Mar 9, 2010

Brett Parker

# Introduction

The goal of this paper is to describe a version of de Rham cohomology for exploded manifolds which extends de Rham cohomology for smooth manifolds. At first glance, the most natural extension of de Rham cohomology would be to take the complex of smooth or C1;1 differential forms on an exploded manifold with the usual differential d. Unfortunately, this naive extension does not have good properties, in a smooth connected family of exploded manifolds, the cohomology defined this way might change. Moreover, the tools of integration and Poincar´e duality are not available for this naive extension.

Instead, we shall use a sub-complex 𝜞 (B) of C1;1 differential forms on B, defined below in definition. In the case that Bis a smooth manifold,𝜞(B) is the usual complex of smooth differential forms. We shall show in Section H𝜞 (B) does not change in connected families of exploded manifolds. (This fact is nontrivial to prove because families of exploded manifolds are not always locally trivial.) As suggested by the names of the sections of this paper, many of the standard tools of de Rham cohomology still apply for 𝜞(B).

From now on, some knowledge of the definitions and notation from [5] shall be necessary to understand this paper. Recall that coordinates on Rn Tm P are given by xj:Rn Tm P !R for 1 j n and zi: Tm P !C t R for 1 i m.

# (A compactly supported form with infinite integral)

T1 1 := T1 [0;1) has a single coordinate z:T1 1 !C t [0;1). Consider the two-form 낎 given by the wedge product of the real and imaginary parts of z 1 d z. Over any tropical point t a 2 T1 1 in the tropical part of T1 1, there is a C worth of points corresponding to a choice of coefficient c of z = ct a. On the C worth of points over each tropical point of T1 1, 낎 is a nonzero C invariant volume-form, so by any straightforward definition of integration, 낎 should have infinite integral. Similarly, if 낎 is multiplied by any continuous function f:T1 1 !R which is nonzero when d z e = 0, the integral of f District 낎 is again infinite.

# (Definition: Let k(B) be the vector-space)

Let k(B)be the vector-space of C1;1 differential k–forms 낎 on B. For Stokes' theorem to work out correctly, we shall also require the following condition: Given any map f:T1(0;1) !B , we shall assume that our differential forms vanish on all of the vectors in the image of Tf.

# Mayer Vietoris sequence

The Mayer Vietoris sequence holds by using partitions of unity, which are constructed in section. Given open subsets U and V of an exploded manifold B, the Mayer Vietoris sequences.

# Integration and Stokes' theorem

We shall show below that if B is oriented and n–dimensional, then the integral of compactly supported forms in 𝜞 (B) is well defined. Any top–dimensional form in 𝜞 (B) will vanish on all strata apart from those strata of B which are smooth manifolds (and therefore have no nonzero integral vectors). We can therefore define the integral of a top–dimensional form 낎 on an oriented exploded manifold B to be the sum of the integrals of 낎 over these smooth strata. This integral is well defined if the integral over each smooth stratum is well defined and the resulting sum of integrals is well defined.

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