If X is an n-dimensional metric continuum, then by a classical result of Menger
and Nobeling in dimension theory [21], X can be embedded into (2n + 1)-dimensional Eu-
clidean space JR . Flores [9] showed that this was the best possible result without
putting additional hypotheses on X by exhibiting an n-dimensional compact polyhedron
which does not embed in IR . Whitney [52] showed that if X is an n-dimensional
smooth manifold, then X smoothly embeds in TR .In the 1950's an elaborate obstruc-
tion theory was developed to determine which n-dimensional compact polyhedra could be em-
bedded in JR"n [42]. Whitneyfs techniques can be used to show that if f: X -* V is a
continuous map of a compact n-manifold into a 2n-manifold which induces epimorphisms of
fundamental groups, then f is homotopic to an embedding. There are counterexamples if
the hypothesis on the fundamental groups is not assumed [53].
The techniques of VJhitney were generalized in the 1960fs by many topologists inclu-
ding Haefliger [10],[11],Irwin [22] and Jfadson [20];one of the main results of this
theory is the theorem that if f: X + V is a continuous map of a closed n-manifold X
into a q-^nanifold V , q - n _ 3 , such that f is (2n - q + 1)-connected, then f is
homotopic to an embedding of X into V .
If X is a compact n-dimensional polyhedron, then the analogue of the latter result
is false. However, Stallings [46] and Wall [50]were able to show that if f: X •+ V is
a (2n - q + 1)-connected map of X into a q-manifold V , q - n ^ 3 , then there exists
a compact n-dimensional polyhedron Y C V and a simple homotopy equivalence h: X * Y
such that the diagram
h \ U
is homotopy commutative. We say that X "embeds up to simple homotopy type" in V .
This result, together with the Casson-Sullivan "Theorem [49], yields another proof of the
embedding theorem when X is a manifold.
Received by the editors March 3, 1981.
Presented to the Society November 10, 1979 and November 14,1980.
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