## DERIVATIVES OF EMBEDDING FUNCTORS I: THE STABLE CASE (2007)

### BibTeX

@MISC{Arone07derivativesof,

author = {Gregory Arone},

title = {DERIVATIVES OF EMBEDDING FUNCTORS I: THE STABLE CASE},

year = {2007}

}

### OpenURL

### Abstract

For smooth manifolds M and N, let Emb(M, N) be the homotopy fiber of the map Emb(M, N) − → Imm(M, N). Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula V ↦ → Σ ∞ Emb(M, N × V). In this paper, we describe the derivatives of this functor, in the sense of M. Weiss ’ orthogonal calculus. Our construction involves a certain space of rooted forests (or, equivalently, a space of partitions) with leaves marked by points in M, and a certain “homotopy bundle of spectra ” over this space of trees. The n-th derivative is then described as the “spectrum of restricted sections ” of this bundle. This is the first in a series of two papers. In the second part, we will give an analogous description of the derivatives of the functor Emb(M, N × V), involving a similar construction with certain spaces

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Citation Context ..., i = 0, 1, 2, . . . fit into a cosimplicial space (resp. spectrum), which in the case when C is a discrete category is the standard cosimplicial object that is used to define the homotopy limit of F =-=[5]-=-. For a general topological category C, we define holim F to be the (derived) total space (or spectrum) of this cosimplicial object. C In practice, we will want to make some assumptions that will guar... |

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Citation Context ... N) is, in addition to being a covariant functor of N, also a contravariant functor of M (where M is considered an object in a suitable category of manifolds over N), one can apply embedding calculus =-=[11]-=- to this functor, and also to the functor Dn Σ ∞ Emb(M, N). Propositioin 1.7 in fact describes the Taylor tower in the sense of embedding calculus of Dn Σ ∞ Emb(M, N). In particular, it shows that Dn ... |

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Citation Context ...r Top ∗. There is an adjunction, where F (−, −) denotes, just for the moment, the space of maps inDERIVATIVES OF EMBEDDING FUNCTORS I: THE STABLE CASE. 15 the category of pointed spaces over B. (see =-=[9]-=-, Proposition 1.2.8). F (K ∧B E1, E2) ∼ = F (E1, Map B(K, E2)) Remark 2.1. It is not difficult to see that there is a homeomorphism Γ(Map B(K, E)) ∼ = Map ∗(K, Γ(E)), Where Γ(E) is the space of sectio... |

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Citation Context ...ion, or fiber homotopy equivalent to a fibration. In the pointed case, one needs to require, in addition, that E is wellpointed in the sense that the section s: B ↩→ E is a fiberwise cofibration (see =-=[6]-=-). In this case, Γ(E) has the expected (pointed) homotopy invariance and excision properties, and so one has some control over the homotopy type of the space of sections. However, the class of fibrati... |

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Citation Context ...rs, and one defines hNat(F, G) := Nat(QF, RG), where Q and R denote cofibrant and fibrant replacements respectively. But in this paper, we use a more direct construction, proposed by Dwyer and Kan in =-=[7]-=-. Dwyer and Kan defined the space of homotopy natural transformation as the homotopy limit over the “twisted arrow category” aC. This is a category whose objects are morphisms x → y in C, and where a ... |

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Citation Context ...ge-wise subdivision of the cosimplicial space defining the right hand side. For the definition of edge-wise subdivision, and a proof of the fact that edge-wise subdivision preserves totalization, see =-=[8]-=-, Section 2, especially Propositions 2.2 and 2.3. □ It is also clear by inspection that there is a homeomorphism NatC(F, G) ≃ lim C⋉F G 3.3.2. Spaces of twisted natural transformations. We would like ... |

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Citation Context ...mersions of M into N, respectively. Let Emb(M, N) be the homotopy fiber of the inclusion map Emb(M, N) ↩→ Imm(M, N). Our goal is to study the space Emb(M, N) using Michael Weiss’s orthogonal calculus =-=[10]-=-. To be more specific, consider the functor from the category of Euclidean spaces to the category of spectra defined by the formula V ↦→ Σ ∞ Emb(M, N × V ). This is a functor to which one can apply or... |

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Citation Context ...rtain local coefficients. The upshot of all this is that while this is not a trivial calculation, quite a lot is known about the equivariant cohomology of the spaces M n+i /∆ n+i M and TΛ (references =-=[2, 3]-=- contain some information about the latter), and so it might be possible to do some interesting calculations with these formulas. We intend to come back to this in a future paper. If one wants to, one... |

4 |
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Citation Context ...rtain local coefficients. The upshot of all this is that while this is not a trivial calculation, quite a lot is known about the equivariant cohomology of the spaces M n+i /∆ n+i M and TΛ (references =-=[2, 3]-=- contain some information about the latter), and so it might be possible to do some interesting calculations with these formulas. We intend to come back to this in a future paper. If one wants to, one... |

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Citation Context ...clidean spaces to the category of spectra defined by the formula V ↦→ Σ ∞ Emb(M, N × V ). This is a functor to which one can apply orthogonal calculus. The Taylor tower of this functor was studied in =-=[4]-=- in the special case when N = ∗ (from a perspective different than the one taken here). The main result there says that the Taylor tower of the functor V ↦→ Σ ∞ Emb(M, V ) rationally splits as a produ... |

4 | constructions for topological operads and the Goodwillie derivatives of the identity - Bar |

2 |
Derivatives of the embedding functor II: the unstable case, in preparation
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(Show Context)
Citation Context ...igure 1, we try to illustrate this space of forests in the case n = 2, together with the fundamental homotopy bundle over it. This viewpoint will be developed further in the second part of the series =-=[1]-=-, where we will give an analogous description of the derivatives of Emb(M, N) as the twisted cohomology of a certain space of connected graphs (as opposed to forests) with points marked by elements of... |

1 | Twisted homotopy theory and the geometric equivariant 1-stem - Cruickshank |