the original proof is that it gives control of the homotopies involved.

### Algebraic topology - Homotopy type of a space

The simplicial set for is also meaningful for any (not necessarily Abelian) group . The simplicial set thus obtained is the standard simplicial resolution of .

I was wondering what tools of algebraic topology are usually used to show that some things have the same homotopy type? Hatcher doesn''t really talk about this in his book even though he defines the concept on page 8. Of course we can compute the homology or homotopy groups of a space, but just showing that they agree is not enough as far as I know.

A class of homotopy-equivalent topological spaces. Two mappings and are said to be mutually-inverse homotopy equivalences if and . If only the first condition is met, is said to be a homotopy monomorphism and is said to be a homotopy epimorphism. A mapping is a homotopy equivalence if and only if it is both a homotopy monomorphism and a homotopy epimorphism. If there exists a homotopy epimorphism , then one says that dominates . If there exists a homotopy equivalence , then and are said to be homotopy equivalent, or spaces of the same homotopy type.

We present a counterexample to the conjecture on the homotopy invariance of configuration spaces. More precisely, we consider the lens spaces L 7 , 6 and L 7 , 7 , and prove that their configuration spaces are not homotopy equivalent by showing that their universal coverings have different Massey products.

The article above concentrates on one aspect of homotopy types, viz. Postnikov towers (or Postnikov decompositions) and gives a great deal of detail concerning the state of affairs in the 6955''s.

The class has the form , where is the fundamental class. The next class has the property that on the fibre of the fibration it cuts out the class , and it is uniquely determined by it. Similarly, the class is uniquely characterized by the fact that, by reducing modulo 7, it transforms to the class . The classes and vanish and the class is uniquely characterized by the fact that on the fibre of the fibration it cuts out the class . Finally, is characterized by the fact that, by reducing modulo 7, it transforms to the class .

$$\{ \chi \text{ and orientability} \} \leftrightarrow \{ \text{homotopy types of closed surfaces} \}$$ but this is very rare.

Theorem. Suppose that $X, Y$ are nilpotent spaces with homotopy groups that are finite dimensional rational vector spaces. Then the following are equivalent for a map $f: X \rightarrow Y$: