The Homotopy Category of Simply Connected 4-Manifolds (London Mathematical Society Lecture Note Series) » holypet.ru

# The Homotopy Category of Simply Connected 4-Manifolds.

particular group contains all the complexity of smooth 4–manifolds with the given fun-damental group, including not just their homotopy types but also their diffeomorphism types. In particular there is a subset of the trisections of the trivial group corresponding to the countably many exotic smooth structures on a given simply connected topo Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share. Sep 10, 2014 · We determine loop space decompositions of simply-connected four-manifolds, n -1-connected 2 n -dimensional manifolds provided n ∉ 4, 8, and connected sums of products of two spheres. These are obtained as special cases of a more general loop space decomposition of certain torsion-free CW -complexes with well-behaved skeleta and some Poincaré duality features. Two simply-connected closed topological 4-manifolds are homeomorphic if and only if they have isomorphic intersection forms and the same Kirby-Siebenmann invariant. Given any even unimodular symmetric bilinear form over there is, up to homeomorphism, a unique simply connected topological 4-manifold with intersection form. 1. Algebraic Topology Thecriticalalgebraictopologicalinformationforaclosed, simplyconnected, smooth 4-manifold Xis encoded in its Euler characteristic eX, its signature σX, and its type tX 0 if the intersection form of X is even and 1 if it is odd.

Lambert [13]. Note that this assertion for Q0 and EQ is not derived from the arguments of the Robertello-Arf invariants of links. Cf. [13], [17]. Example 2.12. For each s > 1 there are compact 4-manifolds W homotopy equivalent to a bouquet of s 2-spheres such that a basis ofH2W; Z is represented by. In the first half of the course we will introduce smooth 4-manifolds. We will briefly survey the topological classification of simply-connected 4-manifolds, and then start to explain how the smooth theory diverges from the topological one, via old tools eg Rochlin's theorem and more recent ones eg Seiberg-Witten leading to a range of. k, then fis a homotopy equivalence. Example 1.1. Cˆ[0;1] the Cantor Set. Let C be the Cantor set with the discrete topology. Then C !Cinduces isomorphisms on all homotopy groups, but it is not a homotopy equivalence, so the CW hypothesis is required. Theorem 1.2 Hurewicz Theorem. Let X be a space and ˇ kX;x = 0 for k

Thus, for a simply-connected Poincaré polyhedron $X$ a PL-manifold of dimension $\geq 5$ homotopy equivalent to it exists if and only a lifting 4 exists. The problem of the existence of topological or smooth manifolds that are homotopy equivalent to an even simply-connected Poincaré polyhedron is still more complicated. In 4 dimensions, simply connected topological manifolds are classified by the intersection form and the Kirby-Siebenmann invariant. The intersection form is obviously defined for any 4-dimensional Poincaré complex. I think Kirby-Siebenmann does as well, but I'm not so sure about that. We give criteria for a closed 4-manifold to be homotopy equivalent to the total space of an S 1 -bundle over a closed 3-manifold.

 Topology Vol. -9. No. 4. pp. 419-440. 1990. 0040- 9383 90 503.00.00 Printed in Great Britain. 1990 Pergamon Press plc ON THE HOMOTOPY THEORY OF SIMPLY CONNECTED FOUR MANIFOLDS TIM D. COCHRANt and NATHAN HABEGGER Received in revised form 14 December 1988 INTRODUCTION AND STATEMENT OF RESULTS THIS PAPER is concerned with the homotopy theory of 1-connected 4-manifolds. THEOREM 3. //x M, M2 are h-cobordant simply-connected 4-manifolds, then for some Jc, M^kiS2x2S2;M 2kS 2x2.S Here k denotes k copies an, dconnected sum. We obtai a numben r of corollaries fo; exampler th, e Grothendieck grou opf oriented simply-connected 4-manifolds is the free abelian o groun Pp an d Q, the complex. The topology of simply-connected four-manifolds is a subject of widespread and enduring interest. They have been classiﬁed up to homotopy type by Milnor [Mi] and up to homeomorphism type by Freedman. May 15, 2002 · Abstract The main theorem asserts that every 2-dimensional homology class of a compact simply connected PL 4-manifold can be represented by a codimension-0 submanifold consisting of a contractible manifold with a single 2-handle attached.

Representing homology classes of simply connected 4-manifolds Article in Topology and its Applications 120s 1–2:57–65 · May 2002 with 27 Reads How we measure 'reads'. Simple homotopy Reidemeister torsion Surgery theory; Modern algebraic topology beyond cobordism theory, such as Extraordinary cohomology, is little-used in the classification of manifolds, because these invariant are homotopy-invariant, and hence don't help with the finer classifications above homotopy. Proposition 8. A path connected space is simply connected if and only if there is only one homotopy class of paths between any two points. Proof. Suppose Xis simply connected. Given two paths f;gbetween x 0 and x 1; we have that f g ’ e g g’ e and so f’ f g g’ g: Conversely, suppose there is only one homotopy class of paths between any.

• Jun 23, 2003 · It deals with the problem of computing the homotopy classes of maps algebraically and determining the law of composition for such maps. This problem is solved in the book by introducing new algebraic models of a 4-manifold. To aid those interested in further reading there is a full list of references to the literature.
• The Homotopy Category of Simply Connected 4-Manifolds London Mathematical Society Lecture Note Series Read more. Surgery on Simply-Connected Manifolds. Read more. The Category of Substance. Report "The homotopy category of simply connected 4-manifolds" Your name. Email.
• London Mathematical Society lecture note series, 297. Other Titles: Homotopy category of simply connected four manifolds: Responsibility: Hans Joachim Baues. More information: Table of contents; Publisher description.
• 297 The homotopy category of simply connected 4-manifolds, H.-J. BAUES 298 Higher operads, higher categories, T. LEINSTER ed 299 Kleinian groups and hyperbolic 3-manifolds, Y. KOMORI, V. MARKOVIC & C. SERIES eds 300 Introduction to Möbius differential geometry, U. HERTRICH-JEROMIN 301 Stable modules and the D2-problem, F.E.A. JOHNSON.

## ON SIMPLY-CONNECTED 4-MANIFOLDS.

Smooth 4-manifolds vs. symplectic 4-manifolds vs. complex surfaces The symplectic geometry part of the course follows the book by Ana Cannas da Silva, Lectures on Symplectic Geometry Lecture Notes in Mathematics 1764, Springer-Verlag; the discussion of Kähler geometry mostly follows the book by R. O. Wells, Differential Analysis on Complex. AMERICAN MATHEMATICAL SOCIETY Volume 219, 1976 CLASSIFICATION OF SIMPLY CONNECTED FOUR-DIMENSIONAL RR -MANIFOLDS BY Gr. TSAGAS AND A. LEDGER ABSTRACT. Let M, g be a Riemannian manifold. We assume that there is a mapping s: M— IM, where IM is the group of isometries of M, g, such that sx = sx, Vx e M, has x as a fixed isolated point. On 2-dimensional homology classes of 4-manifolds - Volume 82 Issue 1 - Selman Akbulut. Journal of the London Mathematical Society, Vol. 91, Issue. 2, p. 439. be sent to your device when it is connected to wi-fi. ‘@’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Don't show me this again. Welcome! This is one of over 2,200 courses on OCW. Find materials for this course in the pages linked along the left. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. No enrollment or registration. Part of the Lecture Notes in Mathematics book series LNM, volume 1346 Keywords Cohomology Class Homotopy Type Homotopy Classification Topological. Classification of simply-connected topological 6-manifolds. In: Viro O.Y., Vershik A.M. eds Topology and Geometry — Rohlin Seminar. Lecture Notes in Mathematics, vol 1346.

### Representing homology classes of simply connected 4-manifolds.

simply connected, PL 4-manifold, then each element of H2W can be represented by a compact PL sub-manifold M ⊂ W such that M consists of a Mazur-like contractible 4-manifold with a single 2-handle attached. Theorem 2. If W is a compact, simply connected, PL submanifold of S4, then each element of H2W can be represented by a locally ﬂat. Please note that terms and conditions apply. The present paper is devoted to a further study of the homotopy invariants of non-simply connected manifolds which correspond to the obstruction to modifying one mani American Mathematical Society. 1326 A. S. MISCENKO. The geometry and topology of three-manifolds is a set of widely circulated but unpublished notes by William Thurston from 1978 to 1980 describing his work on 3-manifolds. The notes introduced several new ideas into geometric topology, including orbifolds, pleated manifolds, and train tracks. Distribution. Copies of the original 1980 notes were circulated by Princeton University. CLASSIFICATION OF CLOSED TOPOLOGICAL 4-MANIFOLDS 3 Then a closed 4-manifold M is topologically s-cobordant to the total space of an F-bundle over B if and only if π1M is an extension of π1B by π1F and the Euler characteristic of M is the product of the Euler characteristics of F and B. References [1] M. Freedman. The Topology of 4-dimensional Manifolds. simply-connected manifolds not di eomorphic to S 4, manifolds with simple in nite fundamen tal group. First examples of rst 3-manifolds not admitting at conformal structure w ere constructed b y W. Goldman in [1]. The ab o v e theorem sho ws that if M admits a at conformal structure, it do es not imply that all comp onen ts of its connected.

Thus, the set of homeomorphism classes of surfaces is a commutative monoid with respect to connected sum, and is generated by and, with the sole relation. Compact 2-manifolds possibly with boundary are homeomorphic if and only if they have isomorphic intersection forms. Cf. the topological classification of simply-connected 4-manifolds. This workshop is principally funded by the Clay Mathematical Institute. Additional support has been received from the London Mathematical Society and the Heilbronn Institute. We also thank the Mathematical Institute, Oxford University, for providing lecture and class rooms.

Cauchy’s theorem is not true for non-simply connected regions in C. The fundamental group measures how far a space is from being simply connected. The fundamental group brie y consists of equivalence classes of homotopic closed paths with the law of composition being following one path by another. However, we want to make this precise in a series. Jan 25, 1991 ·: Geometry of Low-Dimensional Manifolds, Vol. 2: Symplectic Manifolds and Jones-Witten Theory London Mathematical Society Lecture Note Series 9780521400015: Donaldson, S. K.: Books. We want H to be a homotopy from f to r: H inherits the essential properties listed in De nition 1.6, including continuity, from h 1 and h 2. Hence, His a homotopy from fto r, and f’r. De nition 1.11. For f 2PX, the homotopy class [f] of f is the equivalence class of funder the equivalence relation ’. a map f: XY, each homotopy offlA: AY can be extended to a homotopy off: X-r Y. If A has the HEP in X with respect to all spaces Y then A is said to have the absolute homotopy extension property AHEP in X. The following three theorems are well-known in their second formulations. simply-connected minimal symplectic 4-manifold that is homeo there has been a considerable amount of progress in the discovery of exotic smooth structures on simply-connected 4-manifolds with small Euler characteristic. In early 2004, Jongil Park [P2] has constructed the ﬁrst example of exotic smooth. the homotopy exact sequence for a.

The additional structure of coordinates and tangents can be used to revisit homology, gaining additional insight and results. In particular, as we saw in the previous section, the exterior derivative \\mathrmd\ exhibits structure reminiscent of the boundary homomorphism \\partial\ in homology.This can be exploited to build a version of homology based on forms instead of on simplices.