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関手の微積分の応用と例

Goodwillie アイデア している Goodwillie のものを homotopy calculus ということにすると , その 理論 としては Weiss orthogonal calculus Goodwillie Weiss manifold calculus なも のだろう

Manifold calculus , みの 調 べるのに 使 われるので , その embedding calculus ばれているようである ここでは Munson [ Mun ] manifold calculus んだ Boadiva de Brito Weiss [ BdBW13 ] , enrich された version すると , homotopy sheaf いた ている

Steffen Tillmann [ Til ] , manifold calculus simplicial complex への している

これらの として えば 以下 のものがある

Biedermann Dwyer , homotopy n -nilpotent group という する , n -excisive functor loop homotopy n -nilpotent group つことを している また , identity functor Goodwillie tower loop simplicial algebraic theory いた いる

典的 ホモトピ における nilpotency との 調 べているのは , Chorny Scherer [ CS15 ] である LS category LS cocategory との については , Eldred [ Eld16 ] により 調 べられている Costoya Scherer Viruel [ CSV18 ] , fiber とする principal fibration いて “extension by principal fibrations length” という 不変 , homotopy nilpotency との 調 べて いる

Goodwillie Weiss Taylor tower long knot , Bott-Taubes configuration space integral [ BT94 ] をその tower したのは Volic [ Vol06 ] であ Volic long knot “calculus of embedding” いて 調 べることに ついて , [ Vol ] というまとめを いている なども いて ある

Goodwillie-Weiss embedding calculus しようという みもある えば , Poincaré duality space しては , Klein [ Kle17 ] ある

ホモロジ することを えているのは , Kuhn [ Kuh08 KM13 ] である Σ Ω n Σ Ω Goodwillie tower から らえる spectral sequence , それを いて 調 べようとしている

Kuhn [ Kuh15 ] ホモトピ する Whitehead Arone Mahowald [ AM99 ] びつけることに , それにより Whitehead ている Whitehead , 80 Kuhn [ Kuh82 ] ( p = 2) Kuhn Priddy [ KP85 ] (odd prime) により されたもので ある

Goodwillie により , category から category への linear functor generalized homology theory あるいは spectrum , ほとんど じものにな model category -category での spectrum object linear functor として することもできる えば Lurie [ Lur ] Definition 1.4.2.8 など Shimakawa, Yoshida, Haraguchi [ SYH ] , “enriched linear functor” category stable homotopy category モデル として して いる

References

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[AM99]     Greg Arone and Mark Mahowald. The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres. Invent. Math. , 135(3):743–788, 1999, http://dx.doi.org/10.1007/s002220050300 .

[Aro98]     Greg Arone. Iterates of the suspension map and Mitchell’s finite spectra with A k -free cohomology. Math. Res. Lett. , 5(4):485–496, 1998, http://dx.doi.org/10.4310/MRL.1998.v5.n4.a6 .

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[BdBW13]     Pedro Boavida de Brito and Michael Weiss. Manifold calculus and homotopy sheaves. Homology Homotopy Appl. , 15(2):361–383, 2013, arXiv:1202.1305 .

[Beh12]     Mark Behrens. The Goodwillie tower and the EHP sequence. Mem. Amer. Math. Soc. , 218(1026):xii+90, 2012, arXiv:1009.1125 .

[BT94]     Raoul Bott and Clifford Taubes. On the self-linking of knots. J. Math. Phys. , 35(10):5247–5287, 1994, http://dx.doi.org/10.1063/1.530750 . Topology and physics.

[CS15]     Boris Chorny and Jérôme Scherer. Goodwillie calculus and Whitehead products. Forum Math. , 27(1):119–130, 2015, arXiv:1109.2691 .

[CSV18]     Cristina Costoya, Jérôme Scherer, and Antonio Viruel. A torus theorem for homotopy nilpotent loop spaces. Ark. Mat. , 56(1):53–71, 2018, arXiv:1504.06100 .

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[Eld16]     Rosona Eldred. Goodwillie calculus via adjunction and LS cocategory. Homology Homotopy Appl. , 18(2):31–58, 2016, arXiv:1209.2384 .

[GM10]     Thomas G. Goodwillie and Brian A. Munson. A stable range description of the space of link maps. Algebr. Geom. Topol. , 10(3):1305–1315, 2010, arXiv:0910.3561 .

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[Goo92]     Thomas G. Goodwillie. Calculus. II. Analytic functors. K -Theory , 5(4):295–332, 1991/92, http://dx.doi.org/10.1007/BF00535644 .

[GW99]     Thomas G. Goodwillie and Michael Weiss. Embeddings from the point of view of immersion theory. II. Geom. Topol. , 3:103–118 (electronic), 1999, http://dx.doi.org/10.2140/gt.1999.3.103 .

[JM04]     B. Johnson and R. McCarthy. Deriving calculus with cotriples. Trans. Amer. Math. Soc. , 356(2):757–803 (electronic), 2004, http://dx.doi.org/10.1090/S0002-9947-03-03318-X .

[Kle17]     John R. Klein. Embeddings, normal invariants and functor calculus. Nagoya Math. J. , 225:152–184, 2017, arXiv:1408.6469 .

[KM13]     Nicholas Kuhn and Jason McCarty. The mod 2 homology of infinite loopspaces. Algebr. Geom. Topol. , 13(2):687–745, 2013, arXiv:1109.3694 .

[KP85]     Nicholas J. Kuhn and Stewart B. Priddy. The transfer and Whitehead’s conjecture. Math. Proc. Cambridge Philos. Soc. , 98(3):459–480, 1985, http://dx.doi.org/10.1017/S0305004100063672 .

[Kuh82]     Nicholas J. Kuhn. A Kahn-Priddy sequence and a conjecture of G. W. Whitehead. Math. Proc. Cambridge Philos. Soc. , 92(3):467–483, 1982, http://dx.doi.org/10.1017/S0305004100060175 .

[Kuh08]     Nicholas Kuhn. Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology. Algebr. Geom. Topol. , 8(4):2109–2129, 2008, arXiv:0806.3281 .

[Kuh15]     Nicholas J. Kuhn. The Whitehead conjecture, the tower of S 1 conjecture, and Hecke algebras of type A. J. Topol. , 8(1):118–146, 2015, arXiv:1308.4441 .

[Lur]     Jacob Lurie. Higher Algebra, http://www.math.harvard.edu/ ~ lurie/papers/HA.pdf .

[Mac07]     Tibor Macko. The block structure spaces of real projective spaces and orthogonal calculus of functors. Trans. Amer. Math. Soc. , 359(1):349–383 (electronic), 2007, arXiv:math/0404198 .

[Mal15]     Cary Malkiewich. A tower connecting gauge groups to string topology. J. Topol. , 8(2):529–570, 2015, arXiv:1209.1778 .

[Mun]     Brian A. Munson. Introduction to the manifold calculus of Goodwillie-Weiss, arXiv:1005.1698 .

[Mun05]     Brian A. Munson. Embeddings in the 3 4 range. Topology , 44(6):1133–1157, 2005, arXiv:math/0311423 .

[Mun08]     Brian A. Munson. A manifold calculus approach to link maps and the linking number. Algebr. Geom. Topol. , 8(4):2323–2353, 2008, arXiv:math/0702163 .

[Mun11]     Brian A. Munson. Derivatives of the identity and generalizations of Milnor’s invariants. J. Topol. , 4(2):383–405, 2011, arXiv:0909.3074 .

[MV12]     Brian A. Munson and Ismar Volić. Multivariable manifold calculus of functors. Forum Math. , 24(5):1023–1066, 2012, arXiv:0904.4185 .

[MV14]     Brian A. Munson and Ismar Volić. Cosimplicial models for spaces of links. J. Homotopy Relat. Struct. , 9(2):419–454, 2014, arXiv:0906.2589 .

[MW09]     Tibor Macko and Michael Weiss. The block structure spaces of real projective spaces and orthogonal calculus of functors. II. Forum Math. , 21(6):1091–1136, 2009, arXiv:math/0703303 .

[SYH]     K. Shimakawa, K. Yoshida, and T. Haraguchi. Homology and cohomology via enriched bifunctors, arXiv:1010.3336 .

[Til]     Steffen Tillmann. Manifold Calculus Adapted for Simplicial Complexes, arXiv:1702.05608 .

[Vol]     Ismar Volic. Calculus of the embedding functor and spaces of knots, arXiv:math/0601268 .

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