### University of Massachusetts Amherst

Monday, October 29, at pm to pm. Talk 1 - pm. Furthermore, we show that within the current existing framework, this is the limit.

Talk 2 - pm. The Pin 2 -Equivariant Mahowald Invariant Abstract: The existence of Pin 2 -equivariant stable maps between representation spheres has deep applications in 4-dimensional topology. We will sketch the proof for our main result on the Pin 2 -equivariant Mahowald invariants. More specifically, we will discuss the Pin 2 -equivariant Mahowald invariants of powers of certain Euler classes in the RO Pin 2 -graded equivariant stable homotopy groups of spheres. The proof analyzes maps between certain finite spectra arising from BPin 2 and various Thom spectra associated with it.

To analyze these maps, we use the technique of cell-diagrams, known results on the stable homotopy groups of spheres, and the j-based Atiyah-Hirzebruch spectral sequence. MathOverflow is a question and answer site for professional mathematicians.

It only takes a minute to sign up. It is known that the topological classification of a closed Riemann surface is determined by its genus. I was wondering about the similar classification for a general compact four-manifolds possibly with boundaries or even open four-manifolds.

Suppose you can classify all open 4-manifolds. But this classification problem reduces to the word problem on a finitely presented group every such group is the fundamental group of a closed 4-manifold and this is known to have no algorithmic solution. Freedman's work solves the classification problem for closed simply connected 4-manifolds - it says that the intersection form on degree 2 homology together with the Kirby-Siebenmann class provide a complete invariant for such manifolds. Freedman's techniques can also be used to produce complete invariants for closed 4-manifolds with certain prescribed fundamental groups, but this of course depends on the group.

### More about this book

As Paul Siegel pointed out, the fundamental group of a smooth closed orientable 4-manifold can be an arbitrary finitely presented group, and for this reason a general classification is not possible, unless, possibly, for some classes of fundamental groups, e. However, the known classifications, even with this restriction, are in the topological category, while there is no complete classification, even for a fixed topological type, in the smooth or PL category.

If the manifold is closed and connected, then there is such a handle decomposition with only one 0-handle and one 4-handle. A theorem of Poenaru and Laudenbach helps in making this presentation effective: any smooth closed connected oriented 4-manifold can be reconstructed, uniquely up to diffeomorphisms, from a handle decomposition where only the handles up to index 2 are given in other words, you do not need to know 3- and 4-handles to determine the closed manifold.

There is also a nonorientable version of this.

## Topology of gauge theories on compact 4-manifolds - IOPscience

This information can be encoded in a Kirby diagram, which provides a finite presentation of any smooth closed 4-manifolds. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 1 year, 7 months ago. Active 1 year, 7 months ago.

## Vorlesung Topology of 4-Manifolds

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