Question

In Adams, J.F. Infinite Loop Spaces Princ. Univ. Press. page 9 he states Alexander duality theorem

Theorem:[Alexander Duality] $$ H^r(X,G)=H_{n-r+1}(S^n-X,G)$$

for finite CW-complexes with a "nice embedding". That is to say, $S^n-X$ has a CW-complex $Y$ as deformation retract and $X$ is a deformation retract of $S^n-Y$.

My first question is the following: Do you know if every CW-complex admits a nice embedding into some $S^n$? if so, could you give a reference?

Thanks in advance.

Answer 1

1) The answer is yes, at least up to homotopy. This can be found in Wall's paper:

Wall, C. T. C., Classification problems in differential topology---IV. Thickenings. Topology, 1966, 5, 73–94.

Wall argues this in a cell-by-cell way, making use of transversality.

2) Alternatively, if X is a finite simplicial complex, then one can find an actual embedding by general position arguments. See Rourke and Sanderson's book for this. Note it is classical that every finite CW complex is homotopy equivalent to a finite simplicial complex.

Answer 2

Let $X$ be a finite CW-complex. Then there is an embedding $X\to\mathbb{R}^n$ for some $n$. To show this it suffices to consider the case when $X$ is obtained from a CW-complex $Y\subset B^m\subset \mathbb{R}^m$ by attaching a cell: $X=Y\cup_f B^k$ where $B^m$ is the unit ball in $\mathbb{R}^m$ and $f:S^{k-1}\to Y$ is the attaching map.

Let us show that there is an embedding $i:B^m\sqcup B^k\to \mathbb{R}^n$ for some $n$ so that, given $x',x''\in B^m$ and $y',y''\in B^k$, the segments $[i(x'),i(y')]$ and $[i(x''),i(y'')]$ do not intersect unless $x=x''$ or $y'=y''$. Indeed, let's assume both balls live in some $\mathbb{R}^r$ and take $\mathbb{R}^n$ to be the space of polynomials $\mathbb{R}^r\to\mathbb{R}$ of sufficiently high degree (4 will do). Then for any 4 points in $\mathbb{R}^r$ the conditions that an element of $V$ are zero at those points are linearly independent. So for any $x\in B^m\sqcup B^k$ take $i(x)\in V^*$ to be the evaluation function $p\mapsto p(x)$.

Now set $X'$ to be the union of $i(Y), i(B^k)$ and all segments $[i(x),i(f(x))]$ for $x\in S^{k-1}$. there is a natural bijective continuous map $X\to X'$, which is a homeomorphism since $X$ is compact and $\mathbb{R}^n$ is Hausdorff.

Remarks:

1. From this construction it is clear that one can construct an embedding of $X$ that is smooth on the interior of each cell. So by slightly modifying the proof of Whitney's embedding theorem we see that $X$ can be embedded in $\mathbb{R}^{2\dim X+1}$. However, the strong Whitney theorem does not hold. E.g. there are 2-polyhedra that can't be embedded in $\mathbb{R}^4$, see e.g. theorem 3 in this paper http://www.fmf.uni-lj.si/~repovs/clanki/2001/OnReSk01.pdf

2. If $X$ is not assumed finite then I would guess that a necessary and sufficient condition for $X$ to be properly embeddable in some $\mathbb{R}^n$ is that $X$ should be locally finite and should have at most countably many cells.