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For finite dimensional manifolds, there is a lot of theory about when the number of intersections (modulo $2$) of certain objects are preserved under homotopy. I'll give two quick examples:

Let $f:X \to Y$ be a smooth map from a compact manifold $X$ to a connected manifold $Y$ of the same dimension. Then if both $x,y \in Y$ are regular values of $f$, we know that $f^{-1}(x), f^{-1}(y)$ are both 0-dimensional manifolds, compact, and so finite sets. In fact the sizes of these sets are the same modulo $2$.

Let $X,Y \subset Z$ be two transverse submanifolds of $Z$, one of which is compact. The intersection $X \cap Y$ is as above a finite set of points. In fact its size is preserved (modulo $2$) under homotopy of $X,Y$ to $X',Y'$, provided we end with transverse submanifolds $X',Y'$.

I was wondering if any of this theory generalises to infinite-dimensional manifolds? Take the above two theorems for example, do they have analogues versions for $X,Y,Z$ banach manifolds? What about frechet manifolds? I'm aware compactness will be an issue, but can we generalise nonetheless, for example if we assume the intersection number is finite?

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One can speak of transersality of intersections in an infinite dimensional context. If one goes beyond Hilbert manifolds (e.g. Banach, Frechet) one needs to be a bit careful with the definition of transverse, because one needs to impose splitting conditions. For a submanifold one typically demands that the tangent space $T_xX$ admits a closed complement. Then it it is not enough to demand that $T_xX +T_xY=T_xM$ at every point of intersection, but one needs that $T_xX\cap T_xY$ is closed and complemented. To be able to ignore this issue, let me assume that $Z$ is a Hilbert manifold and $X,Y\subseteq Z$ are submanifolds.

Then if $X$ is closed (as a subset of $Z$), is of finite codimension $m$, and $Y$ is compact of finite dimension $n$, and the intersection is transverse then $X\cap Y$ is a compact submanifold of dimension $n-m$. Moreover, if $f:Y\rightarrow Z$ denotes the inclusion, and $g$ is homotopic to $f$ and also transverse to $X$ than $g(Y)\cap Z$ is cobordant to $f(Y)\cap Z$. In particular, this means that the mod $2$ intersection number is well-defined in this context. If the normal bundle of $X$ is oriented, and $Y$ is also oriented, everything works with orientations as well.

A nice class of mappings are Fredholm mappings. Smale famously proved if $f:M\rightarrow N$ is a smooth Fredholm mapping of index $k$, that $f$ has regular values (without Fredholm this can be false), and that the preimage of a regular value is a manifold of dimension equal to $k$. If $f$ is a proper map, then this manifold is compact. The cobordism class of the regular value is independent of the regular value, and the proper Fredholm cobordism class. This can be used to distinguish proper Fredholm mappings. Together with Alberto Abbondandolo we upgraded this invariant to a full invariant (infinite dimensional framed cobordism classes) in the case $N$ is the Hilbert space. This is in this paper here:

MR4058178 Prelim Abbondandolo, Alberto; Rot, Thomas O.; On the homotopy classification of proper Fredholm maps into a Hilbert space. J. Reine Angew. Math. 759 (2020), 161–200. 58B15 (47A53 47H11)

We also discuss the framed cobordism classes of non-positive index in this paper. The index one case, for simply connected Hilbert manifolds, is done in our recent preprint.

https://arxiv.org/abs/2005.03936

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This was supposed to be a comment but got too long.

The general result which encompasses both your examples in finite dimensions is the following: If $Y\subseteq Z$ is a submanifold of codimension $k$ and $f:X\to Z$ is a map transverse to $Y$, then $f^{-1}(Y)\subseteq X$ is a submanifold of codimension $k$.

In your second example $f:X\to Z$ is the inclusion, and so for $f^{-1}(Y)=X\cap Y$ to be a finite set you need $\dim(X)+\dim(Y)=\dim(Z)$, i.e. $X$ and $Y$ to have complementary dimensions. If the manifolds are infinite dimensional, this doesn't seem to make sense.

Your first example, where you have a map $f:X\to Z$ and regular values $y,z\in Z$, does generalize to the setting of proper Fredholm maps. MO user Thomas Rot has done some work on this - see these slides of a talk he gave at the Skye conference in 2018. In particular, if the Fredholm index $$ \dim\ker df_x - \dim\operatorname{coker} df_x $$ of each differential $df_x:TX_x\to TZ_{f(x)}$ is $k$ for all $x\in X$, then the pre-image of a regular value is a well-defined $k$-dimensional unoriented cobordism class. When $k=0$ this is an integer mod $2$.

Surely there is more to say, perhaps Thomas himself will come along and answer.

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    $\begingroup$ That is some nice timing! $\endgroup$
    – Thomas Rot
    Jul 30 '20 at 9:55
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    $\begingroup$ @ThomasRot Ha ha! Nice timing indeed! Now, perhaps someone will deposit £1,000,000 into my account... $\endgroup$
    – Mark Grant
    Jul 30 '20 at 9:57
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    $\begingroup$ I can do an upvote... $\endgroup$
    – Thomas Rot
    Jul 30 '20 at 19:39
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Not exactly what you are after, but for ($C^1$) Fredholm maps of index 0 there is a modulo 2 mapping degree. This can be generalized to an integer-valued degree only for so-called oriented Fredholm maps (of index 0).

For the case that the image manifold is actually a Banach space, there is also a degree for compact and certain non-compact and actually even multivalue perturbations of such maps.

The integer-valued degree (in the oriented case) was developed by M. Furi and P. Benevieri (and for a different notion of orientation also by V. G. Zvyagin and N. M. Ratiner), while I am not sure who was the first to note the much simpler modulo 2 case.

More details and history are in my monograph M. Väth, Topological Analyis, De Gruyter, Berlin, New York 2012.

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  • $\begingroup$ The modulo two case goes back to Cacciopoli probably. Smale also defined it. $\endgroup$
    – Thomas Rot
    Dec 25 '20 at 23:23
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    $\begingroup$ For the history, one probably has to distinguish the $C^2$ and $C^1$ case: The $C^2$ case is completely analogous to the finite-dimensional case. But the $C^1$ case does not follow by approximation as in finite dimensions but requires a rather different approach. My guess is that it was indeed one of the 4 mentioned authors who did the $C^1$ case first. $\endgroup$ Dec 26 '20 at 8:16

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