# 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.