Atlas: A reduction of Whitehead Asphericity Conjecture by K. Salikhov

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Low-Dimensional Topology and Combinatorial Group Theory
July 31 - August 7, 1999
Chelyabinsk State University
Chelyabinsk, Russia

Organizers
Sergei V. Matveev

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A reduction of Whitehead Asphericity Conjecture
by
K. Salikhov
Moscow State University

In 1941 Whitehead stated the following problem: Ïs any subcomplex K of an aspherical 2-dimensional CW-complex L itself aspherical?" In spite of elementarity of this question, there was no significant progress in order to answer it. There are some partial results for the special cases of this problem (e.g. the answer is "Yes", if \pi₁(L) is trivial or finite). Also, it is known that Whitehead Conjecture is equivalent to some purely-algebraic problem on finitely generated free groups. For more information on the history of Whitehead Conjecture we refer the reader to a remarkable survey [, ].

Here we suggest a new, geometrical, point of view on the Whitehead Conjecture, which, may be, will be more productive. The tools we are using here already had played great role in different problems of geometric topology (special polyhedra was used by Casler and Matveev to code PL-manifolds and cell-like resolution was used by Cannon to solve the double suspension problem). Let us introduce some notation.

Definition. [, ]. Let P and Q be polyhedra. A map f:Q --> P is called collapsible if the preimages of points under f are collapsible polyhedra. A collapsible onto map f:Q --> P is called collapsible resolution of P.

Collapsible maps are the PL-analogue of (TOP-)cell-like maps.

Definition. [, ], [, ]. A 2-polyhedron is called a fake surface if any of its points has a neighborhood, homeomorphic either to a 2-disk, a book with 3 sheets of to a cone over 1-skeleton of 3-simplex. A fake surface P is called a special polyhedron if P-P' is a disjoint union of open 2-disks (here P' is a non-manifold set of the polyhedron P).

Special polyhedra have in some sense the ``simplest'' stable singularities. In [, ] it is proven, that any dimensionally-homogenius 2-polyhedron P such that P-P' is a disjoint union of open 2-disks, admits a collapsible resolution f:Q --> P with Q a special polyhedron. Using the slight modification of this result and the fact that a collapsible map is a homotopy equivalence on its image, we prove the following

Theorem. In order to prove or disprove the Whitehead Conjecture, it is suffices to consider only those aspherical 2-dimensional CW-complices L, which are fake surfaces and L-L' is a disjoint union of open 2-disks and open Möebius bands.

I am grateful to S.V. Matveev [, ] for his remark on possibility of this theorem.

References

[]: B.G. Casler, An imbedding theorem for connected 3-manifolds with boundary, Proc. Amer. Math. Soc., 16, (1965), 559-566
[]: S.V. Matveev Special skeletons of PL manifolds, Mat. Sbornik, 92 (1973), 287-293, (In Russian)
[]: W.A. Bogley, J.H.C. Whitehead Asphericity Question, Two-dimensional Homotopy and Combinatorial Group Theory (C. Hog-Angeloni, W. Metzler and A.J. Sieradski, eds.), London Math. Soc., Lecture Notes Series 197 (1993), 309-334
[]: S.V. Matveev, private communication, Moscow, 1998
[]: D. Repovs and K. Salikhov, On cell-like resolutions of 2-polyhedra by special ones, Univ. of Ljubljana, preprint (1998)

Date received: June 15, 1999