Euler characteristic and homotopy cardinality

2018/11/19 by John D. Berman, Berman, John D.
Mathematics · #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.1811.07437

openalex publication_date 2018/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Top^rx containing both these classes of spaces and closed under homotopy pushout squares. In our main result, we compute the K-theory K₀(Top^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 Baez's question globally, but a positive answer when we restrict attention to a prime.

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