nLab H-space

Definition

An $H$-space (“H” for Hopf, as in Hopf construction) is a magma internal to the classical homotopy category of topological spaces Ho(Top), or in the homotopy category $Ho(Top)_*$ of pointed topological spaces, which has a unit up to homotopy.

Similarly:

An $H$-monoid? is a monoid object in Ho(Top), hence an $H$-space is an $H$-monoid if the product of the magma is associative up to homotopy.

An $H$-group is a group object in Ho(Top), so an $H$-monoid is an $H$-group if it also has inverses up to homotopy.

An $H$-ring is a ring object in (pointed) Ho(Top).

To continue in this pattern, one could say that an H-category is a category internal to Ho(Top).

Notice that here the homotopies for units, associativity etc. are only required to exist for an H-space, not required to be equipped with higher coherent homotopies. An $H$-monoid equipped with such higher and coherent homotopies is instead called a strongly homotopy associative space or $A_\infty$-space for short. If it has only higher homotopies up to level $n$, it is called an $A_n$-space.

A better name for an $H$-space might be be $H$-unitoid, but it is rarely used. The $H$ stands for Heinz Hopf, and reflects the sad fact that the natural name ‘homotopy group’ was already occupied; Hopf and A. Borel found necessary algebraic conditions for a space to admit an $H$-space structure.

There are dual notions of $H$-counitoid (or $H'$-space, or co-H-space), $H$-comonoid (or $H'$-monoid) and $H$-cogroup (or $H'$-group) having co-operations with the usual identities up to homotopy.

Examples

Loop spaces

The main example of an $H$-group is the loop space $\Omega X$ of a space $X$, which however is naturally even an A-infinity space.

Suspensions

The main example of an H-cogroup in $Top_*$ is the suspension $S X= S^1\wedge X$ of a pointed topological space $X$.

Spheres

The only n-spheres $S^n$ which have $H$-space structure are those for $n = 0,1,3,7$, and their H-space structure is given by identifying them as the unit spheres in one of the four normed division algebras (real numbers, complex numbers, quaternions, octonions) and taking the product to be that induced by the algebra product.

An example of an H-space that does not lift to an A-infinity space is the 7-sphere $S^7$. It can’t be delooped because its delooping would have cohomology group a polynomial ring on a generator in degree 8, and this is impossible by mod $p$ Steenrod operations for any odd $p$, see Lemma 2, Adams61. The 7-sphere is also not an $H$-group.

Mapping spaces into H-groups

If $K$ is an $H$-group then for any topological space $X$, the set of homotopy classes $[X,K]$ has a natural group structure in the strict sense; analogously if $K'$ is an $H$-cogroup then $[K',X]$ has a group structure. If there is more than one $H$-group structure on a space, then the induced group structures on the set of homotopy classes coincide.

If an $H$-space is equivalent to a deloopable one, then it is a group object in the (∞,1)-category Top. In other words, in that case, the associativity and other axioms hold up to coherent homotopy.

K-Theory space

The classifying space $B U \times \mathbb{Z}$ for (complex) topological K-theory is an H-ring space (p. 205 (213 of 251) in A Concise Course in Algebraic Topology.)

Dwyer-Wilkerson space

See at Dwyer-Wilkerson H-space

Properties

Let $A$ be a connected H-space. Then for every $a:A$, the maps $\mu(a,-),\mu(-,a):A \to A$ are homotopy equivalences.

Nilpotency

Every connected H-space is nilpotent (see there).

Relation to $A_\infty$-spaces

Given an A-∞ space in the (∞,1)-category ∞Grpd/Top, its image in the corresponding homotopy category of an (∞,1)-category Ho(Top) in an $H$-space. Conversely, refining an H-space to a genuine $A_\infty$-space means lifting the structure of a monoid object in an (∞,1)-category from the homotopy category Ho(Top) to the genuine (∞,1)-category ∞Grpd/Top.

Further discussion of this is also at loop space – Homotopy associative structure.

Ring structure on homology

The operations on an H-space $X$ equip its homology with the structure of ring. At least for ordinary homology this is known as the Pontrjagin ring $H_*(X)$ of $X$.

Rational homotopy

Every connected H-space of rational finite type (i.e., whose rational cohomology in each degree is a finite-dimensional rational vector space) is rationally homotopy equivalent to a product of Eilenberg-MacLane spaces (e.g. May & Ponto 2011, Thm. 9.1.1). This means equivalently that its minimal Sullivan model has vanishing differential, see for instance the example of Sullivan models of based loop spaces.

If such an H-space in addition admits the structure of a finite CW-complex then it has the rational homotopy type of a finite product of odd-dimensional rational spheres (Cor. 9.1.2. of May & Ponto 2011).

For example, since smooth manifolds have CW-complex structure (see there) this is the case for all Lie groups (e.g. Lin 1995).

H-Spaces in homotopy type theory

In homotopy type theory, an H-Space consists of

• A type $A$,
• A basepoint $e:A$
• A binary operation $\mu : A \to A \to A$
• A left unitor
  $$\lambda:\prod_{(a:A)} \mu(e,a)=a$$
• A right unitor
  $$\rho:\prod_{(a:A)} \mu(a,e)=a$$

A homomorphism of H-spaces between two H-spaces $A$ and $B$ consists of

• A function $\phi:A \to B$ such that

  • The basepoint is preserved
    $$\phi(e_A) = e_B$$
  • The binary operation is preserved
    $$\prod_{(a:A)} \prod_{(b:A)} \phi(\mu_A(a, b)) = \mu_B(\phi(a),\phi(b))$$

• A function

$$\phi_\lambda:\left(\prod_{(a:A)} \mu(e_A,a)=a\right) \to \left(\prod_{(b:B)} \mu(e_B,b)=b\right)$$

such that the left unitor is preserved:

$$\phi_\lambda(\lambda_A) = \lambda_B$$

• A function

$$\phi_\rho:\left(\prod_{(a:A)} \mu(a, e_A)=a\right) \to \left(\prod_{(b:B)} \mu(b, e_B)=b\right)$$

such that the right unitor is preserved:

$$\phi_\rho(\rho_A) = \rho_B$$

Examples

• There is a H-space structure on the circle. See Lemma 8.5.8 of the HoTT book.

Proof

We define $\mu : S^1 \to S^1 \to S^1$ by circle induction:

$$\mu(base)\equiv id_{S^1}\qquad ap_{\mu}\equiv funext(h)$$

where $h : \prod_{x : S^1} x = x$ is defined in Lemma 6.4.2 of the HoTT book. We now need to show that $\mu(x,e)=\mu(e,x)=x$ for every $x : S^1$. Showing $\mu(e,x)=x$ is quite simple, the other way requires some more manipulation. Both of which are done in the book.

• Every loop space is naturally a H-space with path concatenation as the operation. In fact every loop space is a group.

• The type of maps $A \to A$ has the structure of a H-space, with basepoint $id_A$, operation function composition.

• An A3-space $A$ is an H-space with an equality $\mu(\mu(a, b),c)=\mu(a,\mu(b,c))$ for every $a:A$, $b:A$, $c:A$ representing associativity up to homotopy.

• A unital magma is a 0-truncated H-space.

Related concepts

• topological monoid

• Pontrjagin ring

• group completion

• H-space ring spectrum, H-group ring spectrum

• H-spaceoid

• A3-space

• H-spatial groupoid

• k-tuply associative n-groupoid

References

Introduction and survey:

• Allen Hatcher, §3.C in: Algebraic Topology, Cambridge University Press (2002) [ISBN:9780521795401, webpage]

• Martin Arkowitz, H-Spaces and Co-H-Spaces, chapter 2 in: Introduction to homotopy theory, Springer (2011) 37-70 [doi:10.1007/978-1-4419-7329-0_2]

The terminology “$H$-space” appear sin

• Jean-Pierre Serre, Chapter IV, Section 1 of: Homologie singulière des espaces fibrés. Applications., Ann. Math. 54 (1951), 425–505.

Early discussion:

• William Browder, Torsion in H-Spaces, Annals of Mathematics, Second Series 74 1 (1961) 24-51 [jstor:1970305]

More general notion of internal H-objects:

• Beno Eckmann, Peter Hilton, Group-like structures in general categories I multiplications and comultiplications, Math. Ann. 145, 227–255 (1962) (doi:10.1007/BF01451367)

• Beno Eckmann, Peter Hilton, Group-like structures in general categories III primitive categories, Math. Ann. 150 165–187 (1963) (doi:10.1007/BF01470843)

Original articles:

• Edwin Spanier, Henry Whitehead, On fibre spaces in which the fibre is contractible, Comment. Math. Helv. 29, 1955, 1–8.

• Arthur H. Copeland, On $H$-spaces with two nontrivial homotopy groups, Proc. AMS 8, 1957, 109–129.

• Masahiro Sugawara:

  • $H$-spaces and spaces of loops, Math. J. Okayama Univ, 5, 1955, 5–11;
  • A condition that a space is an $H$-space, Math. J. Okayama Univ. 6, 1957, 109–129; (pdf)
  • A condition that a space is group-like, Math. J. Okayama Univ. 7, 1957, 123–149.

• F. I. Karpelevič, A. L. Oniščik, Algebra of homologies of space of paths, Dokl. Akad. Nauk. SSSR, (N.S.), 106 (1956), 967–969. MR0081478

The theory of $H$-spaces was widely established in the 1950s and studied by Serre, Postnikov, Spanier, Whitehead, Dold, Eckmann and Hilton and many others.

The deloopable case has more coherent structure which has been discovered few years later in J. Stasheff’s thesis and published in

• Jim Stasheff, Homotopy associative $H$-spaces I, II, Trans. Amer. Math. Soc. 108, 1963, 275-312

For a historical account see

• John McCleary, An appreciation of the work of Jim Stasheff (pdf)

The description in terms of groupoid object in an (∞,1)-category is due to

• Jacob Lurie, Higher Topos Theory

see last remark of section 6.1.2 .

Wikipedia's definition (at time of this writing, and phrased in the language of homotopy theory) is rather a unitoid object in the $(\infty,1)$-category Top.

• MathOverflow: homotopy-associative-h-space-and-coh-space

For H-spaces in homotopy type theory:

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)

• Ulrik Buchholtz, J. Daniel Christensen, Jarl G. Taxerås Flaten, Egbert Rijke, Central H-spaces and banded types [arXiv:2301.02636]

See also

• John Adams, Finite $H$-Spaces and Lie groups, Journal of Pure and Applied Algebra 19 (1980) 1-8 (pdf)

• John Adams, The sphere, considered as an $H$-space mod $p$, Quart. J. Math. Oxford. Ser. (2) vol 12 52-60 (pdf)

• Peter May, Kate Ponto. More Concise Algebraic Topology, University of Chicago Press (2011). [doi:10.7208/chicago/9780226511795.001.0001, pdf&rbrack

• James P. Lin. “H-spaces with Finiteness Conditions”. In: Handbook of Algebraic Topology (1995), pp. 1095, 1097-1141. (doi)