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In algebraic topology, a continuous function between topological spaces is called -connected if it induces isomorphisms on all homotopy groups in degree and epimorphisms in degree .
In older literature this is often called an -equivalence, since an -equivalence in this sense is a weak homotopy equivalence.
In terms of the homotopy theory presented by the classical model structure on topological spaces, an -connected function represents an n-connected morphism in the (∞,1)-topos ∞Gpd.
A map of topological spaces is called -connected (e.g. tomDieck 08, p. 144) or an -equivalence (older literature) if the following equivalent definitions hold:
The induced morphism on homotopy groups is, for all
an isomorphism in degree ;
an epimorphism in degree .
for all and all commutative squares
there exists a map such that and is homotopic to relative to .
Hence an -connected map is a weak homotopy equivalence.
For a map and an integer the following conditions are equivalent.
is -connected.
All homotopy fibers of are -connected.
Let and be maps of spaces.
If and are -connected, then so is .
If is -connected and is -connected, then is -connected.
If is -connected and is -connected, then is -connected.
Let
be a commutative diagram of maps of spaces. If is -connected and and are -connected, then the induced map between homotopy pushouts is -connected.
This is (tom Dieck, Theorem 6.7.9).
Let
be a commutative diagram of maps of spaces. If is -connected and and are -connected, then the induced map between homotopy pullbacks is -connected.
Textbook accounts:
Last revised on August 16, 2021 at 12:51:03. See the history of this page for a list of all contributions to it.