Invert quasi-isomorphisms of symmetric cooperads

The theory of symmetric operads in chain complexes (say over a good enough field) is in some sense nice, because we have a well defined homotopy theory. In particular we have a notion of infinity-morphisms of operads (maybe called homotopy morphisms instead), which can be defined as a cooperad map between the appropriate bar constructions … Read more

References for bilinear forms on chain complexes?

I am looking for references that include general results and theorems for bilinear forms defined on chain complexes. That is, bilinear forms ⟨⋅,⋅⟩i:Ci×Ci→K defined for all i on chain complexes ⋯→Ci+1→Ci→Ci−1→⋯ of K-modules over a principal ideal domain K. I am particularly interested in how these descend to homology groups and what invariants/classifications can be … Read more

The homotopy type of the mapping space MapBρ(BS1,BG)Map_{B\rho}(BS^1,BG)? for GG a compact Lie group

Given a homomorphism ρ:S1→G with G a compact Lie group there is an induced map of classifying spaces Bρ:BS1→BG. What is known about the homotopy type of the mapping space MapBρ(BS1,BG)? Integrally the space will be a mess of phantom maps. On the other hand, after rationalisation, BGQ will be a product of even dimensional … Read more

Universal enveloping algebra functor preserves quasi-isomorphism

Let k be a field of characteristic 0. Let DGAk denote the category of DG algebras and DGLAk denote the category of DG Lie algebras. It is well known that there are model structures on them that the weak equivalences are the quasi-isomorphisms and the fibrations are the maps which are degreewise surjective. We have … Read more

Computing the order of elements in a non abelian exterior square of a finite group

If we have an explicit group G, and we pick two elements g,h∈G, could we find the order of the element g∧h∈G∧G? The best thing I could find is Theorem 1.1 in Ellis’ Book (, which tells us that for any finite group G, H2(G) is a finite abelian group with exponent e dividing the … Read more

The order of im(ν′∗)⊆π∗S3im(\nu’_*)\subseteq \pi_*S^3

The 3-sphere S3 has homotopy 2-exponent 4. That is, any 2-torsion element α∈π∗S3 has order at most 4. This bound is sharp, for example the Blakers-Massey element ν′∈π6S3 has order 4 (throughout this question we shall agree to work at the prime 2). Now glancing at the unstable homotopy of S3 up to around π30 … Read more

Classical Morse Theory

In the early sixties Smale has proved that any simply connected, torsion free differential manifold of dimension greater than 5 has a perfect Morse function. This, incidentally, also solves the Poincare’s problem for the higher dimensions. With the results of Perelman, I wonder: has the dimension limitation been now removed? Answer AttributionSource : Link , … Read more

Reference request: Basic H-Space properties of $SO(3)$

I dug into the literature but could not find references for some of the basic H-space properties of $SO(3)$. Basic properties that I am looking for include What H-maps are there $SO(3)\rightarrow S^3$? For what integer $N$ is the $N^{th}$ power map $N:SO(3)\rightarrow SO(3)$ an H-map? What is the group $[SO(3)\times SO(3),SO(3)]$? What is the … Read more

Model structure on spaces with local coefficients

Is there a model structure on the category of topological spaces equipped with a local system (i.e. a functor from the fundamental groupoid to the category of abelian groups), such that the weak equivalences are the isomorphisms in homology with local coefficients? If so, is there a reference in which the model structure is established? … Read more