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

These are two definitions in page 48 of the book an introduction to manifolds by Loring Tu.

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

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 $n$.

In the last lines of page 48, we reed,

Of course, if a topological manifold has several connected components, it is possible for each component to have a different dimension.

But this is a bit strange for me. If a topological manifold has several connected components and each component has different dimension, then how this manifold can be locally Euclidean space, say for example of dimension $n$? That is, by the above definition of topological manifolad, can a non-connected toplogical space be a topological manifold?

• It seems like Definition 5.1 defines locally Euclidean of dimension n, as opposed to just locally Euclidean, which is used in Definition 5.2. Presumably, the definition of locally Euclidean is the same as Definition 5.1, except it allows $n$ to depend on $p$. That, at least, makes it logically consistent. – Jason DeVito May 9 '17 at 16:13

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.

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 $$[0,0,*]$$.

• Ok but to clarify, Eric Wofsey's answer is correct? – Selene Auckland May 10 at 11:54
• @SeleneAuckland: The disagreement here isn’t a matter of correctness. The issue is that are several slightly different well-established definitions of “manifold” in the literature — no one has emerged as the consensus standard. This answer and Eric’s are making different arguments for which definition is best, which is a somewhat subjective question. In any case it seems clear that both notions — the notion where dimension can vary between components, and the notion where it must be globally fixed — both are natural and useful notions, whichever one you choose to call “manifold”. – Peter LeFanu Lumsdaine May 10 at 18:28
• @PeterLeFanuLumsdaine Thanks for replying. I meant to ask if "As written, the term "locally Euclidean" is in fact not even defined at all" is correct and if Def 5.1' and Def 5.2' are likely what was intended. In that case, that is what I mean by Eric Wofsey's answer's being correct. I'm not saying anything like one definition is better than another definition. Prof Tu is saying that we must allow dimensionless manifolds. My response is " 'Ok' (that definition is acceptable) 'but to clarify' are Eric Wofsey's observations and conjectures of intended definitions 'correct' " ? – Selene Auckland May 11 at 8:28
• I think Eric Wofsey's answer is absolutely correct. It is good to first define "locally Euclidean of dimension $n$" at a point. – Loring Tu May 21 at 21:42
• @LoringTu Thanks, Prof Tu! PS I think you need to tag people by using '@' to notify them. – Selene Auckland May 28 at 23:39

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.

• Kinda agree with "a responsible thing to do in this case is to state clearly that the given definition is nonstandard" or at least there is some responsibility here. As a beginner, I don't really care about the definition, but I really cared about the lack of 1. an explicit definition, 2. emphasis such as with... 3. ...examples of manifolds of non-uniform dimensions, in particular the inconsistency with Exercise 9.1, 4. statements of conventions like "From now on, 'manifold' means 'manifold with dimension' ". – Selene Auckland Aug 9 at 1:06
• "In the example with the fixed-point set one can simply say that each connected component is a manifold." Might we say the same about (some if not all) manifolds with boundary? I think one advantage of defining a manifold (without boundary) to have non-uniform dimension is to describe different unions of manifolds: A circle and an open disk could unite to a manifold with (non-empty) boundary, while a circle and a point could unite to a manifold with non-uniform dimension. – Selene Auckland Aug 9 at 1:09