Poincaré complex
In mathematics, and especially topology, a Poincaré complex (named after the mathematician Henri Poincaré) is an abstraction of the singular chain complex of a closed, orientable manifold. The singular homology and cohomology groups of a closed, orientable manifold are related by Poincaré duality. Poincaré duality is an isomorphism between homology and cohomology groups. A chain complex is called a Poincaré complex if its homology groups and cohomology groups have the abstract properties of Poincaré duality.[1]
A Poincaré space is a topological space whose singular chain complex is a Poincaré complex. These are used in surgery theory to analyze manifold algebraically.
Definition
Let [math]\displaystyle{ C = \{C_i\} }[/math] be a chain complex of abelian groups, and assume that the homology groups of [math]\displaystyle{ C }[/math] are finitely generated. Assume that there exists a map [math]\displaystyle{ \Delta\colon C\to C\otimes C }[/math], called a chain-diagonal, with the property that [math]\displaystyle{ (\varepsilon \otimes 1)\Delta = (1\otimes \varepsilon)\Delta }[/math]. Here the map [math]\displaystyle{ \varepsilon\colon C_0\to \mathbb{Z} }[/math] denotes the ring homomorphism known as the augmentation map, which is defined as follows: if [math]\displaystyle{ n_1\sigma_1 + \cdots + n_k\sigma_k\in C_0 }[/math], then [math]\displaystyle{ \varepsilon(n_1\sigma_1 + \cdots + n_k\sigma_k) = n_1+ \cdots + n_k\in \mathbb{Z} }[/math].[2]
Using the diagonal as defined above, we are able to form pairings, namely:
- [math]\displaystyle{ \rho \colon H^k(C)\otimes H_n(C) \to H_{n-k}(C), \ \text{where} \ \ \rho(x\otimes y) = x \frown y }[/math],
where [math]\displaystyle{ \scriptstyle \frown }[/math] denotes the cap product.[3]
A chain complex C is called geometric if a chain-homotopy exists between [math]\displaystyle{ \Delta }[/math] and [math]\displaystyle{ \tau\Delta }[/math], where [math]\displaystyle{ \tau \colon C\otimes C\to C\otimes C }[/math] is the transposition/flip given by [math]\displaystyle{ \tau (a\otimes b) = b\otimes a }[/math].
A geometric chain complex is called an algebraic Poincaré complex, of dimension n, if there exists an infinite-ordered element of the n-dimensional homology group, say [math]\displaystyle{ \mu \in H_n(C) }[/math], such that the maps given by
- [math]\displaystyle{ (\frown\mu) \colon H^k(C) \to H_{n-k}(C) }[/math]
are group isomorphisms for all [math]\displaystyle{ 0 \le k \le n }[/math]. These isomorphisms are the isomorphisms of Poincaré duality.[4][5]
Example
- The singular chain complex of an orientable, closed n-dimensional manifold [math]\displaystyle{ M }[/math] is an example of a Poincaré complex, where the duality isomorphisms are given by capping with the fundamental class [math]\displaystyle{ [M] \in H_{n}(M; \mathbb{Z}) }[/math].[1]
See also
References
- ↑ 1.0 1.1 Hazewinkel, Michiel, ed. (2001), "Poincaré complex", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
- ↑ Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, p. 110, ISBN 978-0-521-79540-1, http://www.math.cornell.edu/~hatcher/AT/ATchapters.html
- ↑ Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, pp. 239–241, ISBN 978-0-521-79540-1, http://www.math.cornell.edu/~hatcher/AT/ATchapters.html
- ↑ Wall, C. T. C. (1966). "Surgery of non-simply-connected manifolds". Annals of Mathematics 84 (2): 217–276. doi:10.2307/1970519.
- ↑ Wall, C. T. C. (1970). Surgery on compact manifolds. Academic Press.
- Wall, C. T. C. (1999), Ranicki, Andrew, ed., Surgery on compact manifolds, Mathematical Surveys and Monographs, 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, http://www.maths.ed.ac.uk/~aar/books/scm.pdf – especially Chapter 2
External links
- Classifying Poincaré complexes via fundamental triples on the Manifold Atlas
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