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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The block structure spaces of real projective spaces and orthogonal calculus of functors


Author: Tibor Macko
Journal: Trans. Amer. Math. Soc. 359 (2007), 349-383
MSC (2000): Primary 57N99, 55P99; Secondary 57R67
Published electronically: August 24, 2006
MathSciNet review: 2247895
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04180-8
PII: S 0002-9947(06)04180-8
Keywords: The block structure space, manifold structure, join construction, orthogonal calculus, the first derivative functor, splitting problem, surgery
Received by editor(s): November 19, 2004
Published electronically: August 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.