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I have an immersion of a 2-simplicial complex S in $\mathbb{R}^3$, and then a piecewise linear motion of that immersion over an interval of time [0,1].

Is there an existing name for the map $f:S\times[0,1]\to\mathbb{R}^3\times[0,1]$?

Update: here is a more detailed definition of f. Let $i_t$ be the immersion of S in $\mathbb{R}^3$ parameterized over the piecewise linear motion: $i:S\times[0,1]\to\mathbb{R}^3$. Now, extrude $\mathbb{R}^3$ into a space-time $\mathbb{R}^3\times[0,1]$. Then the map above that I'm interested in is defined as $f(x,t)=(i_t(x),t)$.

Calling the map a homotopy seems incorrect because then the codomain should really be $\mathbb{R}^3$. I'm interested in looking at the critical phenomena in a Morse theory or singularity theory sense, though I'm relatively ignorant of those fields. Perhaps there is some standard terminology to use from there.

Another possibility which came to mind was thinking of this swept immersion as a cobordism, although that didn't seem quite right since I care about the temporal ordering and resulting causality between the critical phenomena. (e.g. collisions)

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What is the map f? –  Richard Kent Jun 11 '10 at 22:16
Questions: I suppose that $f$ followed by the projection to [0.1] is not just the projection to [0,1]? Is the map $f$ itself an immersion? What do you mean by a PL map being an immersion? Is it in the strict sense -- locally flat? –  Tom Goodwillie Jun 11 '10 at 22:24
Looking again, I see that you did not say S was a manifold, so locally flat would be meaningless. –  Tom Goodwillie Jun 11 '10 at 23:01
I see two questions here: (1) how to precisely distinguish between $f$ and $i$, though they contain equivalent information; (2) what name indicates that $i$ is more special than a homotopy? The first is not a very popular distinction, but there may be an answer. The answer to the second is regular homotopy or isotopy, perhaps with an adjective. –  Ben Wieland Jun 12 '10 at 15:38
I don't know of any standard name for f, but given the homotopy i, I would call f the "associated movie." –  Richard Kent Jun 12 '10 at 21:55

2 Answers 2

up vote 3 down vote accepted

The map $f$ is usually called a level-preserving homotopy (or level-preserving regular homotopy if each $i_t$ is an immersion). At least this seems to be the accepted terminology in Geometric Topology.

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Thanks Mark. Turns out I haven't written the paper yet, so this has been very helpful. –  Gilbert Bernstein Nov 20 '11 at 6:30
Actually, I realized that because I'm actually going to be working in the PL category (with triangle meshes immersed in $R^3$) it might be somewhat misleading to use the smooth terminology. I'm pretty sure I allow for the surface to change its regular homotopy class, since I allow for an arbitrary motion of the vertices, provided it maintains some basic general position conditions. –  Gilbert Bernstein Nov 20 '11 at 6:36
(err, sorry for the excessive comments) I guess I mean to say that the motion is almost always an immersion except for at isolated instants in time. –  Gilbert Bernstein Nov 20 '11 at 6:43
Hi Gilbert. I believe this terminology is used in whatever category (PL, smooth, topological). Also, another name sometimes used for $f$ is the trace of the homotopy $i_t$. –  Mark Grant Nov 20 '11 at 8:36

Looking at the comments on the question, it doesn't seem like this object $f$ is sufficiently interesting to have a standard name.

For the project I'm working on, using a name that's intuitive for computer scientists (who work on computer graphics) has taken precedence over using a mathematically appropriate term. So I've taken to calling this the sweep complex, since (a) the object can be described as "the original 2-simplicial complex swept through space-time by some motion" and (b) I use a triangulated (3-simplicial complex) representation of the object

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