Manifolds with singularities

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[edit] 1 Introduction

Manifolds with singularities are geometric objects in topology generalizing manifolds. They were introduced in ([Sullivan1996],[Sullivan1967]) and [Baas1973]. Applications of the concept include representing cycles in homology theories with coefficients. All manifolds in this article are understood to be smooth.

[edit] 2 Definitions

[edit] 2.1 Cone-like singularities

A manifold with singularities of Baas-Sullivan type is a topological space \overline{A} that looks like a manifold outside of a compact 'singularity set', while the singularity set has a neighborhood that looks like the product of manifold and a cone. Here is a precise definition. Let P_1 be a closed manifold. A manifold with a P_1-singularity (following [Baas1973]) is a space of the form

\displaystyle \overline{A}  = A \cup_{A(1) \times P_1} A(1) \times C P(1)
\displaystyle \partial A    = A(1) \times P_1

Here, A is a manifold with boundary A(1).

More complex singularities occur if, instead of taking a cone over only one manifold P_1, we allow a collection \{P_1,...,P_k\} of several closed manifolds. In this case, we define a a manifold with a \{P_1,...,P_k\}-singularity to be a (second-countable and Hausdorff) topological space \overline{A} locally homeomorphic to one of the spaces \Rr^n , \Rr^{n_1} \times C P_1 , \Rr^{n_2} \times C P_1 \times C P_2 , ....

An alternative approach to manifolds with singularities would be to remove the singular set and to define an equivalence relation on the remaining manifold that 'remembers' the singularities. This view is taken in ([Botvinnik1992],[Botvinnik2001]). We describe it in the next section.

[edit] 2.2 \Sigma-manifolds

Following ([Botvinnik2001], [Botvinnik1992]), an alternative definition can be given. Let (P_1 , ..., P_k) be a (possibly empty) collection of closed manifolds and denote by P_0 the set containing only one point. Then define \Sigma_k := (P_0, P_1, ... , P_k). For a subset I = \{i_1,..., i_q\} \subset \{0,...,k\} define P^I := P_{i_1} \times ...\times P_{i_q}.

Definition 2.1.A manifold M is a \Sigma_k-Manifold if there is given

  1. a partition \partial M = \partial_0 M \cup ... \cup \partial_k M, such that \partial_I M := \partial_{i_1} \cap ... \cap \partial_{i_q} M is a manifold for each I = \{i_1,...,i_q\} \subset \{0,...,k\}, and such that
\displaystyle  \partial (\partial_I M) = \cup_{j \notin I} \partial_j M \cap \partial_I M
  1. for each I \subset \{0,...,k\} a manifold \beta_I M and a diffeomorphism
\displaystyle  \phi_I: \partial_I M \rightarrow \beta_I M \times P^I

such that if J \subset I and \iota: \partial_I M \rightarrow \partial_J M is the inclusion, then the composition

\displaystyle  \phi_J \circ \iota \circ \phi_I^{-1}: \beta_I M \times P^I \rightarrow \beta_J M \times P^J

restricts to the identity on the factor P^J in P^I. The diffeomorphisms \phi_I are called product structures.

On a \Sigma_k-manifold M, there is a canonical equivalence relation \sim: two points x,y \in M are defined to be equivalent if there is an I \subset \{0,...,k\} such that x,y \in \partial_I M and pr \circ \phi_I(x) = pr \circ \phi_I(y), where pr: \beta_I M \times P^I \rightarrow \beta_I M is the projection. Now we can give a general definition: a manifold with a \Sigma_k-singularity is a topological space \overline{A} of the form

\displaystyle  \overline{A} = A / \sim

for a \Sigma_k-manifold A.

The spaces defined above as manifolds with a (P_1,...,P_k)-singularity are contained in this new definition. Given manifolds P_1,...,P_k, set \Sigma_k = (P_0, P_1,...,P_k). Removing a neighborhood of the cone-tips in a manifold with (P_1,...,P_k)-singularity \overline{A} gives a \Sigma_k-manifold M. Now the collapsing of the equivalence relation in M corresponds to the re-attachement of the cone-ends.

When dealing with manifolds with singularities it is convenient to work with the underlying \Sigma-manifold and make sure that all operations one performs on them are compatible with the equivalence relation.

[edit] 3 Some examples and constructions

[edit] 3.1 Intersecting spheres

A basic example is presented by two spheres intersecting each other in a sphere of lower dimension. Choose an embedding i: S^{n-1} \rightarrow S^n. Then define \overline{S} := S^n \cup_{i(S^{n-1})} S^n. Outside of the intersecting sphere i(S^{n-1}) this is an n-dimensional manifold. The intersecting sphere itself has a neigborhood homeomorphic to the product of S^{n-1} and a cone over \Zz_4 . We can write \overline{S} = D^n \times \Zz_4 \cup_{S^{n-1} \times \Zz_4} S^{n-1} \times C \Zz_4.

[edit] 3.2 Inverse images of critical points

Let f:\Rr^n \rightarrow \Rr be a Morse-function with 0 as a single critical point. We can suppose that f(x) = -x^2_1 -...-x^2_k + x^2_{k+1} + ... + x^2_n near 0. Setting M := f^{-1}(\{0\}), we see that the cone C S^{k-1} \times S^{n-k-2} = \{t \cdot x : 0 \leq t \leq 1 , x \in S^{k-1} \times S^{n-k-2} \subset \Rr^n \} provides a neighborhood of 0 in M. It follows that M is of the form N \cup_{S^{k-1} \times S^{n-k-2}} C (S^{k-1} \times S^{n-k-2}).

[edit] 3.3 Structures on manifolds with singularities

Geometric and topological structures that exist for ordinary manifolds can also be defined for manifolds with singularities. This is done in the following way. A manifold with \Sigma-singularity \overline{M} with underlying \Sigma-manifold M carries the structure in question if all manifolds involved in \textbf{Definition 2.1} carry this structure and the product diffeomorphisms preserves it.

For example, \overline{M} is orientable if M as well as all manifolds in \Sigma and the manifolds \beta_I M are orientable and the product diffeomorphisms are orientation-preserving. As another example, \overline{M} becomes a Riemannian manifold with singularities if we put a Riemannian metric on M as well as on the manifolds in \Sigma and on the manifolds \beta_I M in such a way that the product diffeomorphism are isometries.

[edit] 3.4 Bundles on manifolds with singularities

As usual, we define a bundle \overline{E} on a manifold with singularities \overline{M} as a bundle E on the underlying manifold M subject to the following additional condition: there are bundles E_I over the manifolds \beta_I M \times P^I and bundle equivalences E|_{\partial_I M} \rightarrow E_I covering the product diffeomorphisms.

[edit] 4 Bordism theory for manifolds with singularities

A \Sigma_k-manifold M induces the structure of a \Sigma_k-manifold on \partial_0 M. We call \partial_0 M the boundary of M. Given a manifold with a \Sigma_k-singularity \overline{M} := M / \sim, we define \partial \overline{M} = \partial_0 M / \sim to be its boundary. A theory of bordism with \Sigma_k-singularities can now be developed just as for ordinary manifolds.

For illustration we pick up the case of a P_1-singularity considered above. \overline{A} = A \cup_{A(1) \times P_1} A(1) \times C P_1 is bordant to zero if there exists \overline{B} = B \cup_{B(1) \times P_1} B(1) \times C P_1, such that

\displaystyle  \partial B =  A \cup_{A(1) \times P_1} B(1) \times P_1
\displaystyle  \partial B(1) = A(1)

[edit] 5 Invariants


[edit] 6 Classification/Characterization


[edit] 7 Further discussion


[edit] 8 References

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