No Arabic abstract
Baez asks whether the Euler characteristic (defined for spaces with finite homology) can be reconciled with the homotopy cardinality (defined for spaces with finite homotopy). We consider the smallest infinity category $text{Top}^text{rx}$ containing both these classes of spaces and closed under homotopy pushout squares. In our main result, we compute the K-theory $K_0(text{Top}^text{rx})$, which is freely generated by equivalence classes of connected p-finite spaces, as p ranges over all primes. This provides a negative answer to Baezs question globally, but a positive answer when we restrict attention to a prime.
We resolve two long-standing and closely related problems concerning stably free $mathbb{Z} G$-modules and the homotopy type of finite 2-complexes. In particular, for all $k ge 1$, we show that there exists a group $G$ and a non-free stably free $mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k ge 0$, there exists homotopically distinct finite 2-complexes with fundamental group $G$ and with Euler characteristic $k$ greater than the minimal value over $G$. This provides a solution to Problem D5 in the 1979 Problems List of C. T. C. Wall.
We give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the C_2-equivariant p-complete stable homotopy category as a localization of the p-complete cellular real motivic stable homotopy category.
Let F be a field of characteristic different than 2. We establish surjectivity of Balmers comparison map rho^* from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor-Witt K-theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.
In this short note, we provide a calculation of the Euler characteristic of a finite homotopy colimit of finite cell complexes, which depends only on the Euler characteristics of each space and resembles Mobius inversion. Versions of the result are known when the colimit is indexed by a finite category, but the behavior is more uniform when we index by finite quasicategories instead. The formula simultaneously generalizes the additive formula for Euler characteristic of a homotopy pushout and the multiplicative formula for Euler characteristic of a fiber bundle.
We give a new description of Rosenthals generalized homotopy fixed point spaces as homotopy limits over the orbit category. This is achieved using a simple categorical model for classifying spaces with respect to families of subgroups.