Topology reading seminar – University of Copenhagen

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Department of Mathematical Sciences > Research > Topology > Topology reading seminar

Welcome to the topology reading seminar!


Fall 2011: Surgery theory.

An outline of the programme is available here.

We meet on *Wednesdays at 10:15 in room 04.4.01*

Sep 14 Oscar Randal-Williams Smooth manifold techniques
Sep 21 Oscar Randal-Williams The h-cobordism theorem
Sep 28 Angela Klamt Poincare duality and Poincare complexes
Oct 5 Angela Klamt Poincare duality and Poincare complexes, II
Oct 12 Daniella Egas Santander Normal invariants and normal maps
  Handout A Poincare complex inequivalent to a manifold
Oct 19 Oscar Randal-Williams Normal framed embeddings and immersions
Oct 26 Emanuele Dotto Surgery below the middle dimension
Nov 2 Ib Madsen The even-dimensional surgery obstruction
Nov 9 Ib Madsen The even-dimensional surgery obstruction, II
Nov 16 Nathalie Wahl The odd-dimensional surgery non-obstruction
Nov 23 Anssi Lahtinen The surgery exact sequence
Nov 30 Oscar Randal-Williams Dimensions congruent to 2 modulo 4
Dec 7 Oscar Randal-Williams Calculations

 

References:
Algebraic and Geometric Surgery - Ranicki
Surgery on Compact Manifolds - Wall
Surgery on Simply Connected Manifolds - Browder
A Basic Introduction to Surgery Theory - Luck

 


Fall 2010: Equivariant homotopy theory.

Here is a tentative outline . This document will be modified as we go along.

We meet on *Wednesdays at 10:15 in room 04.4.01*

Sep 29 Jesper Grodal Introduction
Oct 6 Richard Hepworth
G-CW-complexes
Oct 13 Alexander Berglund Elmendorf's Theorem
Oct 20 Oscar Randal-Williams Equivariant cohomology theories
Oct 27 Matthew Gelvin Smith theory
Nov 3 Toke Nørgård-Sørensen Self maps of a representation sphere
Nov 10 Bob Oliver Self maps of a representation sphere II
Nov 17 Samik Basu G-equivariant stable homotopy theory
Nov 24 Samik Basu G-equivariant stable homotopy theory (cont.)
Dec 1 Anssi Lahtinen The Atiyah-Segal completion theorem
Dec 8 Otgonbayar Uuye Restriction maps in equivariant K-theory
Dec 15 - -
Jan 12 Oscar Randal-Williams The calculation BU^hZ/2 = BO.
Jan 19 Anssi Lahtinen Change of groups and duality theory
Jan 26 Richard Hepworth Mackey functors



References:

Adams, Prerequisites (on equivariant stable homotopy) for Carlsson's lecture. Algebraic topology, Aarhus 1982 (Aarhus, 1982), 483-532,
Lecture Notes in Math., 1051, Springer, Berlin, 1984.

Greenlees-May, Equivariant stable homotopy theory
Handbook of algebraic topology, 277-323, North-Holland, 1995.

Elmendorf, Systems of fixed point sets ,
Trans. Amer. Math. Soc. 277 (1983), no. 1, 275-284.

Hill-Hopkins-Ravenel,
On the non-existence of elements of Kervaire invariant one
arXiv:0908.3724v1 [math.AT] 

Mandell-May,
Equivariant Orthogonal Spectra and S-modules
Mem. Amer. Math. Soc. 159 (2002), no. 755.

May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, 91.

McClure, Restriction maps in equivariant K-theory , Topology Vol. 25, No. 4,pp. 399 - 409, (1986).


Spring 2010:

We meet on *Tuesdays at 14:15 and Fridays at 13:15 in room 04.4.01*

Feb 3 David Ayala, Derived Koszul Duality in the Algebraic K-Theory of Spaces [BM2]
Feb 5 David Ayala, Derived Koszul Duality in the Algebraic K-Theory of Spaces II [BM2]
Feb 9 Richard Hepworth, Derived Koszul Duality in the Algebraic K-Theory of Spaces III [BM2]
Feb 23 Andrew Blumberg The S.'-construction [BM1]
Mar 2 Richard Hepworth The Blob Complex I
Mar 5 Richard Hepworth The Blob Complex II
Mar 9 Pascal Lambrechts The Goodwillie embedding calculus I
Mar 12 Pascal Lambrechts The Goodwillie embedding calculus II
Mar 16 David Ayala Very Poincaré duality I [DWW, L]
Mar 19 David Ayala Very Poincaré duality II [DWW, L]
Mar 23 Samik Basu A cellular nerve for higher categories I [B]
Mar 26 Samik Basu A cellular nerve for higher categories II [B]
Mar 30 Samik Basu A cellular nerve for higher categories III [B]
Apr 9 Richard Hepworth Iterated wreath products and iterated loop spaces I [B2]
Apr 13 Richard Hepworth Iterated wreath products and iterated loop spaces II [B2]
Apr 16 Richard Hepworth Iterated wreath products and iterated loop spaces III [B2]
Apr 21 Richard Hepworth Iterated wreath products and iterated loop spaces IV [B2]
Apr 23 Samik Basu Iterated wreath products and iterated loop spaces V [B2]
Apr 28 Pascal Lambrechts Iterated wreath products and iterated loop spaces VI [B2]
May 5 Alexander Berglund The Barratt-Eccles operad I [BE, B3]
May 7 Alexander Berglund The Barratt-Eccles operad II [BE, B3]
May 18 Richard Hepworth The additivity theorem [BV], [D]
May 21 Ib Madsen Homology operations
Jun 1 Otgonbayar Uuye Homotopical Algebra for C^*-algebras

 

References:

[BE] M.G. Barratt, P.J. Eccles, Gamma^+-structures I, II, III, Topology, Vol. 13, (1974), 25-45, 113-126, 199-207. 

[B] C. Berger, A cellular nerve for higher categories, Adv. Math. 169 (2002), no. 1, 118-175.

[B2] C. Berger , Iterated wreath product of the simplex category and iterated loop spaces, Adv. Math. 213 (2007), no. 1,230-270.

[B3] C. Berger, Combinatorial models for real configuration spaces and E_n-operads.

[BM1] A. Blumberg, M. Mandell, The localization sequence for the algebraic K-theory of topological K-theory, Acta Math., 200 (2008), 155-179.

[BM2] A. Blumberg, M. Mandell, Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces, arXiv:0912.1670v1[math.KT]

[BV] J.M. Boardman, R.M. Vogt, Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture notes in Math. 347 (1973).

[D] G. Dunn, Tensor product of operads and iterated loop spaces, J. Pure. Appl. Algebra 50 (1988), 237 - 258.

[DWW] W. Dwyer, M. Weiss, B. Williams, A parametrized index theorem for the algebraic K-theory Euler class. Acta Math. 190 (2003), no. 1, 1-104.

[L] J. Lurie, Derived algebraic geometry VI: E_k algebra, arXiv:0911.0018v1 [math.AT]



Fall 2009:

This semester (Fall 2009) we are reading the paper

  F. Waldhausen, Algebraic K-theory of spaces,
  Lecture Notes in Math. 1126, 1985, pp. 318-419

The paper is available at here.

We meet twice a week and go through a part of the paper together. As a platform for discussions, someone presents the material to the others in an informal talk. This task circulates among volunteers.

Sep 30 Alexander Berglund Categories with cofibrations and weak equivalences  
Oct 7 Alexander Berglund S_n C as a Waldhausen category  
Oct 14 David Ayala The additivity theorem  
Oct 21 David Ayala The additivity theorem (continued)  
Oct 23 David Ayala Thomason's interpretation. E(A,C,B) as a Waldhausen cat.  
Oct 28 David Ayala The additivity theorem (continued)  
Oct 30 - -  
Nov 4 Ib Madsen Duality and the S.-construction  
Nov 6 - -  
Nov 11 Otgonbayar Uuye Relative K-theory and cofinality  
Nov 13 Otgonbayar Uuye Relative K-theory and cofinality (continued)  
Nov 18 Ib Madsen Duality and the S.-construction II  
Nov 20 - -  
Nov 25 Alexander Berglund Cylinder functors and the fibration theorem.  
Nov 27 Alexander Berglund Cylinder functors and the fibration theorem (continued)  
Dec 2 Samik Basu The approximation theorem.  
Dec 4 Samik Basu The approximation theorem (continued)  
Dec 9 David Ayala Definition of A(X)  
Dec 11 David Ayala Definition of A(X) (continued)  
Dec 16 Alexander Berglund Waldhausen's S.-construction vs. Quillen's Q-construction  

 


Fall 2008:

Symmetric spectra (original seminar homepage).

Symmetric spectra reading seminar.
This page contains information about the symmetric spectra reading seminar. For information about other activities of the topology group at the University of Copenhagen, please visit this page.

What are symmetric spectra?
Symmetric spectra are used to construct the stable homotopy category in much the same way as the derived category of a ring is constructed from chain complexes. One main feature is that there is a commutative and associative smash product on symmetric spectra that descends to the usual smash product in the stable homotopy category. Another benefit in comparison to other approaches to stable homotopy theory is that defining symmetric spectra does not require a lot of machinery.

What are we doing?
We are reading selected parts of the book (in preparation) [S] by Stefan Schwede and the paper [HSS] by M. Hovey, B. Shipley and J. Smith, and other related papers. Participants take turns in presenting the material to each other.

2008 Sep 16 Antonio Diaz, Spectra and generalized cohomology theories.

Sep 23 Tarje Bargheer, Definitions and basic properties of symmetric spectra. [S]

Sep 30 Alexander Berglund, Examples: Sphere spectrum, Eilenberg-MacLane spectra. [S]

Oct 7 Jens Kaad, Examples: Algebraic K-theory spectrum. [S] [L]

Oct 14 - - Oct 21 Nathalie Wahl, Smash product on symmetric spectra. [HSS]

Oct 28 Antonio Diaz, Survey on model categories.

Nov 4 Alexander Berglund, Stable equivalences of symmetric spectra. [HSS]

Nov 11 Alexander Berglund, Stable equivalences of symmetric spectra (continued).

Nov 18 - - Nov 25 Tarje Bargheer, Stable model structure on symmetric spectra. [HSS]

Dec 2 Tarje Bargheer, Stable model structure on symmetric spectra (continued).

Dec 9 Otgonbayar Uuye, Comparison between symmetric and ordinary spectra. [BF] [HSS] [S]

2009 Jan 20 Otgonbayar Uuye, KK-theory as a non-commutative stable homotopy theory.

Jan 29 Antonio Diaz, Symmetric spectra and Topological Hochschild Homology. [Sh]

Feb 5 - - Feb 12 - - Feb 19 Ib Madsen, From THH to TC.

Feb 26 Alexander Berglund, Smash product on diagram spectra. [MMSS]

Mar 5 Alexander Berglund, Smash product on diagram spectra (continued).

Mar 12 - - Mar 19 Ib Madsen, Equivariant spectra and TC.


References:

[BF]A.K. Bousfield and E. M. Friedlander, Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, Lecture Notes in Math. 658, 1978, pp. 80-130.

[HSS] M. Hovey, B. Shipley and J. Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149-208.

[L] J-L. Loday, K-théorie algébrique et représentations de groupes, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 309-377.

[MMSS] M. Mandell, J.P. May, S. Schwede, B. Shipley, Model Categories of Diagram Spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441-512.

[S] S. Schwede, Symmetric spectra, Book in preparation.

[Sh] B. Shipley, Symmetric Spectra and Topological Hochschild Homology, K-Theory 19 (2000), 155-183.