Can a topological manifold be non-connected and each component with different dimension?

Solution 1:

As written, the term "locally Euclidean" is in fact not even defined at all (only "locally Euclidean of dimension $n$" is defined). What it appears the author really intended is the following pair of definitions:

Definition 5.1'. A topological space $M$ is locally Euclidean of dimension $n$ at a point $p\in M$ if $p$ has a neighborhood $U$ such that there is a homeomorphism $\phi$ from $U$ onto an open subset of $\mathbb R^n$. A topological space $M$ is locally Euclidean if for each $p\in M$, there exists $n$ such that $M$ is locally Euclidean of dimension $n$ at $p$.

Definition 5.2'. A topological manifold is a Hausdorff, second countable, locally Euclidean space. It is said to be of dimension $n$ if it is locally Euclidean of dimension $n$ at every point.

I would add, however, that this definition is not very standard. Most people define manifolds such that they must have the same dimension at every point, even if they are disconnected.

Solution 2:

We must allow a manifold to have connected components of different dimensions because such an object occurs naturally. For example, there is a theorem that the fixed point set of a compact Lie group acting smoothly on a manifold is a manifold (L. Tu, \textit{Introductory Lectures on Equivariant Cohomology}, Annals of Mathematics Studies, Princeton University Press, Th. 25.1, forthcoming). Now consider the action of the circle $S^1$ on the complex projective space $\mathbb{C}P^2$ by $$ \lambda \cdot [z_0, z_1, z_2] = [z_0, z_1, \lambda z_2]. $$ The fixed point set of this action has two connected components, the line $[*,*,0]$ and the single point $[0,0,*]$.

Solution 3:

This is an addendum to Eric's answer. I checked a fair number of books on topology and differential geometry. All but one (Lang's "Differential Manifolds") define manifolds in such a way that all connected components of a manifold have the same dimension (i.e. they define $n$-dimensional manifolds rather than just "manifolds"). Lang's definition is meant to be as general as possible (for instance, he does not assume Hausdorfness): Lang defines manifolds modeled on arbitrary Banach vector spaces, so, in a way, it makes sense for him to allow for different local models.

Remark. I also checked Veblen and Whitehead "Foundations of differential geometry" (first published in 1932), which is the first place where manifolds were rigorously defined (using an atlas of charts with transition maps that belong to a given pseudogroup). However, given their archaic terminology, I find it hard to tell what they meant.

Here is the list of other books that I checked (most are widely regarded as standard references in geometry and topology):

• Kobayashi, Nomizu "Foundations of differential geometry".

• Klingenberg, Gromoll, Meyer, "Riemannische Geometrie im Grossen".

• Helgason, "Differential geometry, Lie groups and symmetric spaces".

• do Carmo, "Riemannian Geometry".

• Bishop and Crittenden, "Geometry of manifolds".

• de Rham, "Differentiable Manifolds".

• Milnor "Topology from the differentiable viewpoint".

• Guillemin and Pollack, "Differential Topology".

• Hirsch, "Differential Topology".

• Lee, "Differential manifolds".

• Lee, "Topological manifolds".

• Hatcher, "Algebraic Topology".

• Massey, "A basic course in algebraic topology".

• Eilenberg, Steenrod, "Foundations of Algebraic Topology".

• Munkres, "Topology".

I stopped at that point.

It is quite clear (say, by looking at this list) that the standard definition is to require a manifold to have constant dimension. Of course, an author is free to give a nonstandard definition, but a responsible thing to do in this case is to state clearly that the given definition is nonstandard. I disagree with Tu's sentiment that

We must allow a manifold to have connected components of different dimensions because such an object occurs naturally.

There are many things which occur naturally. For instance, quotient spaces of finite group actions on manifolds also occur naturally but nobody (as far as I know) calls them manifolds (instead, people call them V-manifolds, orbifolds, stacks...). In the example with the fixed-point set one can simply say that each connected component is a manifold.