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Configuration spaces in algebraic topology
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abstract
These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the integral cohomology of the ordered---and the mod $p$ cohomology of the unordered---configuration spaces of $\mathbb{R}^n$, and the rational cohomology of the unordered configuration spaces of an arbitrary manifold of odd dimension. We also discuss models for mapping spaces in terms of labeled configuration spaces, and we show that these models split stably. Some classical results are given modern proofs premised on hypercover techniques, which we discuss in detail.
Forward citations
Cited by 3 Pith papers
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Homology of configuration spaces in positive characteristic via point-set constructions
Explicit chain complexes and spectral sequences compute homology of configuration spaces in positive characteristic, lifting Knudsen's theorem, with a conjecture on twisted coalgebra equivalences implying homotopy invariance.
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A stable rank filtration on direct sum $K$-theory
A new stable rank filtration on direct sum K-theory is defined via Gamma-sapces, with filtration quotients described as suspension spectra of decomposition posets.
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Configuration spaces and the Arone--Mahowald theorem
Decomposes Cartan-Leray spectral sequence for configuration spaces as direct sum of atomic sequences, recovering Arone-Mahowald vanishing theorem for Goodwillie derivatives.
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