Topology of the Spaces of Functions with Prescribed Singularities on Surfaces
Let be a smooth connected orientable closed surface and a function having only critical points of the -types, . Let be the set of functions having the same types of local singularities as those of . We describe the homotopy type of the space , endowed with the -topology, and its decomposition into orbits of the action of the group of “left-right changings of coordinates”.
MSC 58E05, 57M50, 58K65, 46M18
Let be a smooth connected orientable closed surface and a function having only critical points of the -types, . Let be the set of functions having the same types of local singularities as those of . Denote by the identity component in the group of orientation-preserving self-diffeomorphisms of . The group acts on by “left-right changings of coordinates”. We describe the homotopy type of the space , endowed with the -topology, and its decomposition into -orbits. This result was announced in K:conf:Zlat (); K:conf (). Similar result for a Morse function and was obtained in KP3mn (); KP4msb (); KP4vest ().
Let us give a short historical overview, mostly for the case of a Morse function (see the paper KP3mn () and references therein). A. T. Fomenko posed the question (1997) whether the space is arcwise connected; it was answered affirmatively by the author Kud99 () for , by S. V. Matveev Kud99 () and H. Zieschang in the general case. Open -orbits in were counted by V. I. Arnold Arn () and E. V. Kulinich (1998). Homotopy type of any -orbit in was studied by S. I. Maksymenko Max () (when was allowed to have certain degenerate types singularities) and by the author KP3mn (); KP4msb (); KP4vest (). V. A. Vassiliev Vas () proved the parametric -principle and studied cohomology of spaces of smooth -valued functions not having too complicated singularities on any smooth manifold . However the 1-parameter -principle fails for the spaces of Morse functions on some with CL ().
I Main result
For any function , denote by the set of its ctitical points, and by the set of critical points of the -types, . Recall that, in a neighbourhood of such a point , there exist local coordinates such that for some sign . The integer will be called the level of the point .
Denote by and (respectively ) the set of critical points of of -types, , which are (respectively are not) points of local minima or local maxima. In a neighbourhood of such a point , there exist local coordinates such that where . The integer will be called the level of the point . The subset of degenerate critical points (i.e. those of non-zero levels) in will be denoted by .
Suppose that an action of a group on a topological space , a stratified Whi () orbifold and a continuous surjection are given. If every -orbit in is the full pre-image of a stratum from , we will say that classifies -orbits, while and are the classifying space and map.
The group acts on by the homeomorphisms , . Define the evaluation functional , , and
For every function whose all critical points are of the -types, , there exist smooth manifolds and and surjective submersions , , , such that the diagram
commutes, where is the projection and . Moreover:
(a) the maps are homotopy equivalences and classify - and -orbits in for some stratifications on whose all strata are submanifolds; the map is a fibre bundle with fibres diffeomorphic to ;
(b) the map (resp. ) induces a homotopy equivalence between every -invariant subset (resp. ) and its image, e.g. between every orbit from item (a) and the corresponding stratum;
(c) the group discretely acts on by diffeomorphisms preserving the stratifications from item (a) and the function ; the maps and classify - and -orbits in and for the induced stratifications on and , where and are the projections.
Let us explain the term “submersion” in the case of functional spaces. If are smooth manifolds and , denote by the preimage of under the inclusion , and by the set of maps inducing maps for all . A map will be called a submersion if, for any , there exist a neighbourhood of the point in and a map such that .
Ii Constructing the classifying manifolds and maps
Similarly to KP (), by a framed function on an oriented surface we will mean a pair where has only the -types local singularities and is a closed 1–form on such that (i) the 2-form has no zeros on and defines a positive orientation, (ii) in a neighbourhood of every critical point there exist local coordinates such that either and , or and , or and where , , .
Denote by the space of framed functions such that . Endow this space with the -topology KP (). Consider the right actions of on and by the homeomorphisms and , .
Let be pairwise distinct points. Denote by the identity component of the group , , whence .
Define the classifying manifolds and as , , where and are the universal moduli spaces
of framed functions (resp. framed functions with one marked point) in , . One shows similarly to KP3mn (); KP4msb () that and are orbifolds of dimensions . For every group , we endow and with the stratifications whose every stratum is the full preimage of a point under the projection and .
Due to the -equivariance of the projection and the -invariance of the evaluation functional , , they induce some maps and . Put , .
The projection , , is a homotopy equivalence and has a homotopy inverse map and corresponding homotopies that respect the projections and .
If then is a smooth manifold, while the projection is a homotopy equivalence and has a homotopy inverse map and corresponding homotopies that respect (whence ).
Put . One defines similarly . Define the classifying maps , .
Iii Reducing to the case of Morse functions
If is a Morse function and , then the space from §II coincides with the smooth stratified manifold (the universal moduli space of framed Morse functions) studied in KP3mn (); KP4msb (); KP4vest (). It happens that every and can be described in terms of Morse functions.
Recall that a function is said to be Morse if all its critical points are nondegenerate (i.e. have the -type, cf. §I). Denote by the space of Morse functions on having exactly and points of local minima and maxima and saddle points.
A Morse function will be called -labeled if every its critical point is labeled by an integer and, in the case when this integer does not vanish and , also by a 1-dimensional subspace , moreover of non-critical points of are labeled by non-zero integers in such a way that the level (cf. §I) of every critical point of coincides with the integer label of the corresponding labeled point of , for some bijections , , and a bijection between and the set of labeled non-critical points of .
Denote by the space of framed (cf. §II) -labeled Morse functions. It is not difficult to construct homeomorphisms
Iv Relation with meromorphic functions and the configuration spaces
Suppose that is either a sphere or a torus . If , denote by the space of rational functions on the Riemann sphere such that all poles of the 1-form are simple and have real rezidues, being positive at poles and negative at poles. If , denote by the space of pairs where , , and is a meromorphic function on the torus , whose poles are all simple, all periods of the meromorphic 1-form are purely imaginary, and the residues are positive at poles and negative at poles.
Let be the space of functions or pairs such that has only simple zeros.
Due to (GruKri, , Proposition 3.4), the assignment to a 1-form its poles and residues at them gives a bijection , where is the “labeled configuration space” consisting of -points subsets of equipped by positive and negative real marks with zero total sum. Thus is homeomorphic to the open subset consisting of the “labeled configurations” that correspond to 1-forms without multiple zeros.
It is not difficult to derive from (2) with (cf. (1) and (KP, , Remark 2.6)) that our manifold is homeomorphic to the space of functions or pairs , marked by -labels (cf. §III) at zeros and poles of the 1-form and at some other points, as well as by a “vertical” label consisting of (i) a real label and (ii) either a positive real label in the case of , or integral curves of the field separating the poles from other labeled points. Thus, the manifold can be obtained from the “labeled configuration subspace” by assigning the -labels and the (topologically inessential) “vertical” label.
Acknowledgements.The author wishes to express gratitude to S. Yu. Nemirovski for indicating the paper GruKri (). This work was done under the support of RFBR (grant \No 15-01-06302-a) and the programme “Leading Scientific Schools of RF” (grant NSh-7962.2016.1).
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