The block structure spaces of real projective spaces and orthogonal calculus of functors
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Abstract:
Given a compact manifold $X$, the set of simple manifold structures on $X \times \Delta ^k$ relative to the boundary can be viewed as the $k$-th homotopy group of a space $\widetilde {\mathcal {S}}^s (X)$. This space is called the block structure space of $X$. We study the block structure spaces of real projective spaces. Generalizing Wall’s join construction we show that there is a functor from the category of finite-dimensional real vector spaces with inner product to the category of pointed spaces which sends the vector space $V$ to the block structure space of the projective space of $V$. We study this functor from the point of view of orthogonal calculus of functors; we show that it is polynomial of degree $\leq 1$ in the sense of orthogonal calculus. This result suggests an attractive description of the block structure space of the infinite-dimensional real projective space via the Taylor tower of orthogonal calculus. This space is defined as a colimit of the block structure spaces of projective spaces of finite-dimensional real vector spaces and is closely related to some automorphisms spaces of real projective spaces.References
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Additional Information
- Tibor Macko
- Affiliation: Mathematisches Institut, Universität Münster, Einsteinstrasse 62, Münster, D-48149, Germany – and – Matematický Ústav SAV, Štefánikova 40, Bratislava, SK-81473, Slovakia
- Email: macko@math.uni-muenster.de
- Received by editor(s): November 19, 2004
- Published electronically: August 24, 2006
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 349-383
- MSC (2000): Primary 57N99, 55P99; Secondary 57R67
- DOI: https://doi.org/10.1090/S0002-9947-06-04180-8
- MathSciNet review: 2247895