Welcome to the topology reading seminar!

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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

 

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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).

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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]

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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

 

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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.