This page contains some of the questions mentioned/discussed in the 1st-5th 3J Topology Seminars

1st 3J:

Duan: Chang complexes vs manifolds:

Is it possible to attach a top cell to a Chang complex to get a Poincare complex? In other words, which Chang complexes are almost Poincare complex? How about fake $\mathbb{C}P^3$?

2nd 3J:

There are a lot of questions and open problems raised in the 2nd 3J seminar. Particular interests are on classical unstable homotopy theory and topology of hyperplane arrangements. Here is the brief summary.

Jie Wu’s questions focus on the essentials of EHP sequence in homotopy theory. His main point is about possible generalizations of this powerful tool, that is, to study EHP-type sequences for “general’’ complex rather than spheres. Based on his own research and interests, he then raised/recommended the following (for more details, please refer to the slides of Jie Wu’s talk):

  1. the Hopf invariant problem for spherical fibrations over spheres;
  2. homotopy and geometry of exterior spaces, which serve as candidates of generalization of spheres in EHP-sequences;
  3. homotopy theory of Amin at the critical case;
  4. homotopy exponent problem and homotopy decomposition techniques;
  5. the long standing Barratt conjecture and Cohen’s program.

Ye Liu’s questions focus on the connections between the combinatorial data of hyperplane arrangements and their algebraic topology, especially on how to obtain topological information from the combinatorial aspects of arrangments. He then raised/recommended the following (for more details, please refer to the Ye Liu’s summary on his survey talk):

  1. The face poset of a real hyperplane arrangement can provide more information than the intersection poset. In contrast to Rybnikovs counterexample, can we find a pair of real arrangements, with isomorphic intersection posets, but non-isomorphic face posets?

  2. For central complex arrangement, what topological information of Milnor fibre is determined by the intersection poset? This question is widely open in dimension greater than 2.

  3. The celebrated K(pi, 1)-conjecture and Artin groups.

Others interesting problems include:

  1. the covering type of Karoubi-Weibel is defined in terms of much strong condition than the Lusternik-Schnirelmann category, and is related to minimal triangulations of manifolds. There are a lot of unknown but interesting aspects of this new invariant, including the calculation of its value for many common manifolds and complexes, the possible product inequality and the whole rational homotopy theory of the invariant, etc.

  2. the variation of the second part of Question 13 in (GTM 205, Rational homotopy theory): Which simply connected rational homotopy types occur as the Dold-Lashof classifying space $Baut_1(X)$ for fibrations with fibre the homotopy type of some $X$? This question is long standing and far from being resolved, though there are the models of $Baut_1(X)$ by Sullivan and Schlessinger-Stasheff.