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The homotopy category of the bordism category    h   C  dhas as objects closed oriented    (  d  −  1  )   ‐manifolds and as morphisms diffeomorphism classes of   d  ‐dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of    h   C  dfor    d  =  1   , exhibiting it as a circle bundle overΩ   ∞  −  2CP   −  1   ∞    .  As part of our proof, we construct a quotient    C  1  red    of the cobordism category where circles are deleted. We show that this category has classifying spaceΩ   ∞  −  2CP   −  1   ∞and moreover that, if one equips these bordisms with a map to a simply connected space   X  , the resultingC  1  red    (  X  )can be thought of as a cobordism model for the topological cyclic homology    T  C  (  S  [  Ω  X  ]  )   .  In the second part of the paper, we construct an infinite loop space map    B   (  h   C  1  red   )   →  Q   (   Σ  2    CP  +  ∞   )in this model and use it to derive combinatorial formulas for rational cocycles on    h   C  1  redrepresenting the Miller–Morita–Mumford classesκ  i   ∈  H   (  B   (  h   C  1   )   ;  Q  )    .

The classifying space of the one‐dimensional bordism category and a cobordism model for TC of spaces