What are the 3-dimensional subspaces (or quotient spaces) to which the projections are made in the given figures? (Topological Data Analysis)

[rlamesch]

I've just started my mathematics thesis on Topological Data Analysis last week and I was given a question to resolve for next week:

"What are the 3-dimensional subspaces to which the projections are made in figure A? Are they really subspaces or are they quotient spaces?"

• These are the figures which are mentioned:

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• A link to where the figures are from (p.10): Single-cell transcriptomics reveals distinct
  inflammation-induced microglia signatures
  

I am actually not sure how to proceed to resolve this question. I've skimmed over the paper and I didn't spot anything helpful, except for the figure description:

"leads to nine distinct clusters in a two-dimensional state space inferred by generalized regression modelling"

1. Why do they talk about a two-dimensional state space. The question mentions a three-dimensional space and on the figure we see 3 axes.
2. Using some method we project an n-dimensional space to a 2 (or 3)-dimensional space. What does he mean by "subspaces/quotient spaces to which the projections are made" How would I check this?
3. Do I need to find out how the branching analysis or Generalized Regression Modelling works? By G.R.M. do they mean this: Generalized linear model?

I've been told to ask for further clarification on this problem, but my supervisor is currently travelling and hasn't responded to my e-mails yet. So, I'm just hoping that someone might be able to assist before my next meeting. Thanks for any help given.

Best,
Rik